Recent marine anti-fouling research efforts have sought inspiration from bio-mimetic strategies to develop nontoxic solutions. Surface modifications have shown promising results in their ability to disrupt attachment and growth of early-stage organisms under static immersion conditions but very limited research has attempted to explore the interaction between surface textures and flow under turbulent conditions. The study presented in this article focuses on a simple texture, inspired by the growth rings of the Brill fish Scophthalmus rhombus and developed for applications on the blades of tidal stream turbine. A series of Large Eddy Simulations of fully developed turbulent channel flow were performed to assess the influence of the spacing between the texture elements on turbulent stresses. The aim is to characterize the impact of the texture on turbulent stresses by comparison to a smooth surface and identify whether shelters may form within the gaps between textures. The study clarifies the role of dispersive and Reynolds stresses in terms of their impact on hydrodynamic forces acting on a simplified model of a marine diatom. Reynolds shear stresses predominantly govern the intensity of forces, while dispersive shear stress affects the mean hydrodynamic forces acting on the organism at the early stage of settlement.

Bio-fouling formation and growth can significantly alter the surfaces of most structures, systems, or equipment designed to operate in the marine environment. The process causes the accumulation of organic and inorganic materials, which can add substantial weight to a structure and modify its surface properties. For moving components of ocean energy systems, bio-fouling can impact the dynamic response, energy conversion, and reliability1–7 (see Fig. 1). Many effective anti-fouling solutions have been developed over the years, but in the past, these have mainly relied on paints or coatings designed to leach biocides. Certain biocides have proven highly toxic and harmful to the marine environment and are progressively restricted for usage.8 Environmental concerns and regulatory changes have prompted research into alternative, nontoxic anti-fouling solutions. Among these, bio-inspired micro-scale surface modifications have gained popularity as an anti-fouling solution.9 Surface textures typically rely on roughness ridges arranged in arrays with spacings smaller than the dimensions of fouling species to prevent organisms from forming secure attachments to the substrate.10,11 Although this has proven effective in limiting bio-fouling settlement under static conditions, below a certain spacing, narrow gaps can also be expected to prevent hydrodynamic stresses from reaching initial settlement sites on surfaces.12–14 Without the action of flow-induced stresses, organisms or even passive suspended particles can more easily settle and/or attach. Most existing studies to date have focused on laminar flow or static conditions15–17 and are not directly applicable to applications where surfaces are exposed to flow at high Reynolds numbers, as found on tidal turbine blades or ship hulls. While surface stresses and turbulent fluctuations are known to disrupt early-stage bio-fouling micro-organisms, the nature of turbulent flow interaction with micro-scale textures and its impact on local stresses is still poorly understood, making it difficult to determine which surface modification may be optimal. The narrowest gaps have been shown to create undesirable shelters against hydrodynamic stresses,12,13 and, conversely, increasing the size of the gaps can enhance turbulent fluctuations as a result of spatial variation between flows.18–20 These fluctuations can significantly alter the forces exerted on surfaces and organisms, disrupting early-stage fouling settlements. The aim of the research presented in this article is to explore the interplay between spatial and temporal fluctuations to characterize the effect of surface textures.

FIG. 1.

Photo images of the Magallanes turbine blades25 following immersion and operation at a site in the Orkney Islands, Scotland. A thin bio-fouling cover appears on the surface of the blade throughout its length. There is limited evidence of hard shell organisms growing to mature size except in the lee of the blade near the root.

FIG. 1.

Photo images of the Magallanes turbine blades25 following immersion and operation at a site in the Orkney Islands, Scotland. A thin bio-fouling cover appears on the surface of the blade throughout its length. There is limited evidence of hard shell organisms growing to mature size except in the lee of the blade near the root.

Close modal
FIG. 2.

Image of micro-ridges from the growth rings of the Brill fish species, Scophthalmus rhombus.26 

FIG. 2.

Image of micro-ridges from the growth rings of the Brill fish species, Scophthalmus rhombus.26 

Close modal

The study considers a simple bio-mimetic texture based on arrays of rectangular prisms (see Fig. 4) inspired by the growth rings of the Brill fish Scophthalmus rhombus shown in Fig. 2. The surface features are similar in scale to the diatoms and other marine micro-organism responsible for early-stage biofouling. They have been found from laboratory studies under static conditions to disrupt the attachment of larger organisms by restricting access to secure attachment points both over the texture and within the gaps.21 The narrower the spacing between textures, the less likely are organisms to penetrate within the gaps. Five different texture layouts have been considered here to study the effect of this spacing on local turbulent stresses generated within and above the textures. The smallest gap considered ( S + = 10) approximates the topography of the Brill fish growth rings and have been shown to limit the formation of colonies over the crest of texture elements but is not sufficiently small to entirely block settlement of the smallest organisms within the gaps. The largest spacing ( S + = 80) is closer to the characteristic length of larger early stage micro-foulers. In this case, settlement within the gaps is expected to be unaffected for most organisms except for the larger species, which can range in size from l + = 50 to 200. For these textures, the plan area density ranges from λ p = 0.88 ( S + = 10) up to λ p = 0.457 ( S + = 80). Due to the small dimensions of the gaps, the flow regime predominantly falls into the skimming regime. In the context of 2D roughness definitions, it aligns with d-type roughness ( w / H 1) and semi k-type roughness ( 1 w / H 3), where w represents the stream-wise pitch size of texture elements, and H signifies the element heights.22 A high-resolution Large Eddy Simulation (LES) approach is adopted to simulate turbulent flow and predict stress distributions. Afterwards, a simplified model of a micro-organism modeled based on the Nitzschia Ovalis species23,24 is used to evaluate the impact of the texture on the hydrodynamic forces acting on the early stages of micro-foulers. The results are then compared to the scenario where the organism is positioned on a smooth wall case. All simulations are carried out within the OpenFOAM V6 framework.

Surface roughness, whether it is at micro-scales or in canopy flow over terrain covered by vegetation or flow over the urban landscape, modifies turbulence locally but also in terms of its effect on spatial distributions. Turbulent in-homogeneity in space has motivated researchers to invoke a modified spatially averaged form of the Navier–Stokes (NS) equations.27–29 The governing equations are derived, in this case, using an alternative decomposition of the flow variables. The velocity components for example are decomposed into three distinct terms: u i = u i ¯ + u i ̃ + u i , where u i ¯ represents the spatial and temporal average of the velocity component ui, while u i ̃ and u i represent the spatial and temporal velocity fluctuations, respectively. Spatial averaging is carried out by integrating over a thin volume extruded from a surface parallel to the surface while ensuring sufficient extent in the stream-wise and span-wise directions to eliminate the effects of topological variations caused by obstacles. The Navier–Stokes equations are then spatially and temporally averaged, and the shear stresses are expressed using these three different components. The spatially averaged NS equations are defined by
( u i ¯ ) t + u j ¯ ( u i ¯ ) x j = p x i + τ i j x j + D i ,
(1)
τ i , j ρ = u i u j ¯ u i ̃ u j ̃ + ν u i ¯ x j ,
(2)
D i ρ = 1 V P ¯ n i d S ν V u i ¯ n d S ,
(3)
where u i u j ¯ , u i ̃ u j ̃ , and ν u i ¯ / x j are the spatially averaged Reynolds, dispersive, and viscous stresses, respectively. The dispersive stresses capture the transfer of momentum due to spatial variations in a way similar to that of the Reynolds stresses, which accounts for the transfer of momentum by turbulent velocity fluctuations. The viscous term is known to become negligible at a location just a few y+ from the roughness.18  Di in Eq. (1) accounts for the drag experienced by the roughness, which includes form drag due to the time-averaged pressure P ¯ and friction drag due to wall normal time average velocity gradients u i ¯ / n. The term u i u j ¯ represents the average Reynolds stress across the prescribed horizontal plane and accounts for the transfer of momentum resulting from temporal fluctuations.

The Large Eddy Simulation (LES) approach is used to resolve turbulent flow within and above textures. Various sub-grid scale (SGS) models have been tested to account for the effect of the smallest scales on wall-bounded flows (Samgorinsky, WALE, and k-Equation). Unlike the Smagorinsky model, the k-Equation model operates on a local equilibrium assumption. It solves a transport equation for turbulent kinetic energy, taking into account the historical effects of production, dissipation, and diffusion within the flow. This model has demonstrated its ability to generate better results for both the initial and subsequent statistical moments compared to the Smagorinsky model. Additionally, it has shown good agreement with data from Direct Numerical Simulation (DNS)30,31 and is used in this study.

The study reported in this article considers fully developed turbulent channel flow as a base flow. Six alternative geometries have been considered, including a rectangular smooth wall channel and five one-sided textured wall channels. One additional case incorporates a single bio-fouling organism and positioned within and modeled on the textured surface, which generates the highest turbulent activity. Each simulation geometry case is discussed as follows.

1. Smooth wall channel flow

A fully developed turbulent flow in rectangular channels with smooth walls has been previously studied at a Reynolds number R e τ = 395 using both DNS and LES.32,33 The present study considers the same turbulent flow at the same Reynolds number. The dimensions of the channel in the stream-wise and span-wise direction were chosen to ensure that turbulence could be self-sustaining preventing transition into laminar flow Fig. 3.32–34 The stream-wise, wall-normal, and span-wise dimensions of the base channel are 6 × 2 × 3 m. When scaled in wall units using the viscous length scale from the smooth wall simulations, these correspond to l x + = 3000 and l z + = 1500 for the stream-wise and span-wise lengths, respectively. Periodic conditions are used for the stream-wise (inlet and outlet) and span-wise (right and left) boundaries of the computational domain. The solid upper and lower walls positioned perpendicular to the y direction are modeled with a no-slip condition. The flow and geometrical characteristics for the simulations are summarized in Table I. The driving force for the flow is a uniform negative pressure gradient Π imposed in the stream-wise direction (x) as a source term in Eq. (1).33 

FIG. 3.

The smooth wall channel flow geometry.

FIG. 3.

The smooth wall channel flow geometry.

Close modal
TABLE I.

Geometrical parameters and physical properties of a fully turbulent channel flow at R e τ = 395.

Parameters Symbols Value
Stream-wise channel length  lx  6 ( m ) 
Span-wise channel length  lz  3 ( m ) 
Half-channel height  1 ( m ) 
Bulk velocity  Ub  0.1335 ( m / s ) 
Kinematic viscosity  ν  2 e 5 ( m 2 / s
Reynolds bulk velocity  U b H / ν  13 350 
Target Reynolds friction velocity  U τ H / ν  395 
Parameters Symbols Value
Stream-wise channel length  lx  6 ( m ) 
Span-wise channel length  lz  3 ( m ) 
Half-channel height  1 ( m ) 
Bulk velocity  Ub  0.1335 ( m / s ) 
Kinematic viscosity  ν  2 e 5 ( m 2 / s
Reynolds bulk velocity  U b H / ν  13 350 
Target Reynolds friction velocity  U τ H / ν  395 

2. One-sided textured channel flow

For the one-sided textured channel, the bottom boundary of the smooth wall channel shown in Fig. 3 is covered with arrays of roughness elements. These, as illustrated in Fig. 4, consist of rectangular prisms aligned with the stream-wise x or span-wise z directions. The elements are uniformly spaced with a constant gap size S in both the x and z directions. This spacing or gap between adjacent elements is reported as S+, which is scaled using the viscous length scale of the smooth channel wall (δv). The plan area density (λp) of the texture can be tuned by varying the element dimensions or the gap size.20 The plan area density is defined by the following equation:
λ p = a b L x L z .
(4)
FIG. 4.

Schematic representation of the surface texture made from an array of rectangular prisms applied to the bottom boundary of the channel.

FIG. 4.

Schematic representation of the surface texture made from an array of rectangular prisms applied to the bottom boundary of the channel.

Close modal

Here, a and b represent the dimensions of the texture elements (rectangular bars) in the stream-wise and span-wise directions, respectively. Lx and Lz denote the stream-wise and span-wise dimensions of the area covered by the footprint of an individual element increased by half of the spacing S in both directions x and z (see Fig. 4). Smooth wall units are used to scale all length scales which define the textures. Table II summarizes the geometrical characteristics of the five textures studied. It is worth noting that in order to preserve the geometric periodicity of the computational domain, slight adjustments were made to the width and length of the channels. The roughness Reynolds number h u τ / ν, where h = 0.1 m is the height of individual elements, also called the roughness height, u τ is roughness friction velocity, and ν is fluid kinematic viscosity, remains below 50 for all cases considered. The calculated roughness Reynolds numbers for stream-wise and span-wise directions with λp values of 0.88, 0.77, 0.64, and 0.457 are 38.5, 39.5, 41.5, 48.5, and 45, respectively. This indicates that the flow is transitionally rough in all cases. A no-slip wall condition is applied to all solid walls as is the case with the smooth wall channel flow simulation. Also, the decision to opt for geometry with sharp edges was based on its manifestation of Reynolds number independence, as evidenced in earlier studies.18,35,36

TABLE II.

Geometrical features of the textures.

S+ (gap sizes based on wall units) λp (plan area density) Bars orientation b+ a+
10  0.88  Stream-wise  316  105 
20  0.77  Stream-wise  732  105 
40  0.64  Stream-wise  316  105 
80  0.457  Stream-wise  316  105 
80  0.457  Span-wise  105  316 
S+ (gap sizes based on wall units) λp (plan area density) Bars orientation b+ a+
10  0.88  Stream-wise  316  105 
20  0.77  Stream-wise  732  105 
40  0.64  Stream-wise  316  105 
80  0.457  Stream-wise  316  105 
80  0.457  Span-wise  105  316 

3. Organism model positioned within and above a texture

The Nitzschia are a pennate marine diatom species that are predominantly found in cold water. This group of organisms is known to vary in length, from 1 to 100 μm or more, and in width, from 1 to 15 μm or more. The typical shape of the Nitzschia species is shown in Fig. 5.24 A less complex oval cylinder is recommended by the Baltic Marine Environment Commission to represent the diatom's body for the purpose of evaluating its volume (Fig. 6).23 To avoid the extremely fine meshes that would be needed to capture the higher curvature at the tip with this type of shape, a stadium cylinder was used instead in the present study (Fig. 6). Also, to avoid dealing with fine meshes and high computational costs, a Nitzschia model at the higher end of its size range was chosen. The organism model is specified here both in absolute terms and in viscous length scales based on the wall units calculated in12 for the tip of the tidal turbine where a viscous length scale of 1 μm was determined. The diatom model length, width, and height chosen are d 1 + = 60 , d 2 + = 10, and h + = 5, respectively. This corresponds to d 1 = 60 μ m , d 2 = 10 μ m, and h = 5 μ m when related to the nominal viscous length scale used to represent flow over the turbine blade toward its tip or d 1 = 145 mm , d 2 = 23.7 mm, and h = 12.5 mm based on the smooth wall units for the channel flow at R e τ = 395, when the viscous length scale is 2.53 mm. No-slip boundary conditions are applied on the surface of the organism model.

FIG. 5.

Image of a marine organism belonging to Nitzschia family taken from Ref. 24.

FIG. 5.

Image of a marine organism belonging to Nitzschia family taken from Ref. 24.

Close modal
FIG. 6.

(a) The oval cylinder shape representing Nitzschia suggested by Ref. 23. (b) The stadium cylinder shape used in the present work to represent the Nitzschia.

FIG. 6.

(a) The oval cylinder shape representing Nitzschia suggested by Ref. 23. (b) The stadium cylinder shape used in the present work to represent the Nitzschia.

Close modal

1. Smooth wall channel flow

Because of the simple geometry of the channel flow, structured hexahedral meshes are used to build the computational domain. Three mesh sizes, namely, coarse M1, medium M2, and fine M3, are employed in the present study to examine the impact of grid size on the LES solution. The details of these three meshes are provided in Table III. In Table III, Δ x + and Δ z + represent the scaled mesh sizes using u τ at R e τ = 395. The size of the M2 mesh (medium mesh) is half the size of the coarse mesh (M1), and similarly, the size of the M3 mesh (fine mesh) is half that of M2 (Fig. 7). To enhance the resolution of turbulent structures in the vicinity of the wall and to accurately capture the associated gradients, a biased mesh is defined in the wall normal y-direction. This mesh stretches from the wall toward the center of the channel, with a growth ratio of 1.1.

TABLE III.

The mesh properties employed to evaluate the LES methodology in a smooth channel flow at R e τ = 395.

Mesh case Number of nodes in x × y × z directions First cell height y+ Δ x + Δ z + Total number of cells
M1  60 × 50 × 45  2.2  40  26  135 000 
M2  120 × 100 × 90  1.05  20  13  1 080 000 
M3  240 × 200 × 180  0.5  10  6.5  8 640 000 
Mesh case Number of nodes in x × y × z directions First cell height y+ Δ x + Δ z + Total number of cells
M1  60 × 50 × 45  2.2  40  26  135 000 
M2  120 × 100 × 90  1.05  20  13  1 080 000 
M3  240 × 200 × 180  0.5  10  6.5  8 640 000 
FIG. 7.

Computational mesh used in smooth wall channel flow: (a) coarse mesh, (b) medium mesh, and (c) fine mesh.

FIG. 7.

Computational mesh used in smooth wall channel flow: (a) coarse mesh, (b) medium mesh, and (c) fine mesh.

Close modal

2. One-sided textured meshes

The computational meshes used for the textured cases are illustrated in Fig. 8. These were adapted from Mesh M3 of the validation study for smooth wall channels, using the same block structure. Further refinement was needed to resolve the flow around the textures. The mesh resolution around the prisms was determined from the Kolmogorov length scale η, which was assumed for that purpose to be similar to the values reported in Refs. 12 and 18 at R e τ = 395 as η = 0.0075 l y and η = 0.0039 l y within the roughness gap and in the shear layer over roughness textures, respectively. The five tested meshes were chosen so that the maximum cell length was 10–40 times the expected Kolmogorov length scales. The y+ for the first cell height was chosen to be less than 1 for all cases. Biased non-uniform meshes with a growth ratio of 1.1, emphasizing walls within textures and rectangular sharp edges, are used to resolve regions of higher gradients. The resulting mesh characteristics are given in Table IV.

FIG. 8.

Five different configurations used in simulations: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

FIG. 8.

Five different configurations used in simulations: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

Close modal
TABLE IV.

Mesh characteristics used for the different textures at R e τ = 395.

Mesh case Plan area density (λp) First cell height y+ Δ x + Δ z + Number of cells Number of nodes in x × y × z directions
0.88  0.8  10  17 088 435  456 × 108 × 352 
0.77  1.05  10  11 340 000  350 × 108 × 300 
0.64  1.3  10  7 358, 390  300 × 108 × 231 
0.457 stream-wise  1.5  10  7 358 390  300 × 108 × 231 
0.457 span-wise  1.3  10  7 358 390  300 × 108 × 231 
Mesh case Plan area density (λp) First cell height y+ Δ x + Δ z + Number of cells Number of nodes in x × y × z directions
0.88  0.8  10  17 088 435  456 × 108 × 352 
0.77  1.05  10  11 340 000  350 × 108 × 300 
0.64  1.3  10  7 358, 390  300 × 108 × 231 
0.457 stream-wise  1.5  10  7 358 390  300 × 108 × 231 
0.457 span-wise  1.3  10  7 358 390  300 × 108 × 231 

3. Organism model applied to the λ p = 0.457 stream-wise texture

The candidate organisms were positioned at three distinct locations over the textured surface: (i) location A at the center of the crest of a texture element, (ii) location B on the bottom wall in the center of a stream-wise gap, and (iii) location C on the bottom wall and in the center of a span-wise gap of the stream-wise texture λ p = 0.457. In additional an organism is also placed on the bottom wall of the smooth wall channel to provide a basis for comparison. A similar mesh with a similar cell count to the one-sided texture case is used to mesh this additional case; however, a higher mesh density is used around the organism to better resolve the flow-organism interactions. A total of 16 × 6 × 9 nodes were used to model the geometry of the organism. Once again, the mesh was gradually refined toward the walls of the organism using a biased non-uniform mesh with a growth ratio of 1.07 (Fig. 9). More details on the computational mesh in the vicinity of the organism for the four cases are given in Fig. 10 and Table V. It is worth noting that the organism was oriented in the span-wise direction within the span-wise gap rather than in the stream-wise direction as for the stream-wise gaps. Similar attachment patterns have been observed in lab studies.

FIG. 9.

Surface meshes used on and around the organism.

FIG. 9.

Surface meshes used on and around the organism.

Close modal
FIG. 10.

Overview of computational domain and mesh of channels with model organism with the organism paced (a) on the texture crest, (b) in the stream-wise gap, (c) in the span-wise gap, and (d) on the smooth wall.

FIG. 10.

Overview of computational domain and mesh of channels with model organism with the organism paced (a) on the texture crest, (b) in the stream-wise gap, (c) in the span-wise gap, and (d) on the smooth wall.

Close modal
TABLE V.

Number of mesh nodes used for the λ p = 0.457 stream-wise texture with the organism placed at different locations.

Cases Organism locations Number of nodes in x × y × z directions Number of cells
Texture crest  363 × 140 × 247  12 552 540 
Bottom of the stream-wise gap  363 × 140 × 247  12 552 540 
Bottom of the span-wise gap  317 × 140 × 247  10 961 860 
Smooth wall organism  260 × 200 × 247  6 240 000 
Cases Organism locations Number of nodes in x × y × z directions Number of cells
Texture crest  363 × 140 × 247  12 552 540 
Bottom of the stream-wise gap  363 × 140 × 247  12 552 540 
Bottom of the span-wise gap  317 × 140 × 247  10 961 860 
Smooth wall organism  260 × 200 × 247  6 240 000 

Second-order schemes are used for space and time Discretization along with a segregated solution method based on the PIMPLE pressure velocity coupling of OpenFOAM,37 to solve the unsteady form of the governing equations as detailed in Refs. 12 and 33. The implemented iterative solver relies on two outer iterations, with three pressure correction loops. Two additional non-orthogonal correction loops have been used to maintain accuracy and stability with the mesh in cases where organisms are included, to address the slight deformation of the mesh. The remaining aspects of the simulations are as implemented in the organism-free simulations of Ref. 12. A Courant number of less than 0.5 was used to minimize error propagation, necessitating small time steps of 0.05 (s) for the smooth wall case with fine mesh and 0.01 (s) with the textured wall. The introduction of organisms further reduced the time step to 0.0025 (s). For all cases, a preliminary transient simulation of duration 200 T, where T = l x / U b is used before sampling for turbulent statistics. The averaging phase covers a period of 200 T to 400 T following the procedure adopted in Ref. 18. This duration is necessary to converge the area-averaged statistics, in particular higher-order statistics, to reproduce the linear slope in the reduction of turbulent shear stresses with distance to the wall expected in the outer layer, and to reduce the large spatial variations related to dispersive stresses in the inner layer. The computational domain was partitioned using the Scotch method, integrated into OpenFOAM 6, and executed in parallel on 640 processors. The overall simulations consumed in excess of 2 million CPU hours.

The conducted studies in Refs. 18 and 20 have established the influence of large turbulent scales on turbulent activity near roughened walls. To ensure the quality of the LESs, the Celik LES index38 is employed. This index serves as a measure indicating the portion of turbulent kinetic energy that the LES method can effectively capture. For practical engineering purposes, it is recommended to have an index value surpassing 0.8. In the conducted simulation, the Celik index ranged from 0.92 to 0.95, demonstrating the satisfactory performance of the LES for the engineering application.

The smooth channel flow simulations on three successive meshes M1, M2, and M3 are compared against DNS solutions in Fig. 11 to assess the solution methodology and its sensitivity to the mesh spatial resolution. Profiles of resolved turbulent fluctuations sampled along a line perpendicular to the bottom wall are plotted against the normalized wall normal coordinate y + = y / δ v, where δv represents the viscous length scale for a smooth wall. The velocity fluctuations are defined as u i , rms = u i 2 ¯ and are normalized with respect to the friction velocity of a smooth wall, u τ. The over-bar and angle brackets denote the time and spatial average operator applied to the data. The results are compared with the Direct Numerical Simulation (DNS) solution for a smooth wall at R e τ = 395 as reported in Ref. 32. The results are shown to converge toward the DNS solution as the mesh is refined from the coarse mesh M1 to the finest mesh M3. This convergence is similar to that observed in similar LES studies in Refs. 31 and 39. Additionally, Fig. 11(d) shows the Reynolds shear stress, scaled using u τ 2, where, once again, the fine mesh results are in good agreement with the DNS solution.

FIG. 11.

Profiles of the normalized velocity fluctuation and turbulent shear stress from the smooth wall channel flow plotted against y+. For velocity fluctuations in the (a) stream-wise direction, (b) normal-to-wall direction, (c) span-wise direction, and (d) normalized turbulent shear stress.

FIG. 11.

Profiles of the normalized velocity fluctuation and turbulent shear stress from the smooth wall channel flow plotted against y+. For velocity fluctuations in the (a) stream-wise direction, (b) normal-to-wall direction, (c) span-wise direction, and (d) normalized turbulent shear stress.

Close modal

Given the high spatial variability of turbulent flow within and over textures, it is customary to employ spatial averaging in the data analysis. This introduces additional stresses, known as dispersive stresses. The effect of the surface textures is analyzed here in terms of the wall's normal profile. The resolved spatially averaged profiles for the Reynolds and dispersive stresses are plotted against the dimensionless wall distance y + ε + in Figs. 12 and 13, respectively. Zero-plane displacement ε + was calculated as defined in Refs. 40 and 41. In Figs. 12 and 13, the resolved spatially averaged profiles of Reynolds and dispersive stresses respectively are plotted against the dimensionless wall distance y + ε +. It is customary to present the averaged diagonal stresses u i 2 ¯ as the root mean squared (RMS), represented as V R rms = u i 2 ¯ . The subscript R denotes Reynolds stresses while D refers to dispersive stresses. The calculated RMS are then scaled with the smooth wall mean friction velocity. The mean friction velocity u τ , s at a Reynolds number R e τ = 395 is chosen as the scaling parameter. Also, the non-diagonal Reynolds shear stress is displayed as u v R + = u v ¯ / u τ , s 2. Similar scaling is used for dispersive stresses (Fig. 13).

FIG. 12.

Spatially averaged Reynolds stresses for the textured surface plotted against the scaled wall distance y + ε + compared with smooth wall DNS results:32 (a) scaled stream-wise Reynolds stresses, (b) scaled normal to wall Reynolds stresses, (c) scaled span-wise Reynolds stresses, and (d) scaled Reynolds shear stress.

FIG. 12.

Spatially averaged Reynolds stresses for the textured surface plotted against the scaled wall distance y + ε + compared with smooth wall DNS results:32 (a) scaled stream-wise Reynolds stresses, (b) scaled normal to wall Reynolds stresses, (c) scaled span-wise Reynolds stresses, and (d) scaled Reynolds shear stress.

Close modal
FIG. 13.

Spatially averaged dispersive stresses for the textured surface plotted against the scaled wall distance y + ε + compared with smooth wall DNS results:32 (a) scaled stream-wise dispersive stresses, (b) scaled normal to wall dispersive stresses, (c) scaled span-wise dispersive stresses, and (d) scaled dispersive shear stress.

FIG. 13.

Spatially averaged dispersive stresses for the textured surface plotted against the scaled wall distance y + ε + compared with smooth wall DNS results:32 (a) scaled stream-wise dispersive stresses, (b) scaled normal to wall dispersive stresses, (c) scaled span-wise dispersive stresses, and (d) scaled dispersive shear stress.

Close modal

Figures 12 and 13(a) show that the stream-wise dispersive stress is significantly higher than the Reynolds stress within the gap between textures, with a peak occurring near the surface of the crest. This stream-wise dispersive stress contributes to considerably increased dispersive shear stress compared to the Reynolds shear stress. For the span-wise and wall-normal stresses [Figs. 12, 13(b), and 13(c)], both stresses are of comparable magnitude within the gaps between the textures. In all cases, the dispersive stresses rapidly vanish as the profile approaches and exceeds the position of the texture crest plane. This observation aligns with the findings of Refs. 18 and 20. A closer look at the profiles shows that the texture with the highest area density ( λ p = 0.88) does not generate significant Reynolds and dispersive stresses within the textures. This indicates that stream-wise coherent structures are unable to penetrate these particular textures in any significant way.12 However, the λ p = 0.457 textures whether positioned in the stream-wise or span-wise exhibit notably elevated dispersive and Reynolds stresses within and above the gaps between the textures. This is also the case for λ p = 0.64 although to a lesser extent. For λ p = 0.77, Reynolds stresses within the gaps remain strong but not the dispersive stresses and finally as the gap size is reduced further with λ p = 0.88, stresses almost vanish completely within the gaps.

Figure 14 displays the total turbulent stresses ( ( u i u j ) t + = u i u j ¯ + u i ̃ u j ̃ ). As the viscous stresses become negligible only a few viscous lengths from the walls, this total stress becomes largely dominant. In particular, it can be observed that the presence of textures significantly enhances the stream-wise fluctuations compared to the smooth wall case, particularly for textures with low λp such as 0.457. The observed maximum stream-wise fluctuations plateaus within the gaps with a total stress that is approximately twice the maximum generated above the smooth wall. Transitioning from higher λp to lower texture densities is clearly shown to increase the transfer of turbulent fluctuations within the textures.

FIG. 14.

Normalized total stresses for the textured wall plotted against the scaled wall distance y + ε + compared with smooth wall DNS results:32 (a) scaled total stream-wise stresses, (b) scaled total normal to wall stresses, (c) scaled total span-wise stresses, and (d) scaled total turbulent shear stress.

FIG. 14.

Normalized total stresses for the textured wall plotted against the scaled wall distance y + ε + compared with smooth wall DNS results:32 (a) scaled total stream-wise stresses, (b) scaled total normal to wall stresses, (c) scaled total span-wise stresses, and (d) scaled total turbulent shear stress.

Close modal

A direct comparison of the Reynolds shear stresses (RSS) and dispersive shear stresses (DSS) for λ p = 0.457 with the texture element oriented in both the span-wise and the stream-wise directions is shown in Fig. 15. Inside most of the gaps (except close to the bottom walls), the dispersive shear stresses are clearly dominant and grow at a significantly higher rate with the distance from the wall. Approximately halfway up the gap (around y + ε + = 30 from the bottom wall), the dispersive shear stress is nearly two to three times larger than the Reynolds shear stress. The trend is reversed from y + ε + = 40 45 from the bottom wall, where the maximum dispersive shear stresses are reached so that the difference decreases gradually as the position approaches the crest plane where the two forms of stresses are approximately equal. Beyond this point, the dispersive shear stresses rapidly become negligible.

FIG. 15.

Comparison of spatially averaged Reynolds and dispersive shear stresses diagrams of λ p = 0.457 textures drawn against ( y + ε +).

FIG. 15.

Comparison of spatially averaged Reynolds and dispersive shear stresses diagrams of λ p = 0.457 textures drawn against ( y + ε +).

Close modal

This section examines the spatial variations of stresses over planes parallel to the texture crest planes to identify regions with maximum and minimum stresses. Two positions between the bottom of the gaps and the crest plane at y + = 10 and y + = 30 measured from the bottom wall are considered.

The contours of the Reynolds stresses u i u j ¯ and the dispersive stresses ( u i ̃ u j ̃) on the two planes are shown in Figs. 16–23. The y + = 10 contours provide insights into the turbulent conditions experienced by a fouling organism during initial settlement on the bottom surface while y + = 30 corresponds to the region where the dispersive stresses are at their highest.

FIG. 16.

Contours of stream-wise Reynolds stress ( u u ¯), at the height of y + = 10, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

FIG. 16.

Contours of stream-wise Reynolds stress ( u u ¯), at the height of y + = 10, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

Close modal
FIG. 17.

Contours of stream-wise Reynolds stress ( u u ¯), at the height of y + = 30, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

FIG. 17.

Contours of stream-wise Reynolds stress ( u u ¯), at the height of y + = 30, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

Close modal
FIG. 18.

Contours of stream-wise dispersive stress ( u ̃ u ̃), at the height of y + = 10, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

FIG. 18.

Contours of stream-wise dispersive stress ( u ̃ u ̃), at the height of y + = 10, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

Close modal
FIG. 19.

Contours of stream-wise dispersive stress ( u ̃ u ̃), at the height of y + = 30, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

FIG. 19.

Contours of stream-wise dispersive stress ( u ̃ u ̃), at the height of y + = 30, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

Close modal
FIG. 20.

Contours of stream-wise Reynolds shear stress ( u v ¯), at the height of y + = 10, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

FIG. 20.

Contours of stream-wise Reynolds shear stress ( u v ¯), at the height of y + = 10, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

Close modal
FIG. 21.

Contours of stream-wise Reynolds shear stress ( u v ¯), at the height of y + = 30, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

FIG. 21.

Contours of stream-wise Reynolds shear stress ( u v ¯), at the height of y + = 30, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

Close modal
FIG. 22.

Contours of stream-wise dispersive shear stress ( u ̃ v ̃), at the height of y + = 10, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

FIG. 22.

Contours of stream-wise dispersive shear stress ( u ̃ v ̃), at the height of y + = 10, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

Close modal
FIG. 23.

Contours of stream-wise dispersive shear stress ( u ̃ v ̃), at the height of y + = 30, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

FIG. 23.

Contours of stream-wise dispersive shear stress ( u ̃ v ̃), at the height of y + = 30, above the bottom wall: (a) λ p = 0.88, (b) λ p = 0.77, (c) λ p = 0.64, (d) λ p = 0.457 stream-wise, and (e) λ p = 0.457 span-wise.

Close modal

Figures 16–19 show the stream-wise Reynolds and dispersive stresses at y + = 10 and y + = 30. As λp decreases, the stream-wise Reynolds stresses are shown to increase rapidly in the stream-wise gaps but remain small in the span-wise gaps. The flow in the lee of the textures appears to remain largely sheltered from these stream-wise Reynolds stresses. Meanwhile, the strength of stream-wise dispersive stress u ̃ u ̃ has been clearly shown to fill most of the span-wise gaps, whether it is in the lee of textures or at the intersection with the stream-wise gap. For example, in the case λ p = 0.457 with stream-wise textures, the stream-wise dispersive stress values reach nearly 5 at y + = 10 and 8 at y + = 30, which is much higher than the corresponding Reynolds stresses. This indicates that spatial variations can drive strong momentum transfer in the lee of the textures. Interestingly for the same texture density, λ p = 0.457, but with textures aligned with the span-wise direction the stream-wise dispersive stress is seen to concentrate over the stream-wise gaps instead. This is likely due to the stronger interference from the textures when positioned perpendicular to the flow direction. A comparison between stream-wise stresses for the two textures of area density λ p = 0.457 shown in Figs. 16–19(d), and 19(e), suggests that texture elements which extend further in the span-wise direction reduce the strength of dispersive stresses while providing a strong shelter against Reynolds stresses. As a result, much lower total stresses can be expected in the lee of textures that are aligned with the span-wise direction. Although this does not preclude the use of texture elements that are stretched in the flow direction, in practice it would be difficult to guarantee that the texture orientation is aligned with the flow direction. This suggests that shorter elements should be used to avoid creating shelters. Figures 20–23 present a direct comparison between Reynolds and the dispersive shear stresses generated within the textures at distances y + = 10 and y + = 30 measured from the bottom wall. In the case of textures with λ p = 0.77 , 0.88 and 0.64, Reynolds shear stresses in y + = 10 are found to be negligible, but increased in y + = 30. On the contrary, for both textures with area density λ p = 0.457 (stream-wise and span-wise), Reynolds shear stresses are mainly concentrated in stream-wise gaps. There is also a clear intensification in the Reynolds shear stresses at the junction between the span-wise and stream-wise gaps. This is most likely due to increased mixing as a result of the interaction between stream-wise and span-wise vortexes. As is the case with Reynolds shear stresses, a high stream-wise dispersive stress region can generate a higher dispersive shear stress. This is observed in the span-wise gaps where stream-wise dispersive stress is high. At y + = 10 near the bottom wall, a thin layer of negative dispersive stress is observed in the span-wise gaps on the windward side of the textures. This negative layer increases in magnitude as λp is reduced. Further from the front side of texture elements, a region of positive dispersive stresses forms. For the stream texture with λ p = 0.457, this positive dispersive stress reaches a magnitude of 1.8 when scaled with the smooth wall friction velocity. It concentrates on the lee of the texture element in an attached bubble. For the span-wise texture at the same area density, a similar increase forms near the vertical edges of the texture element but does not fill the backward facing vertical face of the elements. This confirms the earlier suggestion that the element size in the span-wise direction should kept small to avoid creating relative shelters. Again, a reduction in the size of the gap is clearly seen to reduce the strength of the dispersive stresses. For example, with λ p = 0.62, the maximum dispersive stress reaches 0.35 at y + = 30 instead of 1.8 for the lower area density.

In the case of textures with λ p = 0.77 , 0.88 and 0.64, the Reynolds shear stresses at y + = 10 are found to be negligible and increase only slightly at y + = 30. For the two texture types with λ p = 0.457 (stream-wise and span-wise), the Reynolds stresses are of similar magnitude and mainly concentrated in the stream-wise gaps. There is also a notable increase in shear stresses at the junction between stream-wise and span-wise gaps, which is most likely due to the interaction between eddies originating from both gaps. High stream-wise dispersive stresses appear to lead to increased dispersive shear stress in span-wise gaps. On closer observation, at y + = 10 near the bottom wall, a thin layer of negative dispersive stress is observed on the windward side of the textures at the span-wise gaps. This negative layer increases in magnitude as the λp value decreases. As the distance from the textures increases, a positive dispersive stress region forms. For the stream-wise texture with λ p = 0.457, the scaled positive dispersive stress reaches a magnitude of 1.8 at y + = 10 and 4.2 at y + = 30. The region affected by increased stresses forms an attached bubble in the lee of the texture, which extends all the way to the downstream texture. Similar effects are observed with the span-wise texture but in this case the stresses appear to be more diffused and reduce away from the vertical edges of the texture. A thin layer of positive dispersive shear stress, with a significantly smaller value than the stream-wise case (0.35 when scaled with smooth wall friction velocity), is also observed at λ p = 0.62 textures. In this case again, the dispersive shear stresses are found to increase by a factor of two or three as y+ increase from 10 to 30.

The aim of this section is to explore the impact of roughened wall turbulent flow on a rigid obstacle attached to the surface. The obstacle is meant to mimic an organism (Nitzschia) at the initial stage of settlement. The analysis focuses on three locations, on and within the texture elements. The positions shown in Fig. 24 were selected to explore local variations in stresses and forces acting on the organism. Figures 25–27 compare the time history of the hydrodynamic forces in the stream-wise (x), normal-to-wall (y), and span-wise (z) directions at the three prescribed locations(A, B, and C) against equivalent forces acting on an organism settled on a smooth surface. All force components are scaled by a notional force intended to represent the shear force that would be felt by an organism as it settles over the smooth un-textured surface, assuming that the effect of the organism itself is negligible. The force is defined from the mean shear stress τ w , s over the area covered by the footprint of the organism Ao as F ref = τ w A o. This reference force is calculated from the smooth wall simulation as F ref = 0.000187 ( N ). Time is scaled in terms of the turnover time of eddies shed by textures T ref = h / u τ 10 ( s ). Here, h represents the texture heights, and u τ corresponds to the friction velocity of the texture. The forces were sampled over a time period T = 12 T ref, providing sufficient data for averaging. The difference between the mean values during the first and second half of the sampling period was less than 2% of the overall mean. Although both viscous shear stress and pressure act on the surface of the organism, the dominant force, in the streamwise direction, is largely due to viscous shear irrespective of the location on or around the texture elements. In all prescribed locations, the proportion of stream-wise forces acting on the organism is consistently above 85% for viscous forces and less than 15% for pressure forces.

FIG. 24.

Illustration of the locations, A, B, and C, where the organism is positioned (the fluid flows in the direction of the major axis of the rectangle).

FIG. 24.

Illustration of the locations, A, B, and C, where the organism is positioned (the fluid flows in the direction of the major axis of the rectangle).

Close modal
FIG. 25.

The time-history of hydrodynamic forces exerted on the organism model in the stream-wise direction.

FIG. 25.

The time-history of hydrodynamic forces exerted on the organism model in the stream-wise direction.

Close modal
FIG. 26.

The time-history of hydrodynamic forces exerted on the organism model in the wall-normal direction.

FIG. 26.

The time-history of hydrodynamic forces exerted on the organism model in the wall-normal direction.

Close modal
FIG. 27.

The time-history of hydrodynamic forces exerted on the organism model in the span-wise direction.

FIG. 27.

The time-history of hydrodynamic forces exerted on the organism model in the span-wise direction.

Close modal
A summary of the statistical characteristics of these forces is given in Tables VI and VII. This includes the mean forces and the intensity of fluctuations, acting on the organism defined by the following equation:
I i = ( 1 T 0 T η 2 d t ) 0.5 η ¯ ,
(5)
where η is the difference between the instantaneous value of η and the time-averaged value η ¯.
TABLE VI.

Mean hydrodynamic forces acting on the organism in different directions.

Organism location F x ¯ / F ref F y ¯ / F ref F z ¯ / F ref
Organism on the smooth wall  4.0659  1.2267  0.2922 
4.5119  2.2795  0.1462 
3.3954  1.5572  0.1542 
6.5323  1.5102  0.1928 
Organism location F x ¯ / F ref F y ¯ / F ref F z ¯ / F ref
Organism on the smooth wall  4.0659  1.2267  0.2922 
4.5119  2.2795  0.1462 
3.3954  1.5572  0.1542 
6.5323  1.5102  0.1928 
TABLE VII.

Intensity of the hydrodynamic forces acting on the organism in different directions.

Organism location Ix Iy Iz
Organism on the smooth wall  0.5457  2.7738  8.0212 
0.3391  1.8099  20.5726 
0.3732  3.0258  9.6579 
0.3732  2.1920  10.6488 
Organism location Ix Iy Iz
Organism on the smooth wall  0.5457  2.7738  8.0212 
0.3391  1.8099  20.5726 
0.3732  3.0258  9.6579 
0.3732  2.1920  10.6488 

Tables VI and VII provide a summary of the averaged forces acting on the organism model in x, y, and z directions and the corresponding intensities of the force fluctuations as defined by Eq. (5). The data given in Table VI show an increase in the magnitude of the stream-wise force for points A and C by approximately 60% and 11% and a moderate decrease in 16% at B. These increases occur in spite of a substantial reduction in excess of 33% to 50% in the span-wise force component. However, it is noted that this force component makes a minor contribution to the total hydrodynamic force. The increase at location C occurs in spite of what was interpreted as a sheltering effect from the texture when considering the Reynolds stress contours alone. The wall-normal component is also shown to increase by 23%, 27%, and 86% for locations C, B, and A, respectively. The intensity of fluctuations are also an important indicator of the effect of the textures as it reflects the strength of transient fluctuations that organisms are exposed to. A large departure from the mean can cause significant additional instantaneous stresses. The texture is shown to induce a significant decrease in the stream-wise intensity at all locations and a more moderate one for the wall-normal intensity at locations A and C. Interestingly, although the increase in span-wise fluctuations is moderate at locations B and C, a very large increase in approximately 150% is observed at location A. This suggests that the textures have a strong impact on vortices over the crest plane, possibly due to an amplification in eddy shedding in the span-wise and wall-normal directions caused by the sharp edges of the texture.

The height of the organism reaches y + = 4.82 at its apex. Reynolds and dispersive shear stresses contours from velocity fluctuations in the stream-wise direction below this level are given in Figs. 28–30, for two planes at a distance y + = 2 and y + = 4 from the base of the organism. The dispersive stresses can be seen to concentrate within the stream-wise gaps in the vicinity of the texture crest plane, while the Reynolds stresses are mainly significant in the span-wise gaps. Within the span-wise gaps, dispersive stresses are typically one order of magnitude larger than Reynolds stresses at y + = 4 and most likely a better measure of the effect of turbulence on forces experienced by the organisms settled within the gap. This can be explained by reference to Eq. (1), which can simplify to Eq. (6) in the case of a statistically stationary flow,
u v ¯ y = f v + f p ,
(6)
where fv and fp denote the viscous drag and the pressure drag. Increases in dispersive stresses normal to wall result in higher wall normal gradients in the span-wise gap and, hence, higher average stream-wise drag acting on the organism at location C.
FIG. 28.

Contours of the stream-wise Reynolds and dispersive shear stresses around the organism model positioned at location A: (a) Reynolds shear stress at y + = 2, (b) Reynolds shear stress at y + = 4, (c) dispersive shear stress at y + = 2, and (d) dispersive shear stress at y + = 4.

FIG. 28.

Contours of the stream-wise Reynolds and dispersive shear stresses around the organism model positioned at location A: (a) Reynolds shear stress at y + = 2, (b) Reynolds shear stress at y + = 4, (c) dispersive shear stress at y + = 2, and (d) dispersive shear stress at y + = 4.

Close modal
FIG. 29.

Contours of the stream-wise Reynolds and dispersive shear stresses around organism model positioned at location B: (a) Reynolds shear stress at y + = 2, (b) Reynolds shear stress at y + = 4, (c) dispersive shear stress at y + = 2, and (d) dispersive shear stress at y + = 4.

FIG. 29.

Contours of the stream-wise Reynolds and dispersive shear stresses around organism model positioned at location B: (a) Reynolds shear stress at y + = 2, (b) Reynolds shear stress at y + = 4, (c) dispersive shear stress at y + = 2, and (d) dispersive shear stress at y + = 4.

Close modal
FIG. 30.

Contours of the stream-wise Reynolds and dispersive shear stresses around organism model positioned on C location: (a) Reynolds shear stress at y + = 2, (b) Reynolds shear stress at y + = 4, (c) dispersive shear stress at y + = 2, and (d) dispersive shear stress at y + = 4.

FIG. 30.

Contours of the stream-wise Reynolds and dispersive shear stresses around organism model positioned on C location: (a) Reynolds shear stress at y + = 2, (b) Reynolds shear stress at y + = 4, (c) dispersive shear stress at y + = 2, and (d) dispersive shear stress at y + = 4.

Close modal

The effect of the location on the total hydrodynamic forces F t = F x 2 + F y 2 + F z 2 experienced by the organism is illustrated and summarized in Fig. 31 and Table VIII, where it is normalized by Fref. The texture reduces the overall intensity of fluctuations compared to the smooth wall case, but increases the mean total force and maximum force both at the crest and within the span-wise gaps (locations A and C). The increase in the mean total force is approximately 31% and 44%, respectively, compared to the smooth wall case, with a similar increase in the maximum forces. On the contrary, the organism at the base of the stream-wise gap (location B) experiences a slight decrease in 6% in the mean total forces with little change to the maximum. In terms of instantaneous maximum forces, locations A and C exhibit an approximate 25% increase compared to the smooth wall case, while point B does not show a significant change compared to the smooth wall pattern.

FIG. 31.

The time-history of total hydrodynamical forces exerted on the organism model.

FIG. 31.

The time-history of total hydrodynamical forces exerted on the organism model.

Close modal
TABLE VIII.

Total hydrodynamical forces acting on the organism in different cases.

Organism location F t ¯ / F ref It Max F t / F ref
Organism on the smooth wall  5.2574  0.6255  10.8212 
6.9329  0.3514  13.4912 
4.9424  0.4582  10.5037 
7.5855  0.5134  13.6054 
Organism location F t ¯ / F ref It Max F t / F ref
Organism on the smooth wall  5.2574  0.6255  10.8212 
6.9329  0.3514  13.4912 
4.9424  0.4582  10.5037 
7.5855  0.5134  13.6054 

1. Dispersive and Reynolds stresses impact on the exerted forces

The results show that organisms within the groves experience larger forces when located in the lee of the texture rather than in the stream-wise groves. The comparison between forces at these two locations is shown in Fig. 32. This clearly shows that the only significant change comes from the shear dispersive stream-wise force. A similar observation was made with the stream-wise dispersive stresses, which were shown to concentrate around location C (Fig. 30). This suggests that dispersive stresses have the strongest influence on the average force when the organism is located in the span-wise gap. Since the organism is almost entirely in the viscous sub-layer ( y + 5), viscous stresses can be expected to dominate compared to Reynolds stresses (wall-normal dispersive shear stress gradient has mostly influenced the viscous stresses rather than pressure). The comparison also shows how limited the impact of changes in location between the two gaps (B or C) is on the intensity of force fluctuations. Similar observations were made with the Reynolds stresses, which can be expected to be the dominant stress affecting force fluctuations. More variability is seen between forces acting on the organism located at A and C (Fig. 33).

FIG. 32.

Forces applied to the organism at B and C locations within the texture: (a) stream-wise force ( F x / F ref ), (b) normal-to-wall force ( F y / F ref ), and (c) span-wise force ( F z / F ref ).

FIG. 32.

Forces applied to the organism at B and C locations within the texture: (a) stream-wise force ( F x / F ref ), (b) normal-to-wall force ( F y / F ref ), and (c) span-wise force ( F z / F ref ).

Close modal
FIG. 33.

Forces applied to the organism at A and C locations within the texture: (a) stream-wise force ( F x / F ref ), (b) normal-to-wall force ( F y / F ref ), and (c) span-wise force ( F z / F ref ).

FIG. 33.

Forces applied to the organism at A and C locations within the texture: (a) stream-wise force ( F x / F ref ), (b) normal-to-wall force ( F y / F ref ), and (c) span-wise force ( F z / F ref ).

Close modal

This research investigates how changes to the design of a surface texture inspired by the growth rings of Brill fish can influence turbulent stresses at locations where the early-stage settlement of marine diatoms may occur. The aim is to identify how flow-induced stresses and forces may impact the initial stage of bio-fouling formation and assess whether relative shelters may be formed. The study considers changes to local and surface-averaged stress distributions due to changes to the gap size between adjacent textures, making a distinction between Reynolds stresses and dispersive stresses. It includes an analysis of the hydrodynamic forces experienced by an organism positioned at three different locations on and within a single texture. The results are compared to a reference case where the organism has settled on a smooth surface. The study relies on Large Eddy Simulations (LESs) of the turbulent flow validated by reference to DNS simulations of a smooth channel flow. The analysis of turbulence and its sensitivity to the gap size are summarized below:

  • For lower density arrays with λ p = 0.457 where texture elements are aligned with the stream-wise direction, the spatially averaged stream-wise dispersive stresses within the texture gap close to the bottom walls are nearly twice as large as the Reynolds stresses.

  • At higher area densities λp of 0.77 and 0.88, turbulent stresses make little impact on the flow within the gaps between adjacent textures.

  • The span-wise gaps are shown to shelter against Reynolds stresses but not dispersive shear stresses. Observations suggest that the use of elongated texture elements should be avoided to ensure that strong dispersive stresses develop everywhere in the lee of the texture elements.

In the second phase of the study, an obstacle intended to mimic a settled bio-fouling organism was placed at three distinct positions around the texture. The maximum height of the organism is less than y + = 5. The analysis considers a single texture with area density λ p = 0.457 and texture elements aligned with the stream-wise direction. The key results are summarized below:

  • The presence of the prismatic texture led to a significant increase of approximately 60% in the mean stream-wise drag experienced by the organism in the span-wise gap downstream of the texture. This is shown to occur in spite of a reduction in Reynolds stresses but does appear to correlate with an increase in dispersive stresses behind the texture.

  • The hydrodynamic forces acting on the organism at positions B and C (C is located in the region with high dispersive stress) follow the same trend but higher stream-wise forces develop at point C. This indicates that dispersive shear stresses influence the magnitude of the mean force, while the Reynolds stresses contribute to the intensity of fluctuations.

  • The texture induces a significant increase in the mean and maximum of the total force acting on the organisms if positioned on the surface of the texture or in the span-wise gap downstream of the texture. The increases reach 31% and 44%, respectively.

This research is funded by the European Commission under its Horizon 2020 Framework Programme (Grant No. 815278). The authors also acknowledge the support of the Irish Center for High-End Computing (ICHEC through its class B project dceng017b for access to the Kay supercomputer with 1.2 M CPU hours.

The authors have no conflicts to disclose.

Amin Peyvastehnejad: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Resources (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Fiona Regan: Conceptualization (equal); Funding acquisition (lead); Resources (lead); Writing – review & editing (supporting). Chloe Richards: Conceptualization (supporting). Adrian Delgado: Conceptualization (supporting). Philip Daly: Conceptualization (supporting). Javier Grande: Funding acquisition (lead); Project administration (supporting); Resources (lead). Yan Delaure: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

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

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