Mitigation of flow-induced vibrations in high-speed flows using triply periodic minimal surface structures Open Access

Reducing flow-induced vibration (FIV) is crucial for guaranteeing the reliability and operational efficiency of pipelines. Pump-induced pulsations, pressure fluctuation, cross-flows, surface roughness, or geometric variations in shape or cross section induce instabilities that render flows in pipes from laminar to turbulent. Turbulent flows, in turn, cause FIV and noise in extended and/or low-damped pipes. For instance, a fluid flow through a sudden expansion of a pipe is characterized by recirculation zones (Fig. 1) that render a flow turbulent and largely contribute to the excitation of FIV.¹ This and other turbulent phenomena are especially harmful for semiconductor and high-tech systems, affecting their functioning, operational capacity, and lifetime which raises safety issues and exploitation costs.

There exist two key approaches to mitigate flow turbulence that rely on flow laminarization or the reduction of wall friction. Alternatively, existing solutions can be categorized as active and passive, depending on an employed control mechanism.

Active systems use feedback control of FIV and include, e.g., active wall motion,² annular fluid injection, or fluid injection using fluidic oscillators.^3,4 They are highly tunable to suit the needs of specific applications. However, active systems require sensors, actuators, and control algorithms that increase installation and maintenance costs, exploitation requirements, and the risk of failure. In addition, the risks associated with active systems outweigh their potential rewards because of extremely high costs of downtime in the semiconductor industry. Also, active control systems require space for the system components and cannot be used, e.g., in high vacuum or strong magnetic fields. Therefore, passive systems remain a preferred choice in the semiconductor and other high-tech industries.

Passive solutions include custom flow channel designs, flow-control devices, e.g., flow straighteners with cubic or honeycomb grid inserts, and altering internal wall surfaces, e.g., by riblets or superhydrophobic coatings, among others.^3,5–8 Custom flow channel design and flow control devices are especially promising to suppress FIV induced by recirculation due to an abrupt change in the pipe diameter. Custom flow channel designs allow controlling the fluid flow path and significantly reduce recirculation.⁵ Flow-control devices such as flow straighteners with cubic or honeycomb grid inserts suppress transverse velocity components and can be used to precisely control the velocity profile and laminarize internal flows of up to $Re=10 000$ by varying streamwise length over the cross section of an insert.³ In addition, flow control devices using a porous medium can reduce the emergence of recirculation zones. For instance, the incremental reduction of the permeability K of a porous insert enables eliminating the recirculation zones and significantly decreasing turbulent kinetic energy κ. This approach was applied, e.g., to fully laminarize a turbulent flow at Reynolds numbers around 37 000.⁸ 

While the described passive techniques show promising results in FIV mitigation, the research on reducing turbulence in high-speed flows remains limited. In this work, we aim to fill in this gap by studying the applicability of so-called triply periodic minimal surface (TPMS) structures as inserts to reduce turbulence in flows with recirculation zones activated due to variations of the pipe diameter (Fig. 1).

TPMS structures have become increasingly popular because of their unique mechanical, thermodynamic and hydrodynamic properties, which can be tuned to application needs.^9–11 A TPMS is a three-dimensional surface with zero mean curvature at any point,¹⁰ with mean curvature H defined as an average of principal curvatures k₁ and k₂, i.e., $H=k1+k22$⁠.¹² If such a surface has no self-interactions and is periodic in three orthogonal directions, it is called a TPMS.

The commonly used TPMSs include the Schwarz primitive and Schwarz diamond developed by Herman Schwarz and the Schoen Gyroid, Schoen I-Graph and Wrapped Package developed by Alan Schoen.^11,13,14 Their mathematical formulas are given in Table I.

There are two approaches to create a solid structure from a TPMS. The first approach delivers structures formed between two surfaces offset from the TPMS along its normal direction. The resulting designs are referred to as sheet-based TPMS structures. Alternatively, a structure can be formed by a solid bounded by a TPMS and is then referred to as a solid-based TPMS structure. The representative unit cells of the SP, SD, SG, and IW-P TPMS structures obtained in these two ways are given in Fig. 2 and were constructed using MSLattice.¹⁵ 

TPMS structures have gained increasing interest as feasible solutions to various engineering problems because of the growing capabilities of additive manufacturing techniques to reproduce their intricate geometries.^11,16 The high stiffness-to-weight ratio and enhanced mechanical energy absorption make the TPMS structures useful in automotive and aerospace industries as structural components in so-called crumple zones.^17,18 The large surface area-to-volume ratio and highly porous geometries of TPMS elements make them beneficial in thermal management.¹⁹ The resemblance of the trabecular bone topography and equal distribution of stresses make these structures promising as scaffolds and implants in biomedical applications.^20–23 

The studies of the hydrodynamic behavior of TPMSs are centered around their biomedical applications and, thus, are mainly limited to analyzing their properties in low-speed laminar flows, with a mean velocity, e.g., below $1×10−2$ m/s.^9,24 Here, we aim to go beyond this limitation by expanding the available analytical models to high-speed flow regimes and thus assess the turbulence-reducing capabilities of TPMS inserts in flows with a mean velocity of 2–4 m/s.

While the Darcy–Forchheimer model for flow through porous media is already commonly used in chemical, petroleum and environmental engineering applications, its applications in the high-tech industry have been limited.^26–28 In this study, we aim to expand the Darcy–Forchheimer model for flow through TPMSs to the turbulent flow conditions with Reynolds numbers between $Re=$ 10 000 and 20 000 relevant for the semiconductor manufacturing industry. For this, we derive the values for permeability K and inertial drag factor C_F in Eq. (2) as effective characteristics of the TPMS unit cells by simulating high-speed fluid flows through an unbounded TPMS structure. The obtained estimations are used in full-scale finite-element CFD simulations for a pipe with an abrupt change of diameter to analyze the capabilities of the TPMS inserts to reduce turbulence induced by flow recirculation.

The permeability K and inertial drag factor C_F values for Eq. (2) were obtained by running CFD simulations for fluid flows through TPMS inserts using the Fluid Flow module of COMSOL Multiphysics^®. The RANS κ–ε turbulence model was employed because of its numerical robustness.

The homogeneous TPMS designs were formed by periodically tessellating each of the four sheet-based and solid-based unit cells shown in Fig. 2 along the flow direction. The eight geometries were generated using MSLattice¹¹ with four porosity levels, ranging from 50% to 80% with 10% step, and two unit cell sizes, α, of 1 and 2 mm. It results in 64 case studies. Each TPMS insert was represented by 1 × 4 × 1 unit cells, similar to the study of Asbai-Ghoudan et al.⁹ The sensitivity analysis of permeability K to the number of the unit cells along the flow direction showed that increasing this number beyond four barely changes the value of K.

We considered six mean flow velocities ranging from 1 to 2 m/s with a 0.2 m/s step. The inlet velocity profile was set to a fully developed flow boundary condition; the outlet boundary condition was set to a constant ambient pressure. Finally, periodic boundary conditions were applied at the lateral sides of the fluid domain to model an infinite number of the unit cells along the directions perpendicular to the flow.

The pressure drop over a TPMS structure was recorded for each specified values of the mean flow velocity. The values of K and C_F were estimated using a second-order polynomial fit.

To assess the capability of the TPMS inserts to reduce flow turbulence, we used the free and porous media flow studies in COMSOL Multiphysics^® to simulate high-speed flow in a pipe with an abrupt change of diameter (Fig. 1).

The fluid domain consists of a narrow pipe of 5.1 mm in internal diameter and 51 mm in length followed by an abrupt expansion of a pipe diameter to 10.2 mm. The expanded part of the pipe has the length of 112 mm. Right after the expansion, we placed a porous insert of 10 mm long and 10.2 mm in diameter. The pipe diameters and flow conditions were chosen as relevant for semiconductor manufacturing applications.²⁹ 

The flow through the TPMS insert was modeled using the Darcy–Forchheimer model (2) with the values of permeability K and inertial drag coefficient C_F derived from the unit-cell-level studies in Sec. II A. We also modeled the flow in the pipe without a porous insert, used as a benchmark case. In both scenarios, the inlet boundary condition is a fully developed flow with a mean flow velocity v = 4 m/s. All the wall surfaces have a no-slip condition, and the outlet boundary condition is set to ambient pressure.

In the two sets of studies, we employed physics-defined finite-element meshes described by the parameter ‘Element Size’ that ranges in scale from Extremely Course to Extremely Fine, with seven steps in between. For the mesh analysis, we arbitrary chose a gyroid-sheet geometry with porosity $ϕ=70$% and unit cell size α = 1 mm. The simulations were carried out for each type of mesh for both unit-cell-level and macroscopic simulations. The COMSOL Multiphysics^® naming convention and the respective number of mesh elements for each set of simulations are given in Table II. The analysis was limited by the available PC memory (32 GB). The mesh quality was assessed by evaluating the pressure drop over the four unit-cell TPMS insert.

We analyzed eventual patterns in the obtained numerical data by performing a statistical analysis in RStudio^® on statistically independent subsets.

We use a similar approach for creating subset data bases differentiated by a TPMS type, geometry, and porosity.

The results of the mesh quality analysis for the unit-cell-level and Darcy–Forchheimer simulations are presented in Figs. 3(a) and 3(b), respectively. At the unit-cell scale, the pressure drop over the insert converges to around 54 000 Pa, for the fine element size (see Table II). The calculations for this mesh are, however, resource- and time-demanding. Hence, to balance the computation time and preserve the accuracy of results, we chose the mesh with the Coarse Element Size for the unit-cell-level simulations (Table II). For the coarse mesh, computations take roughly four hours for the whole parametric sweep on a desktop-type PC (Intel core i7 7700k, 32 GB).

For the Darcy–Forchheimer simulations, the mesh analysis revealed small variations in the estimated pressure drop depending on the mesh size [Fig. 3(b)]. Therefore, we chose the Extra Course mesh to run these simulations to reduce the simulation time. For this mesh, the simulations took on average three hours on the desktop PC mentioned above.

Figures 4 and 5 show the calculated streamline velocity distributions for the sheet- and solid-type TPMS inserts of porosity 80%, respectively, while Figs. 6 and 7 show the corresponding pressure values. These results reveal distinct characteristics of the flow through the TPMS inserts of different geometries. In particular, the sheet-based Primitive insert substantially accelerates the flow due to the presence of pores with small openings that is accompanied by the largest pressure drop compared to that for the other TPMS inserts. In these other cases, the pressure drop varies from small to large depending on the TPMS geometry, but an adverse pressure gradient is always present due to the inhibition of the flow.

We computed the pressure drop over the TPMS inserts relative to ambient pressure by estimating an average pressure over a cross-sectional cut plane upstream of the TPMS unit cells. Next, we fitted a second-order polynomial to each set of the pressure drop values as shown in Fig. 8.

Figure 8(a) presents an exemplary data for the Diamond Sheet design of 80% porosity and unit cell size α = 1 mm, while Fig. 8(b) presents all the remaining fitted curves. All the curves reveal a nonlinear dependence of the pressure gradient on the velocity, and all the polynomial fits had $R2>0.999$⁠. This confirms that the flow behavior follows the Darcy–Forchheimer law (2). The fitting coefficients corresponding to the permeabilities K and inertial drag factor C_F values are given in Figs. 9 and 10, respectively.

It is worth mentioning that our computations were successful for all except one set of the simulations. For the IW-P Solid geometry of $α=$ 2 mm, the second-order polynomial fit provided a negative linear component. This results in a negative permeability meaning that the substance would hinder or block flow from passing through the insert, which contradicts the concept of permeability. This result was therefore disregarded in subsequent simulations.

Additionally, the dependence of the results on the κ–ε turbulence algorithm was compared to the SST turbulence model. For the Gyroid Sheet geometry with 80% porosity, for instance, the resulting pressure drop was 15 006 Pa for the κ–ε turbulence model and 12 350 Pa for the SST turbulence model and, i.e., about a 20% difference.

The flow through a pipe with an abruptly changing diameter [Fig. 11(a)] was simulated for each of the 64 TPMS inserts at a mean velocity of 4 m/s in the narrow section and 1 m/s in the wider section. Figure 11(b) presents the estimated velocity distribution for the benchmark case without an insert and for the Diamond-Sheet insert of porosity 80%. One can see significant differences in flow characteristics for these two cases. The reference case has substantial recirculation zones that are eliminated in the presence of the Diamond-Sheet insert, for which the flow is smooth and laminar. Further analysis of the distribution of turbulent kinetic energy κ and vorticity ω for these two cases [Figs. 11(d) and 11(e) reveals that the TPMS insert eliminates these turbulence signatures from the flow. However, the turbulence reduction comes at the price of a significant pressure drop created by the low permeability of the TPMS insert [Fig. 11(c)]. The same effects were observed for all the other proposed TPMS inserts.

To quantify the turbulence reduction, we estimated the volume-average turbulent kinetic energy and the volume-average vorticity for a section of the pipe next to the placement of a TPMS insert and compared them to the volume-average turbulent kinetic energy and the volume-average vorticity for the same volume in the benchmark case. The results of this comparison are shown in Fig. 12 for $κ=0.118 m2 s−2$ and $ω=200 s−1$ in the benchmark case. The best-performing geometries are the Diamond Sheet insert of $ϕ=50$% and α = 1 mm and the IW-P Sheet insert of $ϕ=50$% and α = 1 mm as they both reveal 97.6% reduction in turbulent kinetic energy and 32.7% reduction in vorticity.

The key goal of this study was to expand the analytical Darcy's model for laminar flows through a TPMS insert to a turbulent flow regime, in which the Forchheimer nonlinear term cannot be neglected. For this, we analyzed 64 TPMS designs of four types, both sheet-based and solid-based, with four porosity values and two unit cell sizes and numerically estimated permeability K and inertial drag factor C_F that enter the Darcy–Forchheimer Eq. (2). For porous inserts, one expects a positive correlation between porosity and permeability and a negative correlation between porosity and the inertial drag factor, as reducing the porosity restricts fluid flow. The estimated correlations for the TPMS Sheet and Solid designs are 0.992 and −0.917 for the permeability and inertial drag factor, respectively, thus, aligned with the expected trends.

Similarly, the unit cell size should correlate positively with permeability as smaller unit cells have a larger internal surface area. As the flow interactions with internal walls activate viscous drag forces, the increased internal surface areas in smaller unit cells can be associated with a decrease in permeability. Indeed, our results reveal that the mean permeability for the analyzed TPMS inserts of 1 mm unit cell size is $4.96×10−10$ m² or 503 Darcy, while the mean permeability of the inserts of 2 mm unit cell size is $1.55×10−9$ m² or 1571 Darcy. The average correlation between the permeability and the unit cell size of every type of the TPMS inserts is 1, meaning that the permeability is smaller for the smaller unit cells provided the geometry and porosity are the same. The mean inertial drag factors for the two sizes of the unit cells are 0.127 and 0.133, respectively. Despite these quite close values, the average correlation between the inertial drag factor and unit cell size is −1, meaning that the inertial drag factor is smaller for the larger unit cells in every analyzed case.

The average correlations between the permeability and turbulent kinetic energy and the permeability and vorticity for the TPMS inserts are −0.768 and −0.780, respectively. The average correlations between the inertial drag factor and turbulent kinetic energy, and the inertial drag factor and vorticity are 0.987 and 0.985, respectively. These values suggest that to reduce the turbulent kinetic energy and vorticity efficiently, one would ideally reduce the permeability as much as possible while limiting the associated rise in the inertial drag factor. This approach becomes apparent for the Primitive Sheet insert, where the inertial drag factor is much larger than that for the other TPMS geometries of lower porosity levels. We also note that the Primitive TPMS designs of low porosity levels appear to be too restrictive in terms of the inertial drag factor that hinders the turbulence-reduction performance and results in an undesirably high turbulent kinetic energy, as shown in Fig. 12.

Our results illustrate the efficiency of the TPMS inserts in reducing turbulent kinetic energy κ and vorticity ω in a limited way. An alternative approach implies presenting an average change in average turbulent kinetic energy κ divided by the pressure drop induced by the TPMS insert and the change in the volumetric average of the module of vorticity vector ω divided by the pressure drop induced by the insert, as shown in Figs. 13 and 14, respectively. With these alternative measures, it becomes clear that while more “restrictive” unit cells show better absolute results. This is achieved at a high cost in terms of efficiency. Therefore, it could be beneficial to conduct further research about evaluating the efficient use of the TPMS inserts for turbulence reduction by analyzing less restrictive geometries of higher porosities and/or larger unit cell sizes.

Though our results reveal clear advantages of the TPMS inserts in reducing turbulence, some limitations of this study should also be clarified.

The Darcy–Forchheimer model for a high-speed flow through a TPMS medium is validated numerically. One of the conclusions that can be derived from it is that as the unit cell size approaches zero, the influence of the structural design on the flow becomes negligible. For larger unit cells, it remains however unclear whether this model can properly capture the variations in the flow structure governed by the TPMS geometry. To understand this phenomenon better, future studies could consider either experimental validation or direct simulations of larger structures. While the experimental tests require specialized equipment and well-controlled excitation conditions, the direct simulations of flow interactions with the TPMS are very expensive computationally. These two types of studies are out of the scope of this work.

This study proved the relevance of the non-linear Forchheimer term and expanded Darcy's model for internal laminar flows through TPMS inserts to higher speed flow regimes, relevant for industrial applications. By simulating 64 TPMS designs, we derived valuable insights into the effects of the insert geometry on the values of permeability K and inertial drag factor C_F which are the leading terms in the Darcy–Forchheimer model.

As anticipated, the porosity of TPMS inserts positively correlates with their permeability and negatively correlates with the inertial drag factor. Similarly, the unit cell size shows a positive correlation with permeability, so smaller unit cells with higher internal surface areas are characterized by increased viscous drag forces. The observed correlations underscore the intricate interplay between geometry and fluid dynamics within TPMS structures.

Our analysis also revealed significant correlations between permeability and turbulent kinetic energy, as well as between inertial drag factor and vorticity. These findings emphasize the potential for optimizing TPMS designs to simultaneously reduce turbulence and vorticity, albeit with efficiency considerations.

In our simulations, the best performing TPMS inserts yielded reductions in turbulent kinetic energy of 97.6% and reduction in vorticity of 32.7%.

Although our simulations offer promising results, the applicability of the Darcy–Forchheimer model to larger unit cell sizes warrants further investigation, possibly through experimental validation or direct simulation of expanded unit cell arrays.

To sum up, this study contributes to a deeper understanding of the efficacy of TPMS structures in turbulence reduction and opens prospects for future research to focus on refining models and exploring less restrictive geometries for enhanced efficiency and effectiveness in practical applications.

The authors have no conflicts to disclose.

Bastiaan Jan Thomas Piest: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Pablo D. Druetta: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Anastasiia Olexandrivna Krushynska: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Project administration (supporting); Software (supporting); Supervision (lead); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.