12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

AN EXPERIMENTAL STUDY OF PRESSURE DROP AND FLUID FLOW IN
TRIPLY PERIODIC MINIMAL SURFACE (TPMS) SCAFFOLDS

Daejung Kim
Department of Mechnical Engineering
The University of Melbourne
Victoria 3010, Australia
daejungk@student.unimelb.edu.au

Chenxi Peng
Department of Civil and Infrastructure Engineering
RMIT University
Victoria 3000, Australia
S3756293@student.rmit.edu.au

Jonathan Tran
Department of Civil and Infrastructure Engineering
RMIT University
Victoria 3000, Australia
jonathan.tran@rmit.edu.auABSTRACT

Jimmy Philip
Department of Mechnical Engineering
The University of Melbourne
Victoria 3010, Australia
jimmyp@unimelb.edu.aucell penetration and migration.
With the development of additive manufacturing, experimental studies on TPMS scaffolds have proliferated. This advanced technology has enabled the possibilities to customize
pore size, morphology, and porosity that were the big obstacles
encountered by traditional manufacturing processes. A variety of TPMS scaffolds are now manufactured with different
materials, and their mechanical behaviours have been studied
numerically and experimentally (e.g., Abueidda et al., 2017).
Also, some attention have been given to the permeability of
TPMS scaffolds. Bobbert et al. (2017) and Castro et al. (2019)
manufactured different TPMS scaffolds in a size of 15-20 mm
and tested their permeability. They found that gyroid shows
good permeability compared with other scaffolds, but due tothe limited size of their specimen led to uncertainty in theirconclusion, and the flow dynamics inside the scaffolds remains
unknown. Furthermore, due to the primary focus on biomedical applications, the flow rates have been quite low; therefore, fluid behaviour at higher flow rates within these porous
scaffolds has not been studied. This limits the application of
TPMS scaffolds in other fields. As such, our broad aim is acomprehensive study of fluid behaviour in such complex internal architectures.
To achieve this, flow behaviours in different TPMS scaffolds are investigated experimentally. A gyroid is selected
because of its complex internal architecture as a representative of TPMS scaffolds. Also, we choose a TPMS-based
BCC scaffold due to its open cell structure. There are various ways to design TPMS topologies (Nitsche, 1989; Karcher,1988; Brakke, 1992), but we adopt nodal surface approximations because of their mathematical expressions (Von Schnering & Nesper (1991)), which enables us to produce customized
topologies using Matlab. In this method, either a solid or asheet type of topologies are designed by choosing a level-set
constant c (Al-Ketan & Abu Al-Rub (2019)). Since TPMS
sheet scaffolds have large surface area that can be advantageous in many engineering applications, we design TPMS
sheet gyroid (Vijayavenkataraman et al., 2018; Al-Ketan &Abu Al-Rub, 2019) and BCC (Zhao et al. (2018)) in this study.
The equations governing the TPMS geometry of ’gyroid’ andCellular materials in nature have unique structures that show
outstanding properties, which has led many scientists and engineers to manufacture them artificially. Among them, TriplyPeriodic-Minimal-Surface (TPMS) scaffolds have drawn more
attention due to their advantages over other similar lattice
structures. With the development of the additive manufacturing known as a 3D-printing, many researchers have been manufacturing a variety of TPMS scaffolds with different methods, and then studing their mechanical behaviours. However,less attention is devoted to fluid dynamics in TPMS scaffolds.
Here, to understand the flow behaviour in different TPMS scaffolds, measurements of pressure drop are carried out with two
different structures, namely ‘gyroid’ and body-centered cubic(BCC), with three different porosities, and we compare the results with the classical Ergun equation used for flow throughrandom sphere packing. The results from the gyroid scaffolds
agree well with the Ergun equation within the laminar and theturbulent regimes once we introduce an equivalent hydraulicdiameter. The results from the BCC scaffolds, however, deviate within the transitional and turbulent regimes. For further
investigation, we select the highest porosity case of the BCC
scaffolds and match its refractive index with the working fluid
to conduct 2D-2C particle tracking velocimetry (PTV) measurements. We observe flow separation at nodes and recirculation flows between nodes at the different velocities demonstrating the dynamical changes with increasing Reynolds number.

Introduction
TPMS are surfaces generated by mathematical expressions
that have zero mean curvature and three-dimensional (3D)
periodicity. Due to these advantages, TPMS scaffolds havebeen of great interest to engineers and biologists. In structural
engineering applications, TPMS scaffolds show outstanding
mechanical properties compared to conventional lattice structures. Especially, TPMS scaffolds have high ratio of the surface to volume, which brings a lot of advantages in tissue engineering applications. For example, smooth and higher surface
areas enhances cell adhesion, and open cell structures improve1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
different locations. The manometers are carefully designed tominimize the effect of surface tension, and height measurements are carried out with sub-millimeter resolution. Water isused as the working liquid during pressure measurements. For
the 2D-PTV, the same set-up is used as shown figure 2. The
inlet section is maintained, and only the measurement section
is replaced. The BCC scaffold at e = 0.85 is selected for velocity measurements. Spectra Physics EV 25 diode pumped solidstate continuous 532 nm (green) laser is used to illuminate flow
at 1 mm away from the center of the BCC scaffold. A PCOdimax HS4 12 bit high speed camera with a resolution of 2000
× 2000 pixels is used in this experiment, which allows us tocarry out time-resolved PTV measurements. We use (orange)
fluorescent seeding particles combined with a green filter on
the camera to block the reflection from scaffold geometry. To
reduce optical distortions, a 59% of ammonium thiocyanate
solution in water at 17◦ C is developed to match the refractiveindex (RI=1.48) of the BCC scaffold, and used as the working liquid. To reduce the refraction due to curvature of thetube, a 65 mm × 50 mm × 65 mm acrylic box filled with theworking liquid is attached at the end of the 75 cm of the BCC
scaffolds coinciding with the camera field-of-view (c.f.figure 2inset: measurement 02).

’BCC’ are respectively given by:

ugyroid ≡ cos(wx x)sin(wy y) + cos(wy y)sin(wz z)+ cos(wz z)sin(wx x) − c = 0(1)uBCC ≡ cos(2wx x) + cos(2wy y) + cos(2wz z)− 2(cos(wx x)cos(wy y) + cos(wy y)cos(wz z)(2)+ cos(wz z)cos(wx x) − c = 0
Here x, y, and z are spatial coordinates in the three-dimensional
Cartesian coordinate system, and wx , wy , and wz are definedby wi = 2π Lnii (i = x, y, z). The constant value of c plays an important role to generate different thickness. Eq (1) and Eq (2)are solved using Matlab, and its built functions of isosurface
and iscaps are used to generate desired surfaces of TPMS
scaffolds. The x − y view in figures 1(a) to (b) shows thegeometry of gyroid and BCC respectively where its unit cellsize is 12 mm×12 mm×12 mm. A 36 mm×36 mm×120 mmof gyroid and BCC respectively is produced in Matlab
following the described procedure. This geometry is imported
on Autodesk Netfabb software to make a cylindrical sample
as shown in figure 1.

Results
Figure 3(a) shows the result of pressure drop within the gyroidand BCC scaffolds measured across 1.5 m distance. Pressuredrops are non-dimensionalized as shown on the left-hand-sideof Eq (3). During the measurements, water is maintained at atemperature of 22◦ C resulting in constant viscosity and density. The net pressure drop is composed of viscous and pressure drag, each dominating respectively at lower and higher
Re. At ReH /e < 100, both the gyroid and the BCC scaffolds,
although shifted along the ordinate, show a linear relationshipthat follow laminar drag law parallel to the Ergun equation represented in the solid line. In this case, gyroid with e = 0.7and BCC scaffold with e = 0.55 show the closest trend to theErgun values while maintaining the linear relationship, whichsuggests that theses geometries behave more or less similar
to random packed spheres with similar pore diameters. The
majority of the data, however, do not follow the Ergun equation. We, therefore, introduce the concept of ‘an equivalent hydraulic diameter’ dequ . The essential idea is to find the equivalent hydraulic diameter that moves the laminar drag law region of each geometry to that given by the laminar region inthe Ergun equation (for packed spheres). Thus, for each geometry, we are trying to find the equivalent sphere diameterthat gives the same laminar viscous drag. To find the dequ for
gyroid and BCC, a slope in the region of the linear pressure
drop is calculated by fitting the number of experimental data
points in an increment of one data at each time. For example, the first and second data are fitted to find a slope and thenby increasing one data point the first three data points are fitted to find a new slope. A next slope is calculated by fitting
four data points.This process continues until a constant slope
is no longer maintained. Taking the slope in the linear region,dH is varied until a root mean square error (RMSE) between
the laminar drag law and the experimental results become aminimum, that is less than 1% difference. The dequ is thus defined as the final dH obtained at the end of this procedure. All
the data are replotted with dequ and are shown in figure 3(b).
Essentially, dequ becomes a free parameter to make the laminar part of the normalised pressure drop data similar to thatgiven by the Ergun equation. The results with the equivalentdiameters of the gyroid scaffolds at the three different porosities agree well with the theoretical plot unlike that of the BCC

In the following sections, we will present a description ofexperimental set-ups for the measurements of pressure dropand particle tracking velocimetry. Then, the results of pressure drop measurements are compared with the Ergun equation(e.g., Ergun & Orning, 1949; Wood et al., 2020).
 3∆P e2 dH
21=150+ 1.75,
2L ρu2c
3 ReH /e(3)where ∆P is pressure drop across length L of the scaffold, dH
is a hydraulic diameter, uc is superficial velocity defined as thevolumetric flow rate divided by the cross-sectional area of thetube where the scaffolds are placed, and e is porosity definedas the ratio of void volume to total volume.
The original Eq (3) is represented in terms of a pore diameter,
but we replace it with a hydraulic diameter due to non-uniform
pore sizes. We denote LHS of (3) as fH , which is similar tothe friction factor.

Experiments
A schematic of the experimental set-up for the pressure drops
and 2D-PTV is shown in figure 2. For pressure drop measurements, the set-up consists of three main sections: the inlet,
the measurement section, and the outlet. The inlet has a constant pressure head system, and a gravity driven flow is fed to
the nozzle which includes honeycomb as flow straighteners.
The fluid travels 1.3 m of acrylic tube with an inner diameterof 20.6 mm before entering the tube with scaffolds inserted,which covers a tube length of 1.53 m. The entrance lengthof 1.3 m is such that fully developed laminar flow enters thescaffold section. In the measurement section, a total length of
1.53 m of scaffolds at three different porosities (e = 0.85, 0.70,and 0.55) are inserted in separate acrylic tubes and connectedto the inlet tube. For each tube, the first pressure tap is located 2 cm from the start of the scaffold section, the second
tap is 75 cm from the first taps, and the last ones 75 cm fromthe second taps and 1 cm from the end of the scaffold section.
Static pressures are measured using manometers at these three2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 1. Isometric view of one sample of gyroid (a) and BCC (b) (d=20.54mm and L=120mm).
One unit cell in
12 mm×12 mm×12 mm is in the middle highlighted in black rectangle in the x − y view. In this design, one third of the one unit
cell surrounds the one complete unit cell.

Figure 2. Experimental set-up schematic. Two different measurements are conducted, and a real picture of each measurement isshown in the measurement section. A needle valve controls flow rates at the outlet, and a load cell is used to calculate mass flow rates.

Conclusion

scaffolds. At ReH /e > 100, the BCC scaffolds show different relationships, but having a similar pattern for the different
porosities as the flow rates increase. These suggest the ‘equivalence’ of flow dynamics of the gyroid and random packed
spheres because of the intricate geometrical structure of the gyroid. On the other hand, the open structure with sharper middle
structure of the BCC scaffolds deviates from random spheres,
which highlights the large-scale-inhomogeneities in the BCC
scaffolds. To further understand the different behaviour of theBCC flow structure, velocity field corresponding to the highest
Re=1500 for e = 0.85 in figure 4(a) is measured. Figures 4(a)to (c) show different views of the scaffold where velocity ismeasured. Figure 4(d), where several consecutive particle images are overlayed, shows flow recirculation in the pore regions between nodes. The recirculation region occurs owing
to the relatively low velocities in the middle pore section compared to the high flows above and below the nodes as depicted
in figure 4(e). Flow separation occurs at the beginning of thenode, and backflow exists at the end of the node. The backflow is trapped in pore regions between two nodes due to therelatively high velocities passing above and below the nodes.
Recirculation becomes stronger and chaotic as the flow rate increases. Such a flow behaviour is indeed different to what is
expected from traditional porous media with random spheres.

The measurements of pressure drops in the gyroid andBCC scaffolds for three porosities are conducted. Nondimensionalised pressure drops used to define an equivalenthydraulic diameter allows us to make direct comparison withthe standard Ergun equation. This methodology for porous
media is expected to be general, and has similarities to theconcept of ‘equivalent roughness height’ widely used in roughwall flows. Also, the velocity field of the BCC shows interesting results, which have not been reported in TPMS scaffolds.
Due to the recirculation flow at high flow rates and large-scale
flow inhomogeneity, pressure drops in the BCC scaffolds nolonger follow the Ergun equation at higher Re.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 3. (a) Nondimensional pressure gradients of the gyroids and BCC scaffolds at three different porosities. (b) Nondimentionalpressure gradients are replotted with equivalent hydraulic diameters, dequ .

Figure 4. (a) Isometric view of the last BCC scaffold at the porosity of 0.85 in the measurment section. (b) Side view of (a). Solidvertical line represents a laser plane passing through 1 mm away from the center. (c) Front view of (a). (d) Overlay of consecutive 55images to visualize the fluid flow with the masking image. (e) Velocity field (red-high, and blue-low)in the x-direction with the masking
image. Grey regions represent solid parts where the laser plane pass through. The middle gray regions are nodes at the center in (b).

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