materials
Article Time-Resolved PIV Measurements and Turbulence Characteristics of Flow Inside an Open-Cell Metal Foam
Youngwoo Kim 1, Chanhee Moon 1, Omid Nematollahi 1, Hyun Dong Kim 2,* and Kyung Chun Kim 1,*
1 School of Mechanical Engineering, Pusan National University, Busan 46241, Korea; [email protected] (Y.K.); [email protected] (C.M.); [email protected] (O.N.) 2 Rolls-Royce and Pusan National University Technology Centre, Pusan National University, Busan 46241, Korea * Correspondence: [email protected] (H.D.K.); [email protected] (K.C.K.); Tel.: +82-51-510-1536 (H.D.K.); +82-51-510-2324 (K.C.K.)
Abstract: Open-cell metal foams are porous medium for thermo-fluidic systems. However, their complex geometry makes it difficult to perform time-resolved (TR) measurements inside them. In this study, a TR particle image velocimetry (PIV) method is introduced for use inside open-cell metal foam structures. Stereolithography 3D printing methods and conventional post-processing methods cannot be applied to metal foam structures; therefore, PolyJet 3D printing and post-processing methods were employed to fabricate a transparent metal foam replica. The key to obtaining acceptable transparency in this method is the complete removal of the support material from the printing surfaces. The flow characteristics inside a 10-pore-per-inch (PPI) metal foam were analyzed in which porosity is
0.92 while laminar flow condition is applied to inlet. The flow inside the foam replica is randomly divided and combined by the interconnected pore network. Robust crosswise motion occurs inside
Citation: Kim, Y.; Moon, C.; foam with approximately 23% bulk speed. Strong influence on transverse motion by metal foam Nematollahi, O.; Kim, H.D.; Kim, K.C. is evident. In addition, span-wise vorticity evolution is similar to the integral time length scale of Time-Resolved PIV Measurements the stream-wise center plane. The span-wise vorticity fluctuation through the foam arrangement is and Turbulence Characteristics of presented. It is believed that this turbulent characteristic is caused by the interaction of jets that have Flow Inside an Open-Cell Metal different flow directions inside the metal foam structure. The finite-time Lyapunov exponent method Foam. Materials 2021, 14, 3566. is employed to visualize the vortex ridges. Fluctuating attracting and repelling material lines are https://doi.org/10.3390/ma14133566 expected to enhance the heat and mass transfer. The results presented in this study could be useful for understanding the flow characteristics inside metal foams. Academic Editors: Vincenzo Naso, Marcello Iasiello and Thomas Fiedler Keywords: open-cell metal foam; flow characteristics; time-resolved PIV; 3D printing; refractive index matching Received: 3 April 2021 Accepted: 22 June 2021 Published: 25 June 2021
1. Introduction Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in Flow through porous media is a topic of interest in many disciplines of science published maps and institutional affil- and engineering, such as petroleum engineering [1], groundwater hydrology [2], heat iations. exchangers [3], catalyst reactors [4], mixers [5], and filters [6]. Fluid flow through porous media should be studied carefully because the local flow characteristics can affect the global characteristics of the fluid systems [7,8]. Open-cell metal foams are an irregular metallic porous medium [9]. The skeletal part of the metal foam is a Plateau’s border network
Copyright: © 2021 by the authors. and involves nodes and struts [10]. The skeletal part forms trabecular-like bone cells [11], Licensee MDPI, Basel, Switzerland. which deliver desired geometrical features for thermo-fluidic applications, including high- This article is an open access article porosity [12], big specific surface zone [13], twisting flow paths [14], and good strength [15]. distributed under the terms and Metal foams are fascinating because of their applicability in various fields. Many conditions of the Creative Commons experimental/ numerical methods have been employed to study flows in metal foam. Flow Attribution (CC BY) license (https:// visualization methods are favorable approaches to understanding the fluid flow physics creativecommons.org/licenses/by/ of metal foams. Studies including lump parameters inherently neglect the localized flow 4.0/).
Materials 2021, 14, 3566. https://doi.org/10.3390/ma14133566 https://www.mdpi.com/journal/materials Materials 2021, 14, 3566 2 of 17
features. Additionally, numerical studies highly require robust experimental results for validation [16]. To date, several works have been done regarding experimental fluid flow visualization. Hwang et al. [17] qualitatively visualized the flow of inlet and outlet of two different metal foams (porosity: 0.7 and 0.9) using smoke-wire. When the porosity was 0.7, the amount of smoke accumulation upstream was larger and so was the size of eddies downstream from the metal foam. The smoke accumulation and eddies may represent the permeability related to the pressure drop. Eggenschwiler et al. [18] investigated the velocity uniformity of the highly asymmetric flow through a honeycomb monolith and a metal foam. The flow through the metal foam showed much higher velocity uniformity than that through the honeycomb monolith. The homogenization of the velocity profile through the metal foams might have been caused by effects of the structure and the relatively high pressure drop. Hutter et al. [19] investigated the flow and mass transfer in the flow upstream and downstream of three different metal foams with 20, 30, and 45 pores-per-inch (PPI) using particle image velocimetry (PIV) and laser-induced fluorescence (LIF). The flow behind the 20-PPI metal foam had higher turbulent kinetic energy and mixing performance than that behind the 30 and 45-PPI metal foams. This result indicates that the flow behavior in metal foams should be carefully characterized prior to use. Butscher et al. [20] investigated the flow through a foam-like porous structure (poros- ity: 0.78) with periodic and uniform cell topology using two-frame PIV and refractive index matching techniques. In their experiment, the jet flowing into the cells was decelerated by the cell geometry, which enhanced the heat and mass transfer. Their results are helpful for understanding metal foam flow, but the high-porosity metal foam structure cannot be printed by stereolithography 3D printing. Ensemble-averaged flow fields through 4× scale foam were studied by Onstad et al. [21] employing magnetic resonance velocimetry (MRV). A turbulent flow with cell Reynolds of 840 (bulk Reynolds: 7900) was considered when passing through the foam. A strong transverse flow was evident with 20 to 30% of superficial speed. MRV suffers from achieving TR results when flow field is complex and small, such as in metal foam fluid flow. Regarding the literature review, to the best of the author’s knowledge, a detailed characteristics study of the fluid flow inside metal foams has not been published. Therefore, the flow structure inside of those materials is unclear. Several parameters, including laminar inlet condition, temporal features, and the fluid flow evolution inside foams, should be further investigated. Thus, this study described a detailed method for TR PIV measurement of a transparent foam. The spatiotemporal features of fluid flow through the metal foam were investigated. The outcome of this study can be used for both theoretical or numerical research. Theoretical flow structures, phenomena, and evolution of flow through the metal foams will be known, which helps further implementation of these structures into the new application. Numerically, the results of this study will be a robust case study for validation of computational fluid dynamics (CFD) simulations.
2. Materials and Methods 2.1. Fabrication and Refractive Index Matching of Transparent Metal Foam Replica Aluminum foam with 10 PPI (Duocel® Foam, ERG Aerospace Corp., Oakland, CA, USA.) was prepared as a metal foam replica. X-ray tomography with 743 × 740 × 459 voxels was employed to generate a 3D geometry file with spatial resolution of 0.032 mm. In computer-aided design (CAD) software, replica was cropped to dimensions of 10 × 10 × 25 mm. Due to resolution of the printer, the size of the cropped replica was doubled. Then, the final sizes of the model file were 20 × 20 × 50 mm. The detailed geometry information is provided in Table1. Materials 2021, 14, x FOR PEER REVIEW 3 of 17
Table 1. Geometrical characteristics of utilized foam.
Porosity (ε) Strut Diameter ( ) Pore Diameter ( ) Cell Diameter ( ) 0.92 0.8 ± 0.06 mm 3.9 ± 0.33 mm 9.02 ± 1.06 mm
Two 3D printing techniques that can be used to print transparent porous media are the stereolithography and PolyJet methods. In Stereolithography, the surface of liquid resin is irradiated with a UV laser to cure it. In the PolyJet method, fine resin droplets are jetted onto the model surface and cured by UV lamps. The PolyJet method can print dif- ferent materials simultaneously. Stereolithography provides better surface roughness of the printed model, but the PolyJet method is suitable for printing metal foam structures because the struts of the metal foam need support material when they are printed. Figure 1a shows the replica printed by PolyJet 3D printer. Printer materials were Vero Clear resin Materials 2021, 14, 3566 and water soluble support material. 3 of 17 Achieving an optically clear surface is crucial; therefore, before starting the experi- mental campaign, a series of post-processing actions should be performed. This procedure includes: Table 1. Geometrical characteristics of utilized foam. 1. The support material must be removed completely since it is not transparent. This
wasPorosity done (usingε) a water Strut jet Diameter after soaking (ds) the Pore model Diameter for half (dp a) day. Cell Diameter (dc)
2. Wet sanding0.92 must be done 0.8 ± to0.06 obtain mm the best 3.9transparency.± 0.33 mm The sanding 9.02 ± was1.06 mmstarted from a coarse sand to finer one. All the surfaces were sanded, and each time, the model was inspected under a light source to recognize the sandpaper size change whereTwo 3D surface printing defects techniques were no that longer can bevisible used after to print last transparent sanding pass. porous Each media sanding are the stereolithographystep was done perpendicular and PolyJet to methods. the previous In Stereolithography,one. the surface of liquid 3.resin The is irradiatedprinted model with was a UV then laser placed to cure in a it. wa Inter the channel PolyJet with method, water finecirculating resin droplets at am- are jettedbient temperature onto the model for surface48 h. and cured by UV lamps. The PolyJet method can print 4.different For the materials final step, simultaneously. polishing was Stereolithography done by a polishing provides substantial better surfaceto achieve roughness a glossy of the printed model, but the PolyJet method is suitable for printing metal foam structures high-quality surface. because the struts of the metal foam need support material when they are printed. Figure1a showsThe the printing replica orientation printed by PolyJetwas also 3D considered printer. Printer as [20,22]. materials Figure were 1b Vero shows Clear the resin 3D printedand water transparent soluble support foam after material. the completion of every process. As can be seen after post- processing procedures, a glossy surface is achieved.
(a)
(b)
FigureFigure 1. 1. 3D3D printed printed transparent transparent replica replica of of metal metal foam foam (a ()a before) before and and (b ()b after) after post-processing post-processing for for refractiverefractive index index matching. matching.
Achieving an optically clear surface is crucial; therefore, before starting the experi- mental campaign, a series of post-processing actions should be performed. This procedure includes: 1. The support material must be removed completely since it is not transparent. This was done using a water jet after soaking the model for half a day. 2. Wet sanding must be done to obtain the best transparency. The sanding was started from a coarse sand to finer one. All the surfaces were sanded, and each time, the model was inspected under a light source to recognize the sandpaper size change where surface defects were no longer visible after last sanding pass. Each sanding step was done perpendicular to the previous one. 3. The printed model was then placed in a water channel with water circulating at ambient temperature for 48 h. 4. For the final step, polishing was done by a polishing substantial to achieve a glossy high-quality surface. Materials 2021, 14, 3566 4 of 17
The printing orientation was also considered as [20,22]. Figure1b shows the 3D Materials 2021, 14, x FOR PEER REVIEWprinted transparent foam after the completion of every process. As can be4 seen of 17 after post-processing procedures, a glossy surface is achieved. The main issue regarding the flow-field measurement inside a foam replica is to avoid optical distortion. Therefore, the refractive index should be studied. The refractive index of The main issue regarding the flow-field measurement inside a foam replica is to Vero Clear material is 1.515 for a wavelength of 532 nm. A refractive index matched (RIM) avoid optical distortion. Therefore, the refractive index should be studied. The refractive solution based on NaI was provided using the solution formula developed by Gallagher index of Vero Clear material is 1.515 for a wavelength of 532 nm. A refractive index et al. [23] (60.2 wt% NaI, 32.4 wt% water, and 7.36 wt% glycerin). The grades of reagents matched (RIM) solution based on NaI was provided using the solution formula developed were higher than extra pure. Water was degassed in a vacuum chamber before making by Gallagher et al. [23] (60.2 wt% NaI, 32.4 wt% water, and 7.36 wt% glycerin). The◦ grades ofRIM. reagents In addition, were higher the RIMthan wasextra degassed pure. Water by was increasing degassed the in temperature a vacuum chamber to 40 Cbefore and then makingcooled downRIM. In to addition, room temperature. the RIM was Sodium degassed thiosulfate by increasing (0.1 wt%)the temperature was added to to 40 avoid °C the anddiscoloration then cooled [24 down] caused to room by NaI temperature. reacting with Sodium oxygen thiosulfate to form (0.1 triiodide wt%) was ions. added In addition, to avoidit is recommended the discoloration to [24] store caused the solution by NaI reac in ating glass with container, oxygen to because form triiodide sodium ions. thiosulfate In addition,reacts with it is some recommended metals, such to store as aluminum the solution and in stainlessa glass container, steel, to formbecause complexes. sodium To thiosulfatedemonstrate reacts the with suitability some metals, of RIM, such Butscher, as aluminum Hutter, and Kuhn, stainless and von steel, Rohr to form 2012 com- utilized a plexes.sample To text demonstrate below the the specimen suitability to of present RIM, Butscher, the RIM, Hutter, while AycockKuhn, and et al.von 2017 Rohr employed 2012 utilizeda sample a sample gride under text below the replica the specimen to show to the present RIM inthe a qualitativeRIM, while manner.Aycock et In al. the 2017 current employedstudy, the a method sample ofgride Butscher, under the Hutter, replica Kuhn, to show and the von RIM Rohr in 2012a qualitative was applied manner. to confirm In thethe current RIM. Figure study,2 thea,b methodpresents of theButscher, metal Hutter, foam replica Kuhn, withoutand von Rohr and with2012 was index applied matching, torespectively. confirm the AsRIM. evidenced, Figure 2a,b it presents can be seen the metal that using foam abovereplica procedures without and led with to aindex suitable- matching,matched refractiverespectively. index As evidenced, in comparison. it can be seen that using above procedures led to a suitable-matched refractive index in comparison.
(a)
(b)
FigureFigure 2. 2. MetalMetal foam foam replica replica without without (a) (anda) and with with (b) (indexb) index matching. matching.
2.2.2.2. Experiment Experiment Test Test Section Section AA square square acrylic acrylic duct duct was was manufactured manufactured (20 (20mmmm × 20 ×mm20 × mm 500 ×mm),500 as mm), shown as shownin Figurein Figure 3a. 3Non-dimensionala. Non-dimensional sizes sizesbased based on the on duct the width, duct width, which, which,from now from on, now will on,be- will calledbecalled diameter diameter (D) (D)for a for better a better wording wording to co tompare compare with withprevious previous works, works, are indicated. are indicated. InletInlet and and outlet outlet of ofthe the duct duct are located are located on the on upper the upperside of sidethe duct of the to see duct span-wise to see span-wise fluid flow.fluid The flow. model The was model placed was at placed a length at aof length 15 D from of 15 inlet D fromto remove inlet tothe remove entrance the length entrance effects.length effects. A loop was arranged to allow the RIM to flow over the test section. The test loop included gear pump, RIM container, bubble trap, flowmeter, and test section. Silicon hose Materials 2021, 14, x FOR PEER REVIEW 5 of 17
Materials 2021, 14, 3566 connected the components. To maintain the RIM solution temperature, a constant 5tem- of 17 perature hot plate of 25 °C was located under the solution container. In addition, the con- tainer was partly immersed in a water bath.
(a)
(b)
Figure 3. Experimental setup configuration:configuration: ( a) test section details; (b) overall setup configuration.configuration.
FigureA loop 4a was shows arranged the test to section allow thefilled RIM with to the flow prepared over the RIM test solution. section. It The is practically test loop impossibleincluded gear to match pump, the RIM refractive container, index bubble completely, trap, flowmeter,so the many and edges test of section. the metal Silicon foam replicahose connected are still shown. the components. However, it To has maintain acceptable the transparency, RIM solution temperature,as shown in Figure a constant 4b. temperature hot plate of 25 ◦C was located under the solution container. In addition, the container was partly immersed in a water bath. Figure4a shows the test section filled with the prepared RIM solution. It is practically impossible to match the refractive index completely, so the many edges of the metal foam replica are still shown. However, it has acceptable transparency, as shown in Figure4b.
(a) Materials 2021, 14, x FOR PEER REVIEW 5 of 17
connected the components. To maintain the RIM solution temperature, a constant tem- perature hot plate of 25 °C was located under the solution container. In addition, the con- tainer was partly immersed in a water bath.
(a)
(b) Figure 3. Experimental setup configuration: (a) test section details; (b) overall setup configuration.
Materials 2021, 14, 3566 Figure 4a shows the test section filled with the prepared RIM solution. It is practically6 of 17 impossible to match the refractive index completely, so the many edges of the metal foam replica are still shown. However, it has acceptable transparency, as shown in Figure 4b.
Materials 2021, 14, x FOR PEER REVIEW 6 of 17
(a)
(b)
FigureFigure 4. 4. TestTest section section with with refractive-index-matched refractive-index-matched so solutionlution and and transparency transparency of of the the 3D 3D printed printed model:model: ( (aa)) without without laser laser illumination; illumination; ( (bb)) with with laser laser illumination. illumination.
2.3.2.3. PIV PIV Measurement Measurement Setup Setup AA CMOS CMOS high-speed high-speed camera camera (FastCam (FastCam SA1.1, SA1.1, Photron, Photron, 1k 1k× 1k× pixels)1k pixels) and anda 5 W, a 5532 W, nm532 continuous nm continuous wave wave laser laser were were employed employed to toconduct conduct TR TR 2-D 2-D PIV. PIV. Figure Figure 5a5a shows the positionspositions of of the the camera camera and and laser laser sheet. sheet. Positi Positionon 1 1 was was used used to to observ observee stream-wise stream-wise planes, planes, whilewhile positions positions 2 2 and and 3 3 were were used used to to observe observe sp span-wisean-wise planes. planes. The The equivalent equivalent diameter diameter of of thethe struts struts was was less less than than 1 1 mm, mm, so so the the laser laser sh sheetseets thickness thickness was was fixed fixed to to 0.8 0.8 mm. mm. Fluorescent Fluorescent polymerpolymer particles particles (1–20 (1–20 µm)µm) coated coated with with rhodamine rhodamine B Bwere were used used as as the the tracer tracer particles. particles. ChoosingChoosing the the best best time time spacing spacing between between image image pairs pairs (camera (camera framer framer rate) rate) was was de- de- pendentpendent to to the the camera camera sensor sensor size size (pixels), (pixels), real real field field of view of view size size(physical (physical sizes), sizes), and flow and fieldflow velocity. field velocity. Based Basedon these on parameters, these parameters, the frame the rate frame of the rate camera of the was camera set to was 750 set fps, to considering750 fps, considering the flow thevelocity flow velocityin the pores. in the Greyscale pores. Greyscale images imageswith 8-bit with depth 8-bit and depth 10,000 and 10,000 frames were captured and processed by the TR PIV algorithm. The background was frames were captured and processed by the TR PIV algorithm. The background was sub- subtracted using the POD filter developed by Mendez et al. [25]. The velocity vector fields tracted using the POD filter developed by Mendez et al. [25]. The velocity vector fields were obtained by the fast Fourier transform multi-pass algorithm [26]. The interrogation were obtained by the fast Fourier transform multi-pass algorithm [26]. The interrogation sizes of the passes were 64 × 64 pixels, 32 × 32 pixels, and 24 × 24 pixels. sizes of the passes were 64 × 64 pixels, 32 × 32 pixels, and 24 × 24 pixels. The displacement of the particles was about 1/6 of the final size of the interrogation The displacement of the particles was about 1/6 of the final size of the interrogation area [27]. Each interrogation area was overlapped by 50%. Based on the displacement infor- area [27]. Each interrogation area was overlapped by 50%. Based on the displacement in- mation from the overlapped borders and corners of the interrogation area, the interrogation formation from the overlapped borders and corners of the interrogation area, the interro- area was deformed by spline interpolation to interpolate the data between the passes. A gation area was deformed by spline interpolation to interpolate the data between the two-dimensional Gaussian function was used to find the peak intensity of the correlation passes. A two-dimensional Gaussian function was used to find the peak intensity of the matrix [28]. A local median filter was used to validate the velocity vectors [29]. A sample correlation matrix [28]. A local median filter was used to validate the velocity vectors [29]. of the stream-wise instantaneous velocity vector images is shown in Figure5b. The velocity A sample of the stream-wise instantaneous velocity vector images is shown in Figure 5b. The velocity vectors that were filtered by the local median filter are indicated in red. The evidence of suitable camera frame rate was that there were no bad vectors or unusual vectors in Figure 5b; therefore, we could be sure that the selected fps was considered enough to capture the flow field characteristics. Materials 2021, 14, 3566 7 of 17
vectors that were filtered by the local median filter are indicated in red. The evidence
Materials 2021, 14, x FOR PEER REVIEWof suitable camera frame rate was that there were no bad vectors or unusual vectors7 of 17 in Figure5b ; therefore, we could be sure that the selected fps was considered enough to capture the flow field characteristics.
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FigureFigure 5. 5.(a )(a PIV) PIV measurement measurement setup; setup; ( (bb)) instantaneousinstantaneous velocity vectors vectors (stream-wise—position (stream-wise—position 1). 1).
2.4.2.4. Preliminary Preliminary ExperimentsExperiments BeforeBefore starting starting thethe experimentalexperimental campaign, campaign, a asemi-validation semi-validation of ofsetup setup was was done done to to showshow how how the the experimental experimental setup and FPS FPS were were suitable suitable to to capture capture th thee flow flow structure. structure. In In addition,addition, it it can can be be usedused toto demonstratedemonstrate the the development development condition condition of ofthe the velocity velocity profile profile atat the the inlet inlet of of porous porous media,media, asas inin [[20].20]. The The bulk bulk velocity velocity (U (U )b was) was calculated calculated from from the the volumevolume flow flow rate rate as as 0.82 0.82± ± 0.01 m/s within within a a95% 95% confidence confidence level. level. The The Reynolds Reynolds numbers numbers basedbased on on channelschannels hydraulic hydraulic diameter diameter was was 511, 511, while while the Reynolds the Reynolds number number based on based pore on porediameter diameter was was100. 100. These These numbers numbers prove prove that that the theflow flow regime regime was was laminar; laminar; however, however, basedbased on on the the Reynolds Reynolds number number of of the the pore pore diameter diameter following following recent recent publications publications [[30–30–32], the32], flow the regimeflow regime was categorizedwas categorized as unsteady as unsteady laminar, laminar, where where the the laminar laminar wake wake oscillates. oscil- Inlates. addition, In addition, a bubble a bubble trap wastrap installedwas installed to minimize to minimize the the inflow inflow of bubblesof bubbles into into the the test section.test section. Figure Figure6a shows 6a shows the velocity the velocity contour contour superimposed superimposed with with velocity velocity vectors vectors upstream up- stream of the metal foam replica for instantaneous velocity contour (x: −1.75 D~−2.25 D, y: −0.5 D~0.5 D, z: 0 D). The stream-wise velocity U can be defined as: Materials 2021, 14, x FOR PEER REVIEW 8 of 17
U=〈 ( , , )〉 + ( , , , ) (1) where 〈 ( , , )〉is the mean velocity, and ( , , , )is the fluctuation velocity. The other velocity components can also be defined in the same manner. The velocity magni- tude diagram at (−2 D, −0.5 D~0.5 D, 0 D) is shown in Figure 6b. This location is shown as Materials 2021, 14, 3566 a red line in Figure 6a. The velocity profile shows good agreement with the numerical8 of 17 solution for the laminar flow in the same square duct. This result shows that the flow upstream of the metal foam replica is laminar. In addition, it demonstrates that the flow atof the the inlet metal of the foam porous replica material for instantaneous is fully developed. velocity However, contour (x:a little−1.75 deviation D~−2.25 is obvi- D, y: ous,−0.5 which D~0.5 can D, z:be 0 an D). evidence of unsteadiness, which was previously discussed as un- steady laminar flow.
(a)
(b)
FigureFigure 6. 6. InletInlet flow flow conditions: conditions: ( (aa)) velocity velocity contour contour superimposed superimposed with velocity vectors upstreamupstream of ofthe the metal metal foam foam replica; replica; (b )(b the) the comparison comparison of velocityof velocity profiles profiles from from PIV PIV measurements measurements (experimental) (experi- mental)and laminar and laminar solution. solution.
3. ResultsThe stream-wiseand Discussion velocity U can be defined as: 3.1. Velocity Magnitude U = u(x, y, z)i + u0(x, y, z, t) (1) Figure 7a presents the mean velocity contours of u and v at z = 0, which is superimposed withwhere velocityhu(x, y vectors., z)i is theThe mean direction velocity, of flow and isu 0from(x, y, zleft, t) tois theright. fluctuation x-direction velocity. distance The is othernon- velocity components can also be defined in the same manner. The velocity magnitude diagram at (−2 D, −0.5 D~0.5 D, 0 D) is shown in Figure6b. This location is shown as a red line in Figure6a. The velocity profile shows good agreement with the numerical solution for the laminar flow in the same square duct. This result shows that the flow upstream of the metal foam replica is laminar. In addition, it demonstrates that the flow at the inlet of the porous material is fully developed. However, a little deviation is obvious, which can be an evidence of unsteadiness, which was previously discussed as unsteady laminar flow. Materials 2021, 14, x FOR PEER REVIEW 9 of 17
dimensionalized by both pore and channel diameters in the top and bottom of figure, respec- tively. The channel diameter is employed to non-dimensionalized y-direction distance, and the mean velocity is non-dimensionalized by the bulk velocity. Figure 7b shows the mean ve- locity contours overlaid with the metal foam structure. The solid model was cropped by z = ±0.1 D. The main flow upstream of the foam was divided to two or three streams by the foam Materials 2021, 14, 3566 pores when the main flow met the foam structure for first time. Several jets were made9 of at 17 x = 0 to 0.25 D. After passing through the cells, the jets slowed; however, new jets were simultaneously shaped by other pores. This conversion from earlier jets to new ones con- tinued steadily. This separation and combination through the foam led to well-mixing of fluid.3. Results Considering and Discussion that the current study performed a 2D measurement, the number of these3.1. Velocity separations Magnitude and combinations will be greater. TheFigure separation7a presents regions the behind mean velocity the struts contours should be of consideredu and v at sincez = 0, they which affect is superim-pressure drop.posed However, with velocity Figure vectors. 7 does The not directionpresent out-of-plane of flow is from velocity, left to but right. separation x-direction performance distance isis in non-dimensionalized agreement with Onstad by et both al. [21]. pore The and size channel of the separation diameters area in the is supposed top and bottomto depend of onfigure, the metal respectively. foam strut The shape. channel The diameter strut’s shape is employed is circular to or non-dimensionalized triangular dependingy -directionon poros- ity,distance, according and to the Plateau’s mean velocity law, which is non-dimensionalized describes higher porosity by the in bulk a more velocity. triangular Figure strut7b shape.shows The the meancurrent velocity study was contours performed overlaid on a with triangular the metal strut foam shape. structure. Moon et The al. solid[33] studied model thewas effect cropped of shape by z of= strut±0.1 in D. Kelvin cells on heat transfer and pressure drop.
(a)
(b)
Figure 7. Contours of mean velocity magnitude on the plane at z = 0 D with superimposed vector field: field: ( a)) without and (b) with overlay of metal foam structure. (b) with overlay of metal foam structure.
The main flow upstream of the foam was divided to two or three streams by the foam pores when the main flow met the foam structure for first time. Several jets were made at x = 0 to 0.25 D. After passing through the cells, the jets slowed; however, new jets were simultaneously shaped by other pores. This conversion from earlier jets to new ones continued steadily. This separation and combination through the foam led to well-mixing of fluid. Considering that the current study performed a 2D measurement, the number of these separations and combinations will be greater. Materials 2021, 14, x FOR PEER REVIEW 10 of 17
Materials 2021, 14, 3566 10 of 17 3.2. Mean Velocity Profile To extract the velocity profiles on the span-wise flow, the sheet laser was changed Theaccording separation to Figure regions 8a. behind Figure the 8b struts shows should the velocity be considered profiles sinceon the they plane affect at z pres- = 0 D. The sure drop.blue line However, and red Figure line show7 does the not normalized present out-of-planestream-wise velocity,velocity component but separation /U and performancenormalized is in agreement transverse with velocity Onstad component et al. [21].
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(b)
Figure 8. (a) Mean velocity plot location in the test section; (b) mean velocity profiles. Materials 2021, 14, x FOR PEER REVIEW 11 of 17
Figure 8. (a) Mean velocity plot location in the test section; (b) mean velocity profiles.
3.3. Mean Vorticity
Figure 9a,b show the contours of the mean vorticity 〈 〉 on the plane at z = 0 D. The out-of-plane vorticity 〈 〉 in this plane can be defined as: Materials 2021, 14, 3566 11 of 17 =( ) (2) The mean velocity vector field is superimposed on the vorticity contours. It can Itclearly is evident be seen that that the many main shear change layers occurred are formed in
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(b)
Figure 9. Contours of vorticity at z = 0 D with superimposed vector fields: (a) without and (b) with overlay of metal foam structure.
The mean velocity vector field is superimposed on the vorticity contours. It can clearly be seen that many shear layers are formed by the metal foam structures, and these layers are disturbed by the structures and other jets. This increases the flow instability and causes velocity fluctuation. Figure 10a,b show the root-mean-squared (RMS) fluctuation velocity 0 urms created through the foam, which can be defined as: s u0 2 + u0 2 + u02 u0 = 1 2 l (3) rms l Materials 2021, 14, 3566 12 of 17
Materials 2021, 14, x FOR PEER REVIEW 12 of 17
02 where ul refers to the fluctuating part of velocity, when instantaneous velocity vector u is decomposed into a mean velocity u and a fluctuating part u´ (u = u + u´), while l refers to Figure 9. Contours of vorticity at z = 0 D with superimposed vector fields: (a) without and (b) with overlay of metal foam0 1 of 10,000 captured frames. The RMS fluctuation of the transverse velocity (vrms) can be structure. defined in the same manner.
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(b)
FigureFigure 10. Contours 10. Contours of RMS of RMS fluctuation fluctuation velocities velocities at zat= z 0= D0 D with with superimposed superimposed velocityvelocity vectors: (a (a) )stream-wise stream-wise (x-direc- (x-direction) tion) component and (b) transverse (y-direction) component. component and (b) transverse (y-direction) component.
3.4. Integral Time Scale and Length Scale 0 0 At the inlet of the foam model, urms/Ub and vrms/Ub were near zero, but increased The integral time and length scales can be calculated for better understanding. The to 0.1–0.4 through the foam model. After leaving the foam model, they approached temporal auto-correlation function ( ) for the stream-wise velocity component is de- zero once more. The velocity fluctuated in several areas, including the pore outlet, jet fined as [34]: junction, and behind the struts. This demonstrates that the oscillation might be produced by ( , , , ) ( , , , + ) transverse movements created by the foam configuration or unstable shear layers through ( , , , ) = (4) the interaction of jets. However, further〈 investigation( , , , ) ( , , , ) of the〉 origin of these phenomena
is crucial.where τ is the time lag. The integral time scale ( ) of the stream-wise velocity component is defined as: 3.4. Integral Time Scale and Length Scale ( , , ) = ( ) (5) The integral time and length scales can be calculated for better understanding. The where n is the time when reaches the first zero. In this study, n is set to the point 0 temporal auto-correlation function (ρu0 ) for the stream-wise velocity component u is definedwhere as [34 reaches]: a sufficiently low value (0.05) for practical purposes [35]. Considering the Taylor hypothesis, the integral lengthu scale0(x, y (, z , t) )canu0( xbe, y defined, z, t + τas) [34]: ρu0 (x, y, z, τ) = 0 0 (4) ( , , ) = h u ( ,(x ,, y ,)z , t)u( ,(x ,, y), z , t)i (6)
where τ is the time lag. The integral time scale (Tu0 ) of the stream-wise velocity component u0 is defined as: Z n Tu0 (x, y, z) = ρu0 (τ)dt (5) 0 Materials 2021, 14, 3566 13 of 17
where n is the time when ρu0 reaches the first zero. In this study, n is set to the point where ρu0 reaches a sufficiently low value (0.05) for practical purposes [35]. Considering the Taylor hypothesis, the integral length scale (L 0 ) can be defined as [34]: Materials 2021, 14, x FOR PEER REVIEW u 13 of 17
Lu0 (x, y, z) = Tu0 (x, y, z)hu(x, y, z)i (6)
FigureFigure 11 11 shows shows the the calculated calculated integralintegral timetime and length scales of of u 0( t) on on thethe plane at z =at 0z D.= 0 The D. The scales scales were were normalized normalized by theirby th maximumeir maximum values. values. The The bar bar indicates indicates the the line averageline average for the fory-axis. the y-axis. The integralThe integral time time scale scale rapidly rapi decreasesdly decreases logarithmically logarithmically at anat earlyan stageearly from stage 0.5 from D to 0.5 2.25 D to D. 2.25 The D. integral The integral time scale time increasesscale increases slightly slightly from from the outlet the outlet region ofregion the metal of the foam metal at x foam= 2.0 at D–2.25 x = 2.0 D. D–2.25 The integral D. The length integral scale length shows scale a similar shows tendencya similar to thetendency integral to time the scale.integral These time resultsscale. These show results that the show flow that gradually the flow becomes gradually complex becomes as it passescomplex through as it thepasses metal through foam structure,the metal whichfoam structure, may be an which effect ofmay the be evolving an effect transverse of the motionevolving in thetransverse structure. motion in the structure.
FigureFigure 11. 11.Integral Integral timetime scalescale andand length scale. scale.
3.5.3.5. Span-Wise Span-Wise Vorticity Vorticity FigureFigure 12 12 shows shows thethe meanmean vorticity field field to to illustrate illustrate the the evolution evolution of of the the stream-wise stream-wise vorticesvortices through through the the metalmetal foamfoam structures.structures. Th Thee positions positions of of the the laser laser sheet sheet are are the the same same asas in in Figure Figure8a. 8a. To To capture capture the the images,images, bothboth positions of of 2 2 and and 3 3 were were used used to to capture capture sharpersharper images. images. However, it it was was possible possible to totake take images images from from one side. one side.Sections Sections from −0.5 from −D0.5 to D +0.5 to +0.5 D were D were captured captured from from position position 3, wh 3,ile while the others the others were were taken taken fromfrom position position 2. 2.In In Figure Figure 13a, 13 a,there there was was no stream-wise no stream-wise vortex vortex at x = at−0.5x =D −and0.5 0 D D. and At x 0 =D. 0.25 At D,x several= 0.25 D, severalcounter-rotating counter-rotating stream-wise stream-wise vortices vortices were generated. were generated. At x = 2.0At D, xthe= 2.0number D, the of numbervortices of vorticeswas highly was highlyincreased. increased. At x = At2.25x D= 2.25and D2.5 and D, 2.5the D,number the number and vorticity and vorticity of vortices of vortices ap- appearedpeared similar similar to toeach each other. other. At Atx = x2.75= 2.75 D, these D, these vortices vortices began began to dissipate to dissipate and lose and mo- lose momentum.mentum. At At x =x 4.0= 4.0D, the D, thevortices vortices had hadalmost almost dissipated dissipated completely. completely. The tendency The tendency of the of thevortex vortex evolution evolution was was similar similar to that to that of the of integral the integral time timescale scalein Figure in Figure 11. 11. p 〈 〉 TheThe spatially-averaged spatially-averaged vorticity magnitude magnitude ( ( ω 2)) and and span-wise span-wise planes planes RMS RMS vor- vor- x ticity fluctuation ( 0 , ) are revealed in Figure 12b. Vorticity fluctuation is one of the ticity fluctuation (ωx,rms) are revealed in Figure 12b. Vorticity fluctuation is one of the im- 〈ω 〉 important characteristics of turbulence. The valuep of 2 increased while passing portant characteristics of turbulence. The value of ωx increased while passing through through the foam model and then dissipated to the first state once downstream of the0 the foam model and then dissipated to the first state once downstream of the foam. ωx,rms foam. , increased suddenly in the entrance of the foam model then maintained con- increasedstant inside suddenly the metal in thefoam entrance structure. of theIt is foam believed model that then this maintainedturbulent characteristic constant inside wass the metalcaused foam by structure.the interaction It is believedof jets, which that thishave turbulent different characteristicflow directions wass inside caused the metal by the interactionfoam configuration. of jets, which Interestingly, have different at x flow = 2.0 directions D, inside quickly the metaldecreased, foam and configuration. then it 0 , Interestingly, at x = 2.0 D, ω quickly decreased, and then it slowly decreased through slowly decreased through xx ,=rms 2.75–4 D, downstream of the foam replica. , quickly decreased during x = 2.0–2.5. Downstream of the model, a flow separation zone was formed, which decreased the static pressure. Materials 2021, 14, 3566 14 of 17
0 x = 2.75–4 D, downstream of the foam replica. ωx,rms quickly decreased during x = 2.0–2.5. Materials 2021, 14, x FOR PEER REVIEWDownstream of the model, a flow separation zone was formed, which14 decreased of 17 the
static pressure.
(a)
(b)
FigureFigure 12. 12. EvolutionEvolution of span-wise of span-wise vorticity: vorticity: (a) span-wise (a) span-wise vorticity vorticity at the positions, at the positions, and (b) spa- and (b) spatially- tially-averaged vorticity magnitude—RMS fluctuation values. averaged vorticity magnitude—RMS fluctuation values.
3.6.3.6. Finite-Time Finite-Time Lyapunov Lyapunov Exponent Exponent (FTLE) (FTLE) TheThe finite-time finite-time Lyapunov Lyapunov exponent exponent (FTLE) (FTLE) method method was used was to usedidentify to identifythe vortical the vortical structures of smooth time-dependent velocity fields. FTLE obtains Lagrangian coherent structures of smooth time-dependent velocity fields. FTLE obtains Lagrangian coherent structures (LCSs) by measuring the expansion rate of two neighboring particles during a structures (LCSs) by measuring the expansion rate of two neighboring particles during a finite time. FTLE integration during is defined as [36]: finite time. FTLE integration during TF is defined as [36]: FTLE ( , ) = ln ( ) (7) | | r 1 + FTLE (X , t ) = ln λ Ct0 TF (X ) (7) TF 0 0 max t0 0 where is the integration time, and ( , |)T isF| the initial position of the tracking fluid ( ) particle. is the largest singular value of the Cauchy–Green defor- where T is the integration time, and (X , t ) is the initial position of the tracking fluid mation tensor.F The Cauchy–Green deformation0 tensor0 ( ( )) is expressed as [36]: + particle. λ Ct0 TF (X ) is the largest singular value of the Cauchy–Green deformation max t0 0 ( ) = ( ) ∙ t +T ( ) (8) tensor. The Cauchy–Green deformation tensor (C 0 F (X )) is expressed as [36]: t0 0 ( ): ( ) → ( + ) where is the flow map, and the superscript T is the + h + iT + transpose operator. Ct0 TF (X ) = ∇Ft0 TF (X ) ·∇Ft0 TF (X ) (8) t0 0 t0 0 t0 0 Materials 2021, 14, 3566 15 of 17
+ where ∇Ft0 TF (X ) : X(t ) → X(t + T ) is the flow map, and the superscript T is the Materials 2021, 14, x FOR PEER REVIEW t0 0 0 0 F 15 of 17 transpose operator. TF was set to +1 s and −1 s for the forward and backward FTLE, respectively. The forward FTLE indicates the repelling material line for TF < 0, and the backward FTLE was set to +1 s and −1 s for the forward and backward FTLE, respectively. The indicates the attracting material line for TF > 0. The grid resolution is 0.1 mm × 0.1 mm. forward FTLE indicates the repelling material line for < 0, and the backward FTLE Figure 13a,b show the contours of the forward and backward FTLE fields at x = 2.25 D. indicates the attracting material line for > 0. The grid resolution is 0.1 mm × 0.1 mm. Surprisingly,Figure 13a,b many show ridgesthe contours of the of vortical the forward structures and backward were identified. FTLE fields Both at x the = 2.25 attracting D. and Surprisingly,the repelling many material ridges of lines the werevortical considered structures were to contribute identified. toBo theth the heat attracting and mass and transfer theinside repelling the material metal foam lines structure.were considered The FTLE to contribute ridges to that the hadheat lowerand mass integration transfer inside values could thebe metal regarded foam asstructure. fluctuating The regions.FTLE ridges These that regions had lower were integration expected values to contribute could be greatly re- to the gardedheat andas fluctuating mass transfer regions. by These helping regions the mechanical were expected mixing to contribute by the metal greatly foam to the structure. heat This andresult mass is transfer consistent by helping with the mechanical vorticity fluctuation mixing by the shown metal infoam Figure structure. 13. ThisThis alsoresult provides is evidenceconsistent with of the the metal vorticity foam fluctuation causing shown considerable in Figure flow13. This disturbances, also provides evenevidence at a of relatively thelow metal Reynolds foam causing number. considerable flow disturbances, even at a relatively low Reynolds number.
(a)
(b)
FigureFigure 13. 13.ContoursContours of (a) forward of (a) forward and (b) backward and (b) finite-time backward Lyapunov finite-time exponent Lyapunov (FTLE) exponent at x = 2.25 (FTLE) at D. x = 2.25 D.
Materials 2021, 14, 3566 16 of 17
4. Conclusions A methodology has been presented for TR PIV measurements in a metal foam struc- ture. A metal foam sample was scanned using X-ray microtomography and printed with a transparent material using a PolyJet 3D printer. Detailed post-processing methods for the printed model were introduced because conventional methods could not be applied to the small and complex structure. Time-resolved PIV measurements were conducted, and veloc- ity fields were obtained for various regions of interest with inlet laminar flow conditions. The fluid flow entering the foam configuration was very complicated. The pore network formed a jet; thereafter, the flow was divided and combined. Interflowing jets dis- turbed shear layers created by other jets. A significant transverse velocity near 0.23 hvi/Ub was generated, which is in agreement with Onstad et al. [21], who state that significant transverse movement may happen even in the laminar flow inside of metal foams. The span-wise vorticity evolution in the foam configuration was investigated. The span-wise vorticity inside the metal foam was predicted to increase linearly along the stream-wise direction and show a similar tendency to the integral time/length scale at z = 0 D. The span-wise vorticity demonstrated significant oscillation because of complexity of the flow movement inside the foam configuration, and the fluctuation was visualized by the FTLE method. It is believed that this turbulent characteristic was caused by the interaction of jets that have different flow directions inside the metal foam structure. This was expected to enhance the heat and mass transfer inside the metal foam. This study was performed for a Reynolds number and a foam geometry. In addition, channel size was small; therefore, wall effects must be considered. Therefore, there are open doors for future studies on flow features employing different Reynolds numbers and different metal foam arrangements.
Author Contributions: Conceptualization, Y.K. and K.C.K.; methodology, Y.K. and C.M.; software, O.N.; validation, Y.K., H.D.K., and K.C.K.; formal analysis, Y.K., C.M., and O.N.; investigation, Y.K., C.M., and H.D.K.; writing—original draft preparation, Y.K.; writing—review and editing, C.M., O.N., and H.D.K.; visualization, Y.K. and C.M.; supervision, H.D.K. and K.C.K.; project administration, K.C.K.; funding acquisition, K.C.K. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Research Foundation of Korea (NRF) grant, which is funded by the Korean government (MSIT) (No. 2020R1A5A8018822, No. 2021R1A2C2012469). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Conflicts of Interest: The authors declare no conflict of interest.
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