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Mechanical and hydrodynamic analyses of helical strake-like ridges in a glass sponge

Matheus C. Fernandes

Matheus C. Fernandes

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

Wyss Institute for Biologically Inspired Engineering, Harvard University, Cambridge, MA 02138, USA

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Mehdi Saadat

Mehdi Saadat

Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA

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Patrick Cauchy-Dubois

Patrick Cauchy-Dubois

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

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Chikara Inamura

Chikara Inamura

Media Lab, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

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Ted Sirota

Ted Sirota

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

Wyss Institute for Biologically Inspired Engineering, Harvard University, Cambridge, MA 02138, USA

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Hossein Haj-Hariri

Hossein Haj-Hariri

College of Engineering and Computing, University of South Carolina, Columbia, SC 29208, USA

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Katia Bertoldi

Katia Bertoldi

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

Wyss Institute for Biologically Inspired Engineering, Harvard University, Cambridge, MA 02138, USA

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James C. Weaver

James C. Weaver

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

Wyss Institute for Biologically Inspired Engineering, Harvard University, Cambridge, MA 02138, USA

[email protected]

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Published:https://doi.org/10.1098/rsif.2021.0559

    Abstract

    From the discovery of functionally graded laminated composites, to near-structurally optimized diagonally reinforced square lattice structures, the skeletal system of the predominantly deep-sea sponge Euplectella aspergillum has continued to inspire biologists, materials scientists and mechanical engineers. Building on these previous efforts, in the present study, we develop an integrated finite element and fluid dynamics approach for investigating structure–function relationships in the complex maze-like organization of helical ridges that surround the main skeletal tube of this species. From these investigations, we discover that not only do these ridges provide additional mechanical reinforcement, but perhaps more significantly, provide a critical hydrodynamic benefit by effectively suppressing von Kármán vortex shedding and reducing lift forcing fluctuations over a wide range of biologically relevant flow regimes. By comparing the disordered sponge ridge geometry to other more symmetrical strake-based vortex suppression systems commonly employed in infrastructure applications ranging from antennas to underwater gas and oil pipelines, we find that the unique maze-like ridge organization of E. aspergillum can completely suppress vortex shedding rather than delaying their shedding to a more downstream location, thus highlighting their potential benefit in these engineering contexts.

    1. Introduction

    The geometrically complex siliceous skeletal systems of marine sponges have attracted a great deal of attention from the scientific community due to their multi-scale structural hierarchical organization and remarkable damage tolerance [13]. For example, the mineralized tubular skeleton from one such species, Euplectella aspergillum, consists of bundles of individual needle-like elements (spicules) that are cemented together to form a diagonally reinforced square lattice-like structure that is further covered by a series of helical ridge-like features [4,5]. Detailed investigations into the various components of this structural hierarchy have revealed their surprising mechanical benefits.

    First, single-spicule studies have revealed the presence of an underlying laminated architecture consisting of concentric lamellae of consolidated silica nanoparticles separated by thin organic interlayers. The silica layers decrease in thickness from the spicule core to its periphery, resulting in a functionally graded design that effectively retards crack propagation through the spicules, while simultaneously increasing their buckling resistance [3,6].

    At a second level of structural hierarchy, bundles of these laminated spicules are further organized into a square lattice-like structure which is reinforced by pairs of diagonal struts that cross through every other cell of the lattice, creating a checkerboard-like organization. Through a combination of finite element (FE) simulations and direct mechanical testing, it has been demonstrated that this non-intuitive diagonal reinforcement strategy creates a geometry that exhibits a near-optimal strength-to-weight ratio for this specific family of truss structures [7].

    Finally, surrounding the underlying diagonally reinforced square lattice are a series of right- and left-handed helical ridges, which are oriented perpendicular to the surface of the skeletal tube and form a distinctive maze-like organization. While previous studies have speculated that these ridge-like features provided a mechanical benefit to the sponge [4], as has been demonstrated for other reinforcing axial and helical rib-like elements on tubular structures [810], the striking morphological similarity of the sponge ridges to helical strakes used for vortex suppression in cylindrical structures under flow [1114] motivated the present study which explores their potential hydrodynamic functionality.

    Vortex shedding is a topic of great concern to the engineering community, not only for its ability to excite vortex-induced vibrations, potentially leading to resonance structural vibrations [12,15], but also for its periodic forcing effects and noise generation [11]. Fluid flow past bluff cylindrical structures is known to generate vorticity due to the presence of shear in the fluid’s boundary layer. These small vorticity regions coalesce into regions of concentrated vorticity on both sides of the cylinder, leading to a phenomenon known as von Kármán vortex shedding. [12] Because of the implications of von Kármán vortex shedding on the structural integrity of cylindrical forms, the addition of helical strakes (protruding ridge-like elements) is commonly employed as a method to suppress this effect [11,13].

    Here, we describe a computational framework for investigating structure–function relationships of the complex helical ridge system in the skeleton of Euplectella aspergillum. Using an integrated approach that combines FE simulations and computational fluid dynamics, we explore both the mechanical and hydrodynamic effects of these skeletal features and compare these results to alternative ridge geometries employed for similar functions in their synthetic engineering analogues.

    2. Skeletal geometry

    As demonstrated from previous studies [4], the skeletal system of a fully mature specimen of E. aspergillum (figure 1a) is covered with a laminated silica cement, forming a rigid construct, and contains a maze-like network of external helical ridges that extend perpendicular to the skeletal tube. To explore the organizational details of this complex ridge system, we examined ten different sponge skeletal samples and manually mapped each of the different ridge designs, ultimately constructing a series of planar ridge connectivity diagrams, a representative example of which is shown in figure 1b (all 10 maps are shown in electronic supplementary material, figure S1). In this surface map (which corresponds to the upper ca 50% of the skeleton where the ridges are most pronounced and easily identified), the coloured squares represent the ridge locations (the ridge-less unit cells are white), with red squares denoting clockwise ridges and blue squares denoting counterclockwise ridges (corner elements are denoted in purple). From examination of these different sponge specimens, we identified several common ridge design themes, which include the following (and are consistent with observations from previous studies [4]):

    The ridges occur at 45° angles relative to the long axis of the skeletal tube.

    The ridges populate, on average, every other closed square in the skeletal lattice.

    The ridges intersect at 90° angles.

    The ridges never cross each other; however, they may form T-junctions.

    The total number of ridge-filled cells is similar between specimens, as shown by the purple bar in figure 1c, which denotes the mean ridge quantity for 10 different sponge specimens (error bars represent ±1 s.d.).

    The total number of clockwise ridges is similar to that of the total number of counterclockwise ridges, as denoted by the red and blue bars in figure 1c, which correspond to the mean clockwise and counterclockwise ridge lengths for 10 different sponge specimens (error bars represent ±1 s.d.).

    Figure 1.

    Figure 1. Skeletal features of Euplectella aspergillum. (a) Photograph of a cleaned and dried siliceous skeleton of E. aspergillum (measuring ca 25 cm in height) clearly depicting its tapered tubular form, its highly regular diagonally reinforced checkerboard-like square lattice structure, and its complex network of external ridges. (b) Unrolled ridge diagram illustrating the location of ridge elements in relation to the checkerboard-like lattice system. Filled squares denote the presence, and white squares denote the absence of ridge elements. Red squares denote clockwise ridge elements, blue squares denote counterclockwise ridge elements and purple squares denote corner elements (which do not count toward the number of either clockwise or counterclockwise elements). (c) Bar plot showing the mean occupancy fraction for ridge elements (n = 10 from electronic supplementary material, figure S1). Data presented are averages, and error bars represent ±1 s.d. The purple bar shows the total ridge occupancy fraction, which includes all clockwise (red bar), counterclockwise (blue bar) and corner elements. (d) Plot adapted from Weaver et al. [4], which illustrates that the combination of a decreasing volume per unit area for the underlying diagonally reinforced square lattice (orange) with an increasing ridge height (green) along the length of the tapered tube, results in a relatively consistent total material volume of silica per unit area throughout the entire skeletal system (black dotted line). For the orange and green data, the solid lines denote averages and the shaded regions denote ± 1 s.d. for n = 10 specimens.

    Finally, it is important to note that, on average, the height of the ridges linearly increases from ca 1 mm to ca 6 mm from its anchoring point on the seafloor to its apex (illustrated by the green line in figure 1d) [4]. Despite this increasing ridge height, a relatively constant material volume per unit area has been measured along the length of the tapered skeletal network (illustrated by the dotted black line in figure 1d) [4], which results from a simultaneous increase in lattice unit cell size and a relatively constant strut thickness, which lead to a corresponding decreasing strut width-to-unit cell width ratio as one ascends the sponge (illustrated by the orange line in figure 1d).

    3. Model generation

    To construct the three-dimensional models (described in detail in figures 24) used in both our mechanical and hydrodynamic studies, we subdivided the geometry into two primary components: (1) the hollow cylindrical core and (2) the external ridge system.

    Figure 2.

    Figure 2. Workflow for the generation of sponge structural models. (a) Schematics showing the underlying diagonally reinforced square unit-cell geometry, which was tiled to produce the tubular lattice shown in (b). (c) Schematics showing the different ridge elements used to construct the complete skeletal model. Each component is colour-coded in the left map, indicating its location. The dotted lines in each of the four three-dimensional models correspond to the main ridge supporting elements, the brightly coloured dots denote nodes of fusion between the struts, and the black arrows denote the direction of ridge elongation. To construct the complete ridge system for each tested sponge geometry (as shown in (d)), each of these four elements could be either translated, rotated or mirrored (with the exception of the Bidirecitonal design which requires an additional crossing element).

    3.1. Hollow cylindrical core

    3.1.1. Structural analysis

    For the structural analyses, we considered only the load-bearing glassy skeletal components of the sponge. To produce this model, we tiled the diagonally reinforced square lattice geometry described in Fernandes et al. [7] (figure 2a) to generate a tubular lattice (figure 2b), which consisted of 32 horizontal struts, and 32 vertical struts in circumference (creating a cylinder with a height to radius ratio H/R ∼ 6.2). The ratio between the strut diameter and strut separation was Dnd/L = 0.1 (where L is the square grid strut separation) to match the average dimensions found in this species [7] (figure 2a). Overlaid on this square grid is a system of paired diagonal reinforcing struts (figure 2a), the periodic spacing of which creates a checkerboard-like open and closed cell structure, where every open cell contains an octagonal aperture. This geometry leads to two independent pairs of crisscrossing diagonals struts, each with a spacing of

    S=L2+2.3.1
    In this configuration, the pairs of diagonals are thinner than the non-diagonal struts [4,7], with a relationship given by [7]
    Dd=Dnd2.3.2
    Numerical values of these parameters used in our simulations can be found in electronic supplementary material, table S1.

    3.1.2. Hydrodynamic analysis

    For the hydrodynamic analyses, we explored three different geometries for the cylindrical core, all characterized by an external radius Ro, an internal radius Ri = 0.9Ro and a height H = 6.2Ro.

    No Pores: This design consisted of a featureless smooth cylinder (see figure 3, second column), a geometry that has been widely studied in the field of hydrodynamics, and is known to produce von Kármán vortices [16,17].

    Small Pores: Inspired by the anatomy of living specimens of E. aspergillum, small holes (approximated as octagons) populate locations corresponding to the positions of every open square in the underlying diagonally reinforced skeletal lattice (figure 3, third column). In this configuration, the effective surface porosity is given by ϕs = 0.06. For details regarding the geometry and porosity calculations, see electronic supplementary material, figure S2.

    Large Pores: This geometry is very similar to that of the Small Pores tube, but with larger diameter holes (figure 3, fourth column). The porosity of this design approximates that of only the sponge’s skeleton, and the effective surface porosity of this structure is given by ϕs = 0.41. While this specific geometry is not directly relevant to the living sponge, it provides a useful experimental test bed for investigating the effect of pore fraction on hydrodynamic performance. For the details regarding geometry and porosity calculations, see electronic supplementary material, figure S2.

    Figure 3.

    Figure 3. Fully constructed models for structural and hydrodynamic analyses. Schematics showing (a) the ridge-free baseline cylinder, (b) the Unidirectional ridge design, (c) the Bidirectional ridge design and (d) one representative example of the Sponge ridge design. For each ridge geometry, we also considered four variations of the inner tube (left to right): a solid (non-porous) tube, a low-porosity tube (approximating the porosity of the living sponge), a high-porosity tube (approximating the porosity of only the skeleton), and that of only the load-bearing skeletal elements of a sponge. CFD simulations were conducted on the first three tube geometries (solid, low-porosity and high-porosity geometries) and FE simulations were conducted on the last geometry (load-bearing skeletal elements).

    To create the small and large pore models, respectively, octagonal openings measuring with side S0 = 0.16L (figure 4f,g) and S0 = 0.4L (figure 4h,i) were cut through the cylinder-ridge construct.

    Figure 4.

    Figure 4. Hydrodynamic model generation workflow. (a) Schematic showing one of the ten sponge ridge diagrams that provided a road map for construction of a complete ridge network. (b) The apex of the triangular ridges (blue) cross diagonally through the black squares in (a) and their flat sides contact the underlying cylinder (grey) at the geometric centres (denoted by red dots) of the octagonal openings shown in figure 2a. (c) Schematic showing a cross-sectional view of the non-porous cylindrical core (grey) with the labelled radius encompassing the ridge height Rr. (d) Schematic showing the cylinder thickness parameters, with Ro and Ri denoting the outer and inner radii, respectively. For (c,d), the vertical black dotted lines denote the model’s neutral axis. (e) Schematic showing the complete non-porous baseline ridge geometry (with the model height, H, specified), which was generated from the sponge ridge diagram shown in (a). High (f) and low (g) magnification views, showing the geometries of the octagonal holes (red) for the Small Pores model containing a side length of So ≈ 0.16L, which approximates the porosity of a living sponge. High (h) and low (i) magnification views, showing the geometries of the octagonal holes (red) for the Large Pores model containing a side length of So ≈ 0.4L, which approximates the porosity of only the sponge’s glassy skeletal system.

    3.2. External ridge system

    For our structural and hydrodynamic analyses, we considered three different diagonal ridge arrangements (figure 3).

    Unidirectional Ridges: This geometry consists of eight uninterrupted parallel unidirectional helical ridges, each of which measures 32 cells in length, and occupy every other filled square cell of the sponge’s skeletal lattice (figure 3b).

    Bidirectional Ridges: As with the unidirectional ridge model, this design also incorporates eight helical ridges, each of which measures 32 cells in length. However, for this design, four of the helices are right-handed and four are left-handed (figure 3c).

    Sponge Ridges: Inspired by the external ridge structure of E. aspergillum, this geometry is composed of a maze-like combination of right- and left-handed ridges that occupy every other filled square cell of the sponge’s skeletal lattice. One example (from the list of 10 mapped sponges) is shown in figure 3d. The other nine mapped designs can be found in electronic supplementary material, figure S1.

    It is important to note that for the three different ridge configurations described above, the total ridge lengths are nearly identical. Furthermore, in all our models, we chose the height of the ridges to be hr = 1.45L, which is equivalent to the mean ridge height (RrRo in figure 4c,d) across the length of the sponge [4].

    3.2.1. Structural analyses

    To generate the ridges for our structural models, we first identified the four distinctive design elements shown in figure 2c, which formed the basis of a complete ridge system. These elements are a result of ridge continuations (green schematic), 90° turns (purple schematic), T-junctions (orange schematic) and ridge terminations (blue schematic) in the sponge ridge system. Utilizing these elements and their possible rotations, we generated all of the different ridge-containing models. For all of the struts composing the ridge structure, the diameter matched that of the non-diagonals, namely, Drg = Dnd, where Drg is the ridge strut diameter and Dnd is the non-diagonal strut diameter.

    3.2.2. Hydrodynamic analysis

    To generate the ridges for our hydrodynamic models (which were modelled after those found in living examples of E. aspergillum), we began with a planar ridge diagram (figure 4a), which was used as a road map to construct a series of triangular surface ridges (figure 4b) on an extruded 32-sided polygon (which approximated the cylindrical geometry of the sponge). The flat-sided ridges were designed such that their apex ran diagonally through the squares in the planar ridge map, and contacted the cylinder at a location that corresponded to the geometric centres of each of the octagonal openings shown in figure 4a (and denoted by the red dots in figure 4b). It is important to note that while the underlying skeletal geometry of the ridges is incredibly complex (as shown in figure 2), in the living sponge, the organic material conceals much of this underlying structural complexity, and thus only approximates the rough contours of the underlying skeletal system [1820], thus permitting the construction of the simpler (and still biologically relevant) ridge geometry shown in figure 4.

    4. Structural analyses

    4.1. Methods

    In an effort to understand how the ridges affected the sponge’s structural performance, we conducted FE simulations using ABAQUS/Standard (Dassault Systémes SE). For these analyses, the geometries were constructed using Timoshenko beam elements (ABAQUS element-type B22) with circular cross-sections and the material’s response was captured using a linear elastic material model with Young’s modulus Emat. In our simulations, we considered the ten different sponge ridge designs shown in electronic supplementary material, figure S1, and compared their response to the Unidirectional and Bidirectional models. We also considered a model without ridges, where the ridge material was instead allocated to the cylindrical beams of the underlying diagonally reinforced square lattice (we refer to this model as the Reallocated Ridge Mass model).

    For each design, we assumed that the cylindrical sponge structure was fully constrained at its base (i.e. ux = uy = uz = 0), and performed two different analyses: (1) a linear static analysis to extract the stiffness and (2) a buckling analysis (*BUCKLE step in ABAQUS) to obtain the critical buckling force. In the simulations, we considered four loading cases (illustrated in figure 5a).

    Compression: For this case, we applied a vertical displacement parallel to the z-axis (δappl = δz) to all the nodes of the top ring.

    Torsion: For this case, we applied a tangential displacement (δappl=δθ=θapplR) to all the nodes of the top ring.

    Bending: For this case, we applied a displacement in the xy plane (δappl=δx2+δy2) to all the nodes of the top ring. Note that we systematically varied δx and δy to survey the bending behaviour in different directions. Specifically, we considered eight equally spaced loading directions for each structure.

    Pressure: For this case, we applied a radial displacement (δappl = δR) to all nodes belonging to the cylindrical core of the models.

    Figure 5.

    Figure 5. Structural analyses. (a) Schematics illustrating boundary conditions considered in the FE simulations. For all considered loading conditions, the displacement at the bottom was held fixed, namely ux = uy = uz = 0, and a displacement δappl (indicated with arrows) was applied. (b,c) Bar plots showing normalized stiffness RF/(δapplLEmat) and normalized critical buckling force Fcr/(L2Emat) with different colours corresponding to each loading condition. The models considered were the 10 mapped Sponge geometries (see electronic supplementary material, figure S1), the Unidirectional and Bidirectional geometries, the Ridge-Free design, and the Reallocated Ridge Mass design, from left to right, respectively. For the Reallocated Ridge Mass design, we employed the Ridge-Free design, but reallocated the volume from the ridges into the cylinder elements, thus making each element thicker and more robust.

    To implement these boundary conditions for Compression, Torsion and Bending, we constrained the degrees of freedom of the nodes of the top ring to a virtual node using equation constraints. Similarly, for the loading case Pressure, we tied the degrees of freedom of all nodes belonging to the cylindrical core to a virtual node using equation constraints. For all loading cases, we then applied δappl to the virtual nodes and extracted the resulting reaction force RF.

    4.2. Results

    In figure 5b,c, we report the numerically predicted stiffness and critical buckling load for all considered models and loading cases. We find that the addition of ridges increases the overall stiffness of the cylindrical structure for loading conditions Bending and Pressure (figure 5b) and improves the buckling performance for all loading conditions on average by ca twofold over the ridge-free example (figure 5c). However, if we compare the performance between the different ridge-containing geometries, we see that the stiffness values differ negligibly between the different ridge designs. By contrast, if the ridge material volume was instead allocated away from the ridge system and into the main cylindrical truss frame, the resulting structure, on average, measurably outperforms the other designs for most loading conditions in terms of both stiffness and buckling force. This result is not unexpected, since in this scenario, more material is allocated to the truss members aligned parallel to the loading direction.

    These results demonstrate that the unusual maze-like ridge pattern found in E. aspergillum does not likely add any significant additional mechanical benefit (with regards to either stiffness or critical buckling force) compared to the other investigated ridge geometries, which prompted additional studies into its potential hydrodynamic functions (see §5). Further, the observation that reallocating the ridge material into the underlying diagonally reinforced skeletal lattice resulted in a significant enhancement in the structure’s mechanical performance may be functionally relevant to the sponge’s skeletal anatomy. As shown in figure 1d, the sponge’s material volume per unit area remains relatively constant along the length of the tapered skeletal tube, which is achieved due to an increase in ridge height (and volume), from the base of the sponge to its apex, and a corresponding loss in volumetric skeletal density of the underlying diagonally reinforced square lattice (which results from an increase in vertical and horizontal strut spacing). As such, the mechanically most robust portion of the composite skeleton coincides with its thickened and ridgeless connection point to the flexible holdfast apparatus, which secures the sponges into the soft sediments of the sea floor. Based on our results, this transition in the skeletal anatomy from the rigid skeleton to its flexible holdfast apparatus, is likely a point of highest mechanical stresses and a potential location of skeletal failure, an observation consistent with our simulated buckling locations shown in electronic supplementary material, figure S3.

    5. Hydrodynamic analyses

    To explore the potential hydrodynamic benefits of the unique maze-like collection of bidirectional helical ridges located on the skeletal tube of E. aspergillum, we developed a computational fluid dynamics (CFD) framework, and compared their performance to alternative equal-length ridge geometries that are commonly employed in modern engineering applications for vortex suppression and drag reduction [1113].

    Since the natural habitat of the genus Euplectella can vary widely (occurring in low and temperate latitudes at depths ranging from 36 to 5050 m [18]), these sponges are likely to experience a wide range of flow patterns and velocities that must be considered. As such, in our simulations, we considered non-dimensional Reynolds numbers Re ranging from 5500 to 60 000, where Re was defined as

    Re=ulν,5.1
    with u being the flow speed, l the characteristic length scale (2Ro), and ν the fluid kinematic viscosity. Assuming a representative sponge with diameter 2Ro of ca 35 mm [4], living in water with temperatures ranging from ca 2 to 10°C, this Reynolds number range encompasses a flow velocity u ranging between a biologically relevant ca 200 mm s−1 and an extremely fast ca 2500 mm s−1. It is important to note that the length scale l for all computations does not include the dimensions of the ridges (relating instead to the cylinder outer diameter only), and the resulting Re range encompasses a large flow regime, where vortex shedding behind cylindrical structures is known to occur.

    5.1. Methods

    To model fluid flow around the different cylindrical geometries, we used the CFD package ANSYS CFX, Release 18, which employs a hybrid finite-volume/finite-element approach to discretizing the Navier–Stokes equations governing fluid flow. The systems of equations were solved using an unsteady fully-implicit, fully-coupled multi-grid second-order backward Euler solver in the laboratory frame of reference. The shear stress transport turbulence model [21], which combines the kω model near the wall and the kε model away from the wall, was used throughout this study. This choice of turbulence model ensured accurate prediction of onset and amount of flow separation under adverse pressure gradient conditions. This approach also allowed the model to handle transitions of the flow from laminar to turbulent, accurately refining the flow profile around the complex geometry of the ridges. It is important to note that in our CFD simulations, we did not account for interactions between the flow and the mechanical deformation of the sponge (i.e. we did not perform fluid–structure interaction simulations).

    In the hydrodynamic simulations, we focused on comparing one representative sponge ridge design (from the ten mapped), specifically Sponge 1 shown in figure 3c, to the Unidirectional and Bidirectional configurations for cylindrical cores with both Small and Large Pores. All models with a porous cylindrical core were placed at the bottom of a rectangular prism-shaped fluid domain, with size 40Ro × 80Ro × 20H, to which symmetry condition on the top face and a slip condition on the bottom face were applied (see electronic supplementary material, figure S4, for details). The resulting configuration thus consisted of a porous hollow cylinder with a closed bottom and an open top, functionally analogous to the geometry of the native sponge. The use of a finite length cylinder in a large domain for the porous hydrodynamic simulations was chosen in order to simulate the flow profile through the sponge interior, thus permitting the quantification of any changes in induced flow. To facilitate direct comparisons to previously published results [15,22], the models with a non-porous cylindrical core were placed in a fluid domain with size 40Ro × 80Ro × H, to which we applied periodic boundary conditions at the top–bottom faces (see electronic supplementary material, figure S5, for details). For all analyses, we then imposed a uniform flow velocity perpendicular to one of the side walls, zero pressure to the opposite face and slip conditions on the remaining two side walls. Finally, we imposed no-slip conditions on the boundaries of the cylindrical structure. For each model, we conducted simulations to calculate the following.

    Lift coefficient: For each timestep, we obtained the coefficient of lift as

    CL=2FLρAu2,5.2
    where FL was obtained by integrating the perpendicular-to-flow component of the pressure acting on all external faces of the cylindrical structure, ρ was the fluid density, A was the surface area of all external faces of the cylindrical structure, and u was the flow velocity.

    Shedding frequency: Once CL was computed as a time series, we performed a power spectral density (PSD) analysis on the CL data to obtain a power spectrum of frequencies for the lift force. To obtain the PSD for the time series, we computed the Fourier decomposition of the signal and analysed the relative magnitudes of each of the coefficients. In this analysis, the shedding frequency is identified as the highest power frequency, if shedding exists. Note that in order to ensure that the vortex shedding results were reproducible for perturbations within the sponge ridge design space (in addition to Sponge 1), we also performed a the PSD analysis on two additional sponge ridge configurations (Sponges 2 and 3 in electronic supplementary material, figure S3).

    Drag coefficient: To measure the drag on the structure, we computed the coefficient of drag for each time-step, which was defined as

    CD=2FDρAu2,5.3
    where FD was obtained by integrating the parallel-to-flow component of the pressure acting on all external faces of the cylindrical structure. Note that in order to validate our simulations, we compared our results for a cylinder without pores and without ridges to the drag coefficient data obtained from Blevins [15] (see electronic supplementary material, figure S6(b), for more information).

    Flow profile through the openings in the cylindrical core: To investigate the potential role of the ridges as a mechanism for increasing/redirecting fluid flow through the sponge for feeding/gas exchange purposes, we created a closed cylindrical boundary matching the geometry of the sponge’s interior and computed the time-averaged flow profile through the openings in the sponge’s body wall by averaging the flow over three cycles of shedding. Using this flow profile, we integrated the absolute value of the flow rate over the area A of the cylindrical boundary and top of the sponge defined by

    V=A12|V˙A|dA.5.4
    Note, that because of fluid incompressibility and conservation of mass, the volume flow in the boundary Vin must equal the volume out Vout leading to a division by 2 to obtain total volume exchanged.

    To balance model accuracy and computational efficiency, in all our simulations we refined the mesh near the region of interest (in the sponge’s vicinity) and kept the far-field meshing coarse. For each of the models, we produced a mesh consisting of more than 2 million elements, an illustration of which can be found in electronic supplementary material, figure S7.

    5.2. Results

    In figures 6, 8 and 10, we report the lift coefficient, shedding frequency and drag coefficient (together with numerical snapshots that show the vorticity fields) at Re = 5500 for designs comprising a hollow cylindrical core with No Pores, Small Pores (ϕs ≈ 0.06) and Large Pores (ϕs ≈ 0.41), respectively. For all three cases, we considered three different ridge geometries (Unidirectional, Bidirectional and Sponge configurations) and compared their performance to a ridge-free control.

    Figure 6.

    Figure 6. Hydrodynamic results for the different non-porous model geometries, at Re = 5500. (a) Plot showing transient lift coefficient Cl for the different geometries. (b) Magnified view of transient lift coefficients, showing the initial shedding behaviour for the different model geometries. (c) Plot showing power spectral density (PSD) as a function of Strouhal number for the different model geometries. The vertical dashed black line shows the empirical Strouhal number for a smooth cylinder using a Re = 5500 flow regime. Note that the last 20 s of the transient data (when the flow reaches a periodic state) were used to compute the PSD. (d) Plot showing transient drag coefficient Cd for the different model geometries. The horizontal black dashed line corresponds to empirical drag coefficient data for a smooth cylinder. (e) Vorticity fields illustrating shedding for the different model geometries.

    As shown in figure 6a,b we find that for the geometries with non-porous cylindrical cores and either Bidirectional ridges or no ridges, there exists large cyclic lift forcing acting on the structures. By contrast, we see that for the Sponge and Unidirectional ridge designs, the cyclic forcing is quickly suppressed. The PSD analysis performed on these lift data reveals that the Sponge and Unidirectional ridge designs are effective at suppressing vortex shedding forcing on the cylindrical structure, as evident by the lack of a localized peak in figure 6c. By contrast, we observe vortex shedding behind the cylinder for both the ridge-free and Bidirectional ridge configurations, with the vortices actually amplified by the bidirectional ridge design (as indicated by a higher PSD peak in figure 6c). It is also important to note that while the Unidirectional ridge design shifts the shedding downstream, the Sponge ridge design completely suppresses the vortex shedding behind its cylindrical structure (see snapshots in figure 6e), demonstrating a more efficient vortex suppression mechanism. While the Sponge ridge design outperforms the Unidirectional ridge design in vortex suppression, it does, however, lead to a slightly increased drag force on the structure (figure 6d).

    To validate our results, we first compared both the location of the PSD peak and the drag coefficient obtained for a non-porous cylindrical core without ridges to existing empirical data for a cylinder in a uniform two-dimensional flow at Re = 5500 (black dashed lines in figure 6c,d—for more information, see electronic supplementary material, figure S6) [15,22]. The good agreement between these two datasets confirmed the suitability of our numerical approach. To further validate the effects of vortex shedding observed for our selected sponge ridge geometry with a non-porous cylindrical core, in figure 7, we analysed the PSD for two additional Sponge ridge designs (Sponge 2 and Sponge 3 from electronic supplementary material, figure S1), in addition to the original sponge design (Sponge 1), the Unidirectional, the Bidirectional, and the ridge-free geometries. In this plot, we observe that all three of the sponge-inspired designs efficiently suppress vortex shedding compared to the other more symmetrical ridge geometries. These results thus demonstrate that the observations presented in figure 6 are likely not specific to the unique sponge ridge geometry investigated.

    Figure 7.

    Figure 7. PSD analysis of additional Sponge ridge geometries. PSD plot comparing the different considered structures (from figure 6) to ridge designs obtained from two additional Sponge samples (see electronic supplementary material, figure S1). It should be noted that all three of the Sponge data lines are indistinguishable and superimposed. Results presented for Re = 5500.

    In figure 8, we report results for designs comprising a hollow cylinder with small pores. For this case, we find that large cyclic lift forcing acting on the structure exists only for the model without ridges (figure 8a). We also note that for the Bidirectional and Sponge ridge geometries, the lift coefficient does not average to zero, since the ridges break the flow symmetry and bias the lift coefficient to a particular direction (see electronic supplementary material, figure S8, for more information). By taking the data from the lift coefficient and computing the PSD, we observe that for this level of biologically relevant porosity, prominent vortex shedding still exists behind the ridge-free geometry (figure 8c), while all three models with ridges are capable of suppressing vortex shedding. As such, these results demonstrate that for ϕs ≈ 0.06, the introduction of helical ridges is sufficient to suppress vortex shedding regardless of the specific ridge arrangement. However, although all three ridge geometries suppress the vortices over the long term, it is important to note that some geometries provide a faster dissipation mechanism than others. As shown in figure 8b, the Unidirectional and Bidirectional ridge designs exhibit cyclical lift forcing as soon as flow is initiated, indicating an early onset of vortex shedding, and whose amplitude is partially dissipated over time. The Sponge ridge design, in contrast, transitions much earlier to a steady flow profile with no indication of cyclical lift forcing. Such rapid suppression of vortex shedding is of biological relevance to the sponge, since its local environment can be prone to fluctuations in flow velocity depending on the depth at which it occurs. Finally, the results shown in figure 8d demonstrate that the choice of ridge design has also implications on the drag imposed on the cylindrical structure, with the Bidirectional design increasing the drag substantially compared to the other ridges geometries.

    Figure 8.

    Figure 8. Hydrodynamic results for the different model geometries exhibiting biologically relevant (small) porosities, approximating the anatomy of living sponges, at Re = 5500. (a) Plot showing transient lift coefficient Cl for the different model geometries. (b) Magnified view of transient lift coefficients, showing the initial shedding behaviour for the different model geometries. (c) Plot showing power spectral density (PSD) as a function of Strouhal number for the different model geometries. Note that the last 20 s of the transient data (when the flow reaches a periodic state) were used to compute the PSD. (d) Plot showing transient drag coefficient Cd for the different model geometries. (e) Vorticity fields illustrating shedding for the different model geometries.

    While the hydrodynamic results reported thus far are for Re = 5500 (a value relevant to the natural habitat of the sponge), in figure 9, we also investigated the effect of other Reynolds numbers on these observed behaviours. Specifically, we considered Re = 15 000 and Re = 60 000 and, due to the computational complexity of these simulations, compared the hydrodynamic performance of a cylindrical tube with Small Pores and Sponge ridge design to that of a cylindrical tube with Small Pores and No Ridges. We find that for both considered Reynolds numbers, the trends observed in figure 8 are unchanged for lift coefficients, shedding frequencies, and drag coefficients.

    Figure 9.

    Figure 9. Hydrodynamics results for the low-porosity ridge-free and Sponge geometries, at Re = 15 000 and Re = 60 000. (a) Plot showing transient lift coefficient Cl as a function of time at Re = 15 000. (b) PSD plot, with frequency converted to Strouhal number. Note that the last 10 s of the transient data were used to compute the PSD at Re = 15 000. (c) Plot showing transient drag coefficient Cd as a function of time at Re = 15 000. (d) Plot showing transient lift coefficient Cl as a function of time at Re = 60 000. (e) PSD plot, with frequency converted to Strouhal number at Re = 60 000. Note that the last 5 s of the transient data were used to compute the PSD. (f) Plot showing transient drag coefficient Cd as a function of time at Re = 60 000.

    Finally, in figure 10, we report the hydrodynamic results for models with a large surface porosity (ϕs ≈ 0.42), approaching that of a cleaned (bare) sponge skeleton. The results show two key trends. First, we find that at this level of porosity, the existence of von Kármán vortices is not prominent for any of the models, as indicated by the lack of cyclic lift forcing acting on the structures shown in figure 10a,b. These results therefore suggest that the introduction of ridges is unnecessary for this level of porosity and that there exists a transition in the flow for surface porosities somewhere between ϕs ≈ 0.06 and ϕs ≈ 0.42. Second, the results shown in figure 10c also demonstrate that for large surface porosities, the drag coefficient remains dependent on the ridge arrangements, and that the dependency is similar to that observed in figure 8d for ϕs ≈ 0.06.

    Figure 10.

    Figure 10. Hydrodynamic results for the different model geometries exhibiting (large) porosities, approximating only the skeletal anatomy of tissue-free sponges, at Re = 5500. (a) Plot showing transient lift coefficient Cl for the different model geometries. (b) Magnified view of transient lift coefficients, showing the initial shedding behaviour for the different model geometries. (c) Plot showing power spectral density (PSD) as a function of Strouhal number for the different model geometries. Note that the last 20 s of the transient data (when the flow reaches a periodic state) were used to compute the PSD. (d) Plot showing transient drag coefficient Cd for the different model geometries. (e) Vorticity fields illustrating shedding for the different model geometries.

    Beyond the vortex suppression and drag reduction properties of the sponge’s external ridges, we also investigated their potential role as a mechanism for increasing/redirecting fluid flow through the sponge for feeding/gas exchange purposes. Towards this end, in figure 11a,b, we report the time averaged flow rate over the area of the cylindrical core (defined as in (5.4)) at Re = 5500 for all considered different geometries with a cylindrical core with both Small and Large Pores, respectively. These results demonstrate that the flow through the walls of the cylinder (and out the terminal opening) for the Sponge design is almost identical to that of the model without ridges, whereas the Unidirectional or Bidirectional ridges lead to a slight increase in flow. Similar trends were also observed for Re = 15 000 (figure 11c) and Re = 60 000 figure 11d in the case of Small Pores geometry, suggesting that the presence of the Sponge ridge design does not offer significant added mass flow through the walls of the main cylinder across a wide range of flow regimes. It should be noted that the path of water flow through the body wall of living specimens of Euplectella aspergillum follows a more convoluted trajectory than the simple linear channel modelled in the present study, thus potentially reducing the overall flow rate of water through the sponge in comparison to our predicted values. While this may be the case, these induced flow effects, which have been well documented in previous studies [23,24] are largely driven by flow-induced pressure differentials across the sponge’s tubular form, and as such, the general trends reported here are likely to be unaffected by slight variations in through-wall flow rates.

    Figure 11.

    Figure 11. Quantification of flow rate through the different investigated model geometries. Bar plots showing time-averaged flow rate through the walls of the (a) low-porosity and (b) high-porosity models, at Re = 5500, and the low-porosity models at (c) Re = 15 000 and (d) Re = 60 000.

    6. Conclusion

    To summarize, using a combination of FE simulations and computational fluid mechanics, in this study we explored the potential multi-functionality of the complex network of external maze-like ridges found in the skeletal system of the marine sponge, Euplectella aspergillum. Using this integrated approach, we demonstrate that this complex ridge system offers the sponge an efficient mechanism for rapidly suppressing von Kármán vortex shedding and reducing lift forcing oscillations (which may be essential for keeping the sponge anchored into the soft sediments of the sea floor), while at the same time, improving its skeleton’s mechanical performance. Beyond their ability to suppress vortex-induced vibrations, which could weaken the attachment point between the sponge’s holdfast apparatus and its surrounding substrate, the ridges also likely play a critical role in preventing substrate-associated turbulence and subsequent sediment excavation (see electronic supplementary material, video 1), as has been demonstrated for other bottom-anchored cylindrical geometries under flow [25,26]. The prevention of sediment transport from the down-stream side of the sponge is likely of critical importance to the sponge’s survival, since excavation near its holdfast apparatus would also compromise its ability to remain anchored into the soft sediments of the sea floor.

    To explore the additional benefits of its maze-like organization, we also compared the sponge ridge geometry to helical strake-based vortex shedding suppression systems commonly employed for above ground (e.g. antennas, and smokestacks) and underwater (support beams, vertical pipelines) high aspect ratio cylindrical structures. From these studies, we find that the sponge’s ridge design can completely suppress vortex shedding, a behaviour which differs from that of more symmetrical helical strake configurations, which either delay shedding downstream (Unidirectional ridge design) or amplify the shedding magnitude (Bidirectional ridge design). Further, we demonstrate that the Sponge ridge design continues to effectively suppress vortex shedding across a wide range of flow regimes.

    While the findings of this study provide new insight into the multi-functionality (mechanical and hydrodynamic) of the complex external ridges in the skeleton of E. aspergillum, it is important to note that the flow regimes considered in this study (Re between 5500 and 60 000) are also relevant for many man-made cylindrical structures including smokestacks, antennas, submerged piping and offshore floating platforms. As such, the geometric features identified by this study could be directly used in engineering applications where effective dissipation of vortex shedding is necessary, and where vortex-induced vibrations are to be avoided.

    Data accessibility

    Data and supporting materials are available at https://git.fer.me/sponge-ridges.

    Authors' contributions

    M.F., K.B. and J.W. conceived the study and developed the methods. M.F., P.C., C.I., T.S. and J.W. created all CAD models for this study. G.M. and J.W. calculated the volume fractions of the different skeletal features. M.F. and P.C. carried out the finite-element modelling. M.F. and M.S. carried out the hydrodynamic modelling. M.F., M.S., H.H., K.B. and J.W. contributed to the analysis of results. M.F., M.S., K.B. and J.W. contributed to the writing of the manuscript. M.F., H.H., K.B. and J.W. secured funding for this project. All authors provided valuable feedback and approved the final manuscript.

    Competing interests

    We declare we have no competing interests.

    Funding

    This work was supported by NSF-GRFP Fellowship (grant no. DGE-1144152) (M.C.F.), a GEM Consortium Fellowship (M.C.F.) and the Harvard Graduate Prize Fellowship (M.C.F.), and was partially supported by the NSF through the Harvard University Materials Research Science and Engineering Center (grant no. DMR-2011754) and NSF DMREF (grant no. DMR-1922321).

    Acknowledgements

    We thank George Lauder, Mark Scott, Jochen Mueller, Joanna Aizenberg, Chris Rycroft and Joost Vlassak for insightful discussions.

    Footnotes

    Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.5581086.

    Published by the Royal Society. All rights reserved.

    References