Threshold of motion and orientation of bivalve shells under current flow

Abstract

Disarticulated shells of three bivalve mollusk species (Anomalocardia brasiliana, Codakia orbicularis, and Divaricella quadrisulcata) were experimentally tested in laboratory flumes to determine the threshold of motion and final orientation of the valves. A total of 150 current flow experiments were conducted on single shells resting on a fixed sand bed. This study demonstrated that shells in the convex-up position are more resistant to flow when the umbo is pointing downstream rather than upstream. Moreover, species with higher frontal areas were more likely to be entrained at lower flow velocities. Results of dimensionless shear stress exhibited values far below the threshold of grains movement for beds of uniform roughness (Shields curve). It was observed that circular shells in convex-up positions were mostly orientated with the umbo pointing downstream. Conversely, elliptical shells in convex-up position tended to align their longer axis parallel to the flow with the posterior side of the valve pointing downstream. These results are not only directly applicable in interpretations of incipient shell motions and in paleocurrent analyses from field and sample data, but also support construction of accurate geological models.

KEYWORDS:
bivalve shells; shell orientation; threshold of motion; flume experiments; unidirectional flow

INTRODUCTION

Dense shell accumulations (i.e., coquinas, shell pavements, or shell beds) have been studied in recent decades by many researchers because of their importance as petroleum reservoirs and recurrent occurrence in both ancient deposits (Carvalho et al. 2000, Tavares et al. 2015, Muniz and Bosence 2018, Oliveira et al. 2019, Rodrigues et al. 2022) and modern coastal settings (Calliari and Klein 1993, Jahnert et al. 2012, Weill et al. 2013, Charó et al. 2014). The genesis of mollusk-dominated, shell-rich deposits is frequently associated with coastal and shallow water sedimentary environments in both marine and lacustrine settings, where waves and currents are the main processes responsible for sediment reworking and transport (Fick et al. 2020). The threshold of motion and shell orientation provide important information about how these skeletal grains are transported and sorted by flow. In paleoenvironmental analysis, the shell position and its degree of fragmentation have been used to infer the direction (Nagle 1967, Wendt 1995, Radley 2011) and energy (Trewin and Welsh 1972, Schwartz and Graham 2015) of the paleoflow.

The physical parameters of the shells, such as mass, shape, size, and surface roughness, control their movement behavior (Vogel 1994). Chattopadhyay et al. (2013) revealed that fresh shells with rough surfaces are more easily transported than older reworked shells. The asymmetry of the shells is another distinguishable property. Shape asymmetries contribute to ease of transport because they generate unstable pressure fields around the shell (Olivera and Wood 1997). The orientation of the shell convexity in a flow dictates its stability; the convex-up posture is more stable than the convex-down one (Nagle 1967, Brenchley and Newal 1970, Allen 1984, Dey 2003, Diedericks et al. 2018, Fick et al. 2020). The initial umbo position also affects the entrainment of the shells (Dey 2003, Chattopadhyay et al. 2013, Diedericks et al. 2018). Futterer (1982) reported that circular-shaped shells transported in a convex-up position by currents require a greater range of transport velocities when the umbo faces downstream than when it points upstream.

To better understand the hydrodynamic behavior of shells, this study aimed to investigate the incipient motion conditions and orientation patterns of bivalve shells under currents through physical modeling experiments. Based on the results, insights on taphonomic signatures (shell orientation, sensu Kidwell et al. 1986), bioclastic sediment sorting, and paleocurrent behavior are proposed and discussed in a geological context.

Threshold of motion

The study of the threshold of motion is associated with the critical condition under which a grain is likely to begin moving. This condition is reached when the sum of the flow drag (F_D) and lift (F_L) forces are equal (or on the threshold) to the particle's submerged weight (W). In addition, the bed slope angle (ψ) and the repose angle (ψ_r) of the bed particles must be considered. The shear stress near the bed (τ_b) can be related to F_D through the Eq. 1:

(1)$${\text{F}}_{\text{D}}={\text{C}}_1{\tau }_{\text{b}}{\text{D}}^2$$

Where:

C₁ = a coefficient that considers the drag and lift effects;

τ_b = the shear stress near the bed (Pa);

D = the particle diameter (m).

The submerged weight (W) is given by the Eq. 2:

(2)$$\text{W}={\text{C}}_2({\gamma }_{\text{s}}-\gamma ){\text{D}}^3$$

Where:

C₂ = a geometric coefficient;

γ_s = the specific weight of the grain (N m⁻³);

γ = the specific weight of the water (N m⁻³).

Substituting W and F_D for their respective expressions, the balance of these forces for the threshold of motion condition is given by the Eq. 3:

(3)$${\text{C}}_2({\gamma }_{\text{s}}-\gamma ){\text{D}}^3\cos \text{ψ}\text{tg}{\text{ψ}}_{\text{r}}={\text{C}}_1{\tau }_{\text{b}}{\text{D}}^2+{\text{C}}_2({\gamma }_{\text{s}}-\gamma ){\text{D}}^3\text{sen}\text{ψ}$$

For a horizontal bed, the above equation can be rewritten as follows (Eq. 4):

(4)$$\frac{{\tau }_{\text{b,cr}}}{({\gamma }_{\text{s}}-\gamma )\text{D}}=\frac{{\text{C}}_2}{{\text{C}}_1}\text{tg}{\text{ψ}}_{\text{r}}$$

The left term of the expression in the above equation is the nondimensional Shields parameter (θ_cr), which expresses the ratio between the hydrodynamic forces and the submerged weight.

Shields (1936), based on unidirectional flow experiments with spherical inorganic (barite and granite sand) and organic (amber and coal fragments) grains, defined an experimental curve of the θ_cr parameter versus the grain Reynolds number (Re*). Re* is a dimensionless parameter that expresses the effects of particles on the flow boundary layer, given by the Eq. 5:

(5)$$\mathrm{Re}*=\frac{\text{u*D}}{\text{v}}$$

Where:

u_* = the shear velocity (= $\sqrt{\frac{{\tau }_{\text{b}}}{\rho }}$);

ν = the kinematic viscosity (m² s^-1).

Ever since it was developed, the Shields (1936) curve has been used for siliciclastic grains (Miller et al. 1977, Buffington and Montgomery 1997), calcareous fragments (Paphitis et al. 2002, Smith and Cheung 2004), and nonfragmented shells (Diedericks et al. 2018, Fick et al. 2020) in initiation of movement studies.

MATERIALS AND METHODS

The flume experiments performed in this work were carried out under two different scenarios: single-shell and shell-bed experiments. A summary of the methodology is provided in Table 1.

Table 1.
Methodology summary for the single-shell and shell-bed experiments.

	Single-shell experiments	Shell-bed experiments
Goals	Shell final orientation Threshold of motion condition	Shell final orientation
Valves initial position	Convex-up: umbo downstream Convex-up: umbo upstream	Concave-up: umbo randomly oriented
Species	Anomalocardia brasiliana Codakia orbicularis Divaricella quadrisulcata	Anomalocardia brasiliana

Single-shell experiments

Disarticulated and nonfragmented valves of three bivalve shell species were used in the single-shell experiments. The shell species were Anomalocardia brasiliana, Codakia orbicularis, and Divaricella quadrisulcata. These species are frequently found along the southeastern Brazilian coast. All specimens were obtained from Holocene back-barrier deposits (Fornari 2010) on the Jaguaruna coastal plain, Santa Catarina, Brazil.

A group of valves of different sizes was selected for each species (Fig. 1). Moreover, Anomalocardia brasiliana (elliptical asymmetric shape) shells were identified as left and right valves (Fig. 2). For Codakia orbicularis and Divaricella quadrisulcata, this discrimination was not needed because they are circular shells with both valves bilaterally symmetrical.

Figure 1.
Shells used in single shell experiments. (A) Anomalocardia brasiliana, left valves; (B) Anomalocardia brasiliana, right valves; (C) Codakia orbicularis; (D) Divaricella quadrisulcata.

Figure 2.
Shape of bivalve shells. (A) Scheme of shell axis orientation. (a) and (b) are the base axes, (c) is the shell height. (B) Scheme of the adopted convention for asymmetrical valve sides. (C) Cross sections on the plane formed by the axes (b) and (c). From left to right: Anomalocardia brasiliana specimen (D_sv = 17.31 mm), Codakia orbicularis specimen (D_sv = 12.75 mm), Divaricella quadrisulcata specimen (D_sv = 15.99 mm). The scale in the lower left corner only applies for the cross sections.

Some physical shell properties were accurately determined and are presented in Table 2. Shell mass (m), volume (V), and density (ρ) were measured with a scale (0.01 g precision) and a pycnometer (Hubbard-type, 25 ml). The shells’ longest, intermediate, and minor axes (a, b, and c, respectively) were measured with a caliper (0.02 mm precision). The sieve diameter (D_sv) was calculated using the Church et al.’s (1987) formula (Eq. 6):

(6)$$\frac{{\text{D}}_{\text{sv}}}{\text{b}}=\frac{1}{\sqrt{2}}{\left[1+{\left(\frac{\text{c}}{\text{b}}\right)}^2\right]}^{0.5}$$

Table 2.
Mass, density, shell's axes sizes, sieve (D_sv) and nominal (D_n) diameters, and projected area (S_ac) on a plane composed by the axes (a) and (c).

Species	Sample number	Valve side	Mass (kg × 10-3)	Density (kg m-3 × 103)	Axis (mm)			Dsv (mm)	Dn (mm)	Sac (mm2)
					a	b	c
Anomalocardia brasiliana	1	Left	3.06	2.62	28.5	23.2	9.0	18.9	13.2	166.2
	2		2.08		26.1	21.1	8.0	17.1	11.5	130.5
	3		1.20		21.5	17.2	6.7	14.0	9.6	98.4
	4		0.66		18.3	15.0	5.3	11.9	7.8	62.1
	5		0.53		16.0	13.0	4.8	10.4	7.3	50.6
	6		0.16		12.2	8.9	3.2	7.1	4.9	27.6
	7		0.01		7.7	6.0	2.5	5.0	1.8	4.4
	8	Right	3.15		29.0	24.0	9.0	19.4	13.3	186.7
	9		1.67		26.4	21.8	7.8	17.4	10.7	132.4
	10		1.13		21.5	17.2	6.8	14.1	9.4	100.1
	11		0.72		18.7	14.5	5.5	11.7	8.0	73.6
	12		0.56		15.8	12.7	5.0	10.4	7.4	56.6
	13		0.13		12.0	8.7	3.0	6.9	4.6	24.8
	14		0.01		7.8	6.3	2.5	5.2	2.0	6.6
Codakia orbicularis	15		0.64	2.66	17.2	17.3	5.5	13.5	7.7	61.8
	16		0.60		16.3	16.0	5.1	12.5	7.6	59.0
	17		0.46		15.3	14.7	4.5	11.4	6.9	53.0
	18		0.36		12.8	12.3	4.1	9.7	6.3	43.0
Divaricella quadrisulcata	19		1.05	2.61	22.3	21.5	7.1	16.9	9.0	110.8
	20		0.79		20.0	19.7	6.1	15.3	8.2	89.1
	21		0.60		19.1	18.5	5.7	14.3	7.5	72.7
	22		0.48		17.2	16.5	4.6	12.6	6.9	53.3
	23		0.41		17.7	16.6	4.9	12.8	6.6	57.8
	24		0.28		15.8	14.9	4.3	11.4	5.8	46.4
	25		0.16		14.6	12.5	3.7	9.6	4.9	29.5

The nominal diameter (D_n), which is the diameter of a sphere with the same volume as the shell, was calculated using the Eq. 7:

(7)$${\text{D}}_{\text{n}}={\left(\frac{6\text{V}}{\pi }\right)}^{\frac{1}{3}}$$

The projected area of the shell (S_ac) on the plane formed by axes (a) and (c) was measured by a photograph analysis using AUTOCAD^® software.

To illustrate the convexity and thickness distribution through the shell, one valve of each species was cut into five sections (Fig. 2C). A valve from Anomalocardia brasiliana (D_sv = 17.31 mm) had an average thickness of 1.8 mm, and the anterior side of the shell was 2.4 mm thick on average. The Codakia orbicularis (D_sv = 12.75 mm) and Divaricella quadrisulcata (D_sv = 15.99 mm) specimens had average thicknesses of 1.06 and 0.94 mm, respectively; both species had slight thickness variations between the posterior and anterior sides (less than 0.3 mm).

Experiments were conducted on a masonry flume (Fig. 3) measuring 11 m long, 0.4 m wide, and 0.5 m in height. The flume bed had a gentle slope (4/1,000), and its surface was a compound of well-sorted fixed sand (mean D_sv = 0.8 mm). At the upstream end, the flume was equipped with a rectangular weir (Fig. 3B) to determine the discharge (Q). A depth gauge (0.0001 m precision) was coupled to the weir to determine the height of water (wh) flowing over it. The discharge (Q) value is given by the Eq. 8:

(8)$$\text{Q=}\frac{\left[0.8919+0.137752\left[\left(\frac{\text{wh}}{100}\right)+0.011\right]\right]}{{(\text{wh+0}\text{.01})}^{-1.5}}$$

Figure 3.
Cross section of the flume used on single shell experiments (not to scale): (a) water feeding valve, (b) rectangular weir, (c) initial shell position, (d) downstream floodgate. Vertical black arrows show the position of depth gauges. Horizontal black arrow points in the flow direction.

Four depth gauges (0.001 m precision) were fixed at 1.5, 4, 4.8, and 6 m (black arrows in Fig. 3) from the beginning of the bed to measure the flow depth (h).

The shell's incipient motion condition was tested in two initial positions: convex-up with the umbo facing upstream and convex-up with the umbo pointing downstream. For each experiment, the flume was filled with water at a low current velocity, and a single shell was positioned at a specific point (Fig. 3C). By opening an upstream valve (Fig. 3A), the flow was increased until the shell started its movement. Subsequently, the flow depth and discharge were measured. This procedure was repeated three times for each shell at both the initial positions. The adopted criterion for incipient motion was considered to have been reached when the shell moved at least 0.1 m.

In preliminary experiments, shells, when entrained with the umbo initially facing upstream, turned on their horizontal plane to reach a more stable position and then stopped moving. In these cases, the final orientation of the valves was determined using a protractor. These measurements were taken along the shell's intermediate axis, and the recorded data showed the direction of the umbo. In experiments where the umbo was facing downstream, it was not possible to measure the shell's final orientation because the shells did not stop their movement until reaching the downstream end of the flume.

All experiments were conducted in a uniformly stable flow regime. The mean flow velocity (U) for the threshold of motion conditions was calculated using continuity (Eq. 9):

(9)$$\text{U}={\text{QA}}^{-1}$$

Where:

A = the wet area (m²);

Q = the discharge (m³ s^-1).

In addition, the mean bed shear stress for the incipient motion condition (τ_b,cr) was determined using the Eq. 10:

(10)$${\tau }_{\text{b,cr}}=\gamma \frac{\text{A}}{\text{P}}\text{S}$$

Where:

γ = the specific weight of water (Nm⁻³);

P = the wet perimeter (m);

S = the flume slope.

Shell-bed experiments

In the shell-bed experiments, an amount of 1 kg sample of nonfragmented valves from the Anomalocardia brasiliana species was used. The sample D_sv value ranged from 3.8 to 22.25 mm (Fig. 4).

Figure 4.
Sieve grain size distribution of a sample of Anomalocardia brasiliana valves.

Experiments were conducted in a 4 m long, 0.12 m wide, and 0.5 m tall recirculating glass flume (Fig. 5). The recirculating system connected the pump to a lower reservoir. In addition, the flume had a trough in its center (0.02 m deep and 0.6 m long). The unconsolidated material did not return to the pipe owing to the presence of a net in the downstream box. A calibrated valve and two staff gauges (0.001 m precision) measured the discharge (Q) and flow depth (h), respectively.

Figure 5.
Cross section of the shell bed experiments flume (not to scale): (a) pump, (b) lower reservoir, (c) calibrated valve, (d) upstream valve, (e) staff gauges, (f) trough. Black arrow indicates the flow direction.

Before each experiment, the flume was completely filled with water, and then the entire 1 kg Anomalocardia brasiliana valve sample was lowered into the trough area, filling all available space. Because of this procedure, the valves lay on the bed in a concave-up position (Middleton 1967, Allen 1984, Li et al. 2020). Subsequently, the water level was slowly dropped to the predetermined flow height. The experiment started when the upstream valve (Fig. 5D) was opened with the calibrated valve (Fig. 5C) already adjusted. The experiment lasted for 20 min. This time was established in preliminary tests, in which it was verified that there was no significant transport after this period.

After each run, a top-shell-bed photograph was taken. These photographs were obtained using a digital camera equipped with a CARL ZEISS T*1.4/85 mm planar lens. Subsequently, all shells were collected for use in the next experiment.

To investigate the orientation pattern of the right and left Anomalocardia brasiliana valves, eight experiments were conducted. The mean flow velocities used were 0.48, 0.52, 0.54, and 0.57 ms^-1 with each velocity having one repetition.

The shell-bed photographs taken after the experiments were analyzed to measure the shell orientation. In this analysis, the edges 0.1 m from both ends were discarded due to the transition between fixed and mobile beds. The bed area was split into two in a checkered pattern (Fig. 6). The measurements were computed for all shells in a convex-up position that were partially or fully inside the brown squares. The orientation data were taken along the shell's intermediate axis with the recorded data exhibiting the direction of the umbo.

Figure 6.
Shell bed photograph of the central trough area. The brown squares represent the analysis area.

RESULTS

Threshold of motion

The critical bed shear stresses for the single-shell experiments were calculated using Eq. 3. The relationship between the particle size (D_n and D_sv) and this parameter is shown in Fig. 7. The mean flow depth and velocity values are presented in Table 3. There were significant disparities in the critical shear stress values for the experiments, in which shells with the umbo facing upstream had lower values than those facing downstream, demonstrating that shells in the second situation exhibited major resistance to transportation.

Figure 7.
Critical shear stress as a function of the shell's diameter. From left to right the graphs are for the species Anomalocardia brasiliana, Codakia orbicularis, and Divaricella quadrisulcata, respectively. Upper panel: shells size represented by the nominal diameter (D_n). Lower panel: shells size represented by the sieve diameter (D_sv). Solid symbols represent the umbo initially pointing downstream and open symbols for the umbo pointing upstream.

Table 3.
Mean values of velocity (U), flow depth (h), and critical shear stress (τ_b,cr).

Sample number	Umbo upstream			Umbo downstream
	h	U	τb,cr	h	U	τb,cr
	(m)	(ms-1)	(Pa)	(m)	(ms-1)	(Pa)
Anomalocardia brasiliana
1	0.024	0.43	0.82	0.064	0.71	1.84
2	0.019	0.35	0.67	0.061	0.70	1.79
3	0.017	0.29	0.59	0.055	0.66	1.64
4	0.017	0.31	0.61	0.046	0.61	1.42
5	0.020	0.36	0.70	0.046	0.61	1.42
6	0.018	0.34	0.64	0.021	0.37	0.74
7	0.016	0.28	0.56	0.017	0.27	0.61
8	0.026	0.45	0.88	0.059	0.69	1.74
9	0.016	0.29	0.58	0.060	0.70	1.76
10	0.019	0.34	0.65	0.047	0.62	1.46
11	0.017	0.29	0.59	0.045	0.60	1.39
12	0.020	0.36	0.70	0.046	0.61	1.42
13	0.017	0.29	0.59	0.024	0.40	0.81
14	0.015	0.25	0.53	0.017	0.27	0.61
Codakia orbicularis
15	0.027	0.44	0.91	0.045	0.60	1.39
16	0.028	0.44	0.94	0.044	0.60	1.37
17	0.024	0.40	0.82	0.044	0.58	1.32
18	0.018	0.34	0.62	0.020	0.35	0.69
Divaricella quadrisulcata
19	0.018	0.34	0.64	0.033	0.52	1.09
20	0.019	0.34	0.65	0.031	0.48	1.01
21	0.017	0.33	0.61	0.029	0.46	0.97
22	0.018	0.34	0.62	0.030	0.48	0.99
23	0.016	0.28	0.56	0.024	0.42	0.83
24	0.017	0.33	0.61	0.018	0.34	0.64
25	0.017	0.33	0.61	0.018	0.33	0.62

The regressions showed similar trends for both characteristic diameters, as shown in the graphs. A positive correlation existed between the size and critical shear stress (τ_b,cr) when the umbo was facing downstream. Under these conditions, Anomalocardia brasiliana and Divaricella quadrisulcata showed strong correlations, with the regression analyses (τ_b,cr versus D_n) having coefficients of determination (R²) of 0.91 and 0.87, respectively. With a Pearson coefficient of 0.88 (R² = 0.77), the species Codakia orbicularis had a lower correlation with the critical shear stress than did the other two species, yet the (R²) value is still reasonable.

For the situation in which the umbo faces upstream, the graphs (Fig. 7) from Anomalocardia brasiliana and Divaricella quadrisulcata show that the thresholds of motion were not a function of the shell size. Although the correlation was still positive, small and large shells had similar τ_b,cr values.

For small shells (D_sv from 4 to 8 mm), the τ_b,cr values were very similar for both the initial umbo orientations tested in this work. However, in the case of larger shell sizes, the τ_b,cr ratios between the umbo pointing downstream and pointing upstream varied considerably among the different species: 2.67 for Anomalocardia brasiliana (D_n = 11.5 mm/D_sv = 17.06 mm); 1.53 for Codakia orbicularis (D_n = 7.7 mm/D_sv = 13.47 mm); and 1.71 for Divaricella quadrisulcata (D_n = 9 mm/D_sv = 16.86 mm).

The size range that includes all these species (D_sv from 9.5 to 14 mm) allows comparisons between individual species with respect to their thresholds of motion. In this range, the average τ_b,cr values acquired with the umbo facing upstream show no important differences between species: 0.63 Pa for Anomalocardia brasiliana, 0.82 Pa for Codakia orbicularis, and 0.60 Pa for Divaricella quadrisulcata. However, when the umbo was facing downstream for the Divaricella quadrisulcata species, the average τ_b,cr value (0.77 Pa) was significantly lower than that of the other species. Anomalocardia brasiliana and Codakia orbicularis had average τ_b,cr values of 1.49 and 1.19 Pa, respectively.

Shields parameter

The critical Shields parameter for the incipient motion condition (θ_cr) was calculated from the left term of Eq. 4 using the shell D_sv as the characteristic dimension of the shell, which was then plotted (Fig. 8) as a function of the grain Reynolds number (Re*). Considering that Re* is related to the viscous effects of the flow developed on a bed of uniform roughness, Wiberg and Smith (1987) and Ramsdell and Miedema (2010) recommended that the mean diameter of the bed be used to calculate the grain Reynolds number in Eq. 5. For all experiments performed in this work, the mean sieve diameter (D_sv = 0.8 mm) of the glued sand bed was used in the Re* evaluation, as previously described by Diedericks et al. (2018).

Figure 8.
Shields diagram (θ_cr versus Re*) of the threshold of motion datasets from the bivalve shells Anomalocardia brasiliana, Codakia orbicularis, and Divaricella quadrisulcata. The data points include the experimental results for nonfragmented bivalve shells under currents from Dey (2003) and Diedericks et al. (2018). The Ramsdell and Miedema (2010) and Fick et al. (2020) curves (D/k_s = 6 and 20) are plotted together with the Shields (1936) curve.

As observed in the critical shear stress results, the θ_cr values were also larger for the case when the umbo faced downstream, ranging from 3.37×10^-3 to 8.19×10^-3, whereas in the upstream case, the values varied from 1.98×10^-3 to 6.73×10^-3. Notably, all data points plotted well below the Shields (1936) curve and Fick et al. (2020) envelope.

Regarding the highest critical Shields parameter values for the different species, Divaricella quadrisulcata was less resistant to the flow action, registering a θ_cr value equal to 4.71×10^-3. Anomalocardia brasiliana and Codakia orbicularis, by contrast, were hydrodynamically more stable, as expressed by correspondingly higher motion threshold values (θ_cr = 8.19×10^-3 and 6.96×10^-3, respectively).

The results of θ_cr were also compared with the Ramsdell and Miedema (2010) curves for the prediction of the incipient motion of nonuniform particles in a uniform sand bed with variable roughness. The curves indicate that the ratios between the nonuniform particle diameter (D) and average bed roughness (k_s) are equal to 6 and 20. Following Fisher et al. (1983) and Diedericks et al. (2018), D was chosen to be equal to some characteristic shell diameter (shell D_sv = D), and k_s is equal to the diameter of the sand bed particles (bed particles k_s = 0.8 mm). The D/k_s ratios between all shell sizes studied in this research and the fixed sand bed stayed between 6.23 and 24.20. Almost all shell data points were above the lowest curve (D/k_s = 20) of Ramsdell and Miedema (2010), and only for Divaricella quadrisulcata did no θ_cr value lie above the upper curve (D/k_s = 6).

In addition, Fig. 8 presents the θ_cr values calculated from the critical shear stress reported by Dey (2003) and Diedericks et al. (2018) for the threshold of motion of single shells under currents on a fixed sand bed (convex-up position/umbo facing downstream). All data from these authors have D/k_s ratios higher than 15 and θ_cr values far below the Shields (1936) curve.

Orientation: single-shell experiments

The valve orientations in the single-shell experiments, in which the umbo was initially pointing upstream, were computed for all the investigated species. In these experiments, the umbo orientation was controlled by the flow and shell morphology because the sand bed roughness was insignificant when compared to the shell size.

The rose diagrams (Fig. 9) show the final umbo orientations for all the specimens. The circular-shaped shell species (Codakia orbicularis and Divaricella quadrisulcata) presented a random orientation pattern. In contrast, the asymmetrical elliptical species, Anomalocardia brasiliana, exhibited a bimodal pattern when the orientation of the right valves was compared with that of the left valves.

Figure 9.
Umbo orientation of single shells resting in a fixed sand bed. Top, the elongate species Anomalocardia brasiliana discriminated between right and left valves. Below, the diagrams from the circular shells species: Codakia orbicularis and Divaricella quadrisulcata. n: number of measures; 0°: initial umbo position (upstream). The thin black arrows next to the shells depict the orientation convention. The large black arrow in the upper right corner indicates the flow direction.

Anomalocardia brasiliana showed a trend in which the right valve rotated counterclockwise, with the umbo orientation tending to point toward the right margin of the flume (270°), whereas the left valve rotated in the opposite direction and pointed the umbo toward the left margin. As shown in Fig. 9, all observations fit this pattern. Although the rotation motion occurred chaotically for the circular shells, it can be seen that Codakia orbicularis was mostly located in the second and third quadrants (Fig. 9), indicating that the umbo tended to point in a downstream orientation. In contrast, Divaricella quadrisulcata remained in a position near the starting umbo orientation (0°), probably because the shells of this species stopped their motion before a larger rotation.

Orientation: shell-bed experiments

The final orientation of the Anomalocardia brasiliana valves in the shell-bed experiments presented a more random picture compared to the single-shell experiments. This probably occurred because in this scenario, the hydrodynamically stable position of the shell was impaired by the neighboring shells that compounded the bed.

The rose diagrams (Fig. 10) show the umbo shell orientations that were in a convex-up position on the shell bed. Although these data have a large range of umbo directions, they agree with the tendency observed in the single-shell experiments. Thus, 62% of the umbo orientations (Table 4) on average point toward the right flume margin (third and fourth quadrants). This tendency was below 50% only in experiments 5 and 7. The largest number (77%) of right valves that followed this behavior was observed in experiment 2. In addition, as seen in the single-shell experiments, the results for the left valves show that, on average, 76% of the umbos pointed toward the left flume margin (first and second quadrants).

Figure 10.
Umbo orientation of the shells in a compound bed of Anomalocardia brasiliana right and left valves for the eight experiments. n: number of measurements; 0°: upstream flume side. The thin black arrows next to the shells depict the orientation convention. The large black arrow in the upper right corner indicates the flow direction.

Table 4.
Umbo orientation results for the Anomalocardia brasiliana: left and right valves.

Shell-bed experiment	Flow velocity (ms-1)	Left valves		Right valves
		n	Quadrants first and second (%)	n	Quadrants third and fourth (%)
1	0.57	15	73.33	09	66.67
2		11	81.82	13	76.92
3	0.48	22	81.82	18	61.11
4		25	56.00	28	64.29
5	0.54	21	95.24	11	45.45
6		17	82.35	31	74.19
7	0.52	20	80.00	17	47.06
8		28	60.71	24	58.33
Average			76.41		61.75
Standard deviation			12.73		11.40

DISCUSSION

Threshold of motion

As reported by Futterer (1982) for some bivalve shells (Glycymeris glycymeris and Mytilus edulis), the results of this laboratory investigation demonstrated that shells in a convex-up position were hydrodynamically less stable when the umbo was pointing upstream. In this scenario, the shells began to move downstream while, at same time, rotating around their vertical axes to adopt the more stable umbo-downstream position. Dey (2003) linked this rotational movement to the fact that an inertial movement is induced by an imbalance due to the protruding umbo.

Concerning the differences in the thresholds of motion between species, Divaricella quadrisulcata had smaller values of τ_b,cr and θ_cr in comparison to the other shells. One of the causes that can explain this behavior is the relationship between the shell mass (m) and its projected area facing the flow direction (S_ac). Because the drag forces are a function of this area (Eq. 1), heavier grain particles with the same projected area will be more stable. This relationship (Fig. 11) showed upward curving trends, with that of Divaricella quadrisulcata having the least steepness (R² = 0.97; m = 0.0013 S_ac^1.42). Anomalocardia brasiliana (R² = 0.99; m = 0.0007 S_ac^1.62) and Codakia orbicularis (R² = 0.97; m = 0.0008 S_ac^1.60) showed similar trends but with greater steepness than Divaricella quadrisulcata.

Figure 11.
Shell mass as a function of projected shell areas in the direction of the flow (S_ac).

These thresholds of motion differences between species may be related to the results of Fornari (2010), who investigated a concentration of disarticulated, convex-up shells in a lagoonal facies of a Holocene back-barrier system located in the southern Brazilian coastal plain. In this case, the highest shell concentration was composed of Anomalocardia brasiliana valves, which is an autochthonous species in that habitat.

The species Divaricella quadrisulcata and Codakia orbicularis (both native in shallow marine environments) occurred in overall smaller number wherein the former occurred more frequently than the latter. The fact that Divaricella quadrisulcata has lower values of critical shear stress may be one of the factors that explains its higher concentration among the allochthonous species, as the latter shallow marine species are considered to have been transported by the tidal flood currents.

Regarding the Shields parameter, all thresholds of motion data acquired in the present study were obtained from a transitional flow regime (5 ≤ Re* ≤ 70) with θ_cr values far below those of the critical Shields (1936) curve. Incipient movement studies on beds composed of heterogeneous calcareous sand (Prager et al. 1996) and shell fragments (Paphitis et al. 2002, Weill et al. 2010, Rieux et al. 2019) have reported critical Shields parameter (θ_cr) between 8×10^-3 and 4×10^-2 for transitional flow regimes. These θ_cr values, which were also below the Shields curve, could be a consequence of the platy shape and density of some bioclastic particles (Prager et al. 1996). It should be noted that terrigenous materials composed of platy particles have lower thresholds of motion than the corresponding spherical ones, as observed by Mantz (1977) and Magalhães and Chau (1983) in cohesionless mica flakes and shale particles, respectively.

Fick et al. (2020) integrated some of these unidirectional flow data with their threshold of movement results for disarticulated bivalve and gastropod shells under oscillatory flow, generating an incipient shell bed motion envelope. The results from the present investigation, as well as those of Dey (2003) and Diedericks et al. (2018), showed that single shells in a bed composed of smaller particles (and roughness) had lower thresholds of motion under currents with θ_cr values (Fig. 8) lying below the Fick et al. (2020) envelope.

Theoretical models and experimental data (Egiazaroff 1965, Andrews 1983, Fisher et al. 1983, Wiberg and Smith 1987, Ramsdell and Miedema 2010) have indicated that for spherical grains, if the D/k_s ratios are lower than 1, an increase in the critical Shields parameter is obtained. When this ratio is greater than 1, the θ_cr values are displaced below the Shields (1936) curve. The present θ_cr values, which were below the Fick et al. (2020) envelope, can be explained by this relationship. In the case of Ramsdell and Miedema (2010), this is caused by the fact that the Shields parameter is inversely related to the particle diameter, while the boundary shear stress is only influenced by the change in velocity distribution. In addition, for natural sand, crushed quartzite, and glass spheres, Miller and Byrne (1966) experimentally verified that the repose angle (ψ_r) decreases with an increase in the D/k_s ratio. According to Eq. 4, this relationship suggests that the critical Shields parameter will be lower for larger D/k_s ratios.

The θ_cr scatter points for all three bivalve shells (Fig. 12) tested in this work, where the ratio (D/k_s) was between 15 and 25, show a good positive correlation (R = 0.93), with a coefficient of determination (R²) of 0.86, and the thresholds of motion given by θ_cr = 9×10^-6 Re*^1.83. The results are very similar to those reported by Dey (2003) (R² = 0.80; θ_cr = 9×10^-6 Re*^1.88) under the same conditions. These trends are steeper than the curve (D/k_s = 15) by Ramsdell and Miedema (2010), crossing it at Re* ~ 25.

Figure 12.
Critical Shields parameter (θ_cr) as a function of the grain Reynolds number (Re*). (A) The results of this work discriminated between the shell species (Anomalocardia brasiliana, Codakia orbicularis, and Divaricella quadrisulcata), initial umbo position (u.d.: umbo downstream; u.u.: umbo upstream) and D/k_s ranges (6–15, 15–25). Blue dashed line is the regression curve (y₂) for D/k_s between 6 and 15, red dashed line is the regression curve (y₁) for D/ks between 15 and 25. (B) Results from Dey (2003) for the Coquina clam shells, Ponderous Ark, and transported in a convex-up/umbo upstream position with D/k_s ratio between Cross-barred Chione 6 and 15. Red dashed line is the regression curve (y₃). The Ramsdell and Miedema (2010) curves for D/k_s equal to 6, 15, and 20 are plotted on both graphs.

The data points for the shells studied in this research, in which 6 < D/k_s < 15 (Fig. 12), showed a larger scatter (R² = 0.33; θ_cr = 10^-4 Re*^1.15). Almost half of these data were between the regression curves (D/k_s = 6 and 15) of Ramsdell and Miedema (2010); however, there were also θ_cr values above the upper limit (D/k_s = 6).

Data from the literature show that beds made of carbonate particles have lower values of the critical Shields parameter than spherical siliciclastic grains when using the shell D_sv for the grain characteristic diameter. However, the results of this work, as well as the data from Dey (2003) and Diedericks et al. (2018), revealed that the same did not hold for nonuniform particles when comparing them with those of Ramsdell and Miedema (2010).

Shell orientation

The shell orientation results showed that elliptical and asymmetrical valves (Anomalocardia brasiliana) that moved in a convex-up position tended to point the right valve umbo toward the right side of the flume and the left valve to the opposite side, thus producing a bimodal orientation pattern. This behavior, already described by Kidwell et al. (1986), did not occur in circular shells. From another point of view, Anomalocardia brasiliana specimens have a longer axis parallel to the flow with the posterior end pointing downstream.

In addition, Newell et al. (2007) reported that the posterior sides of bivalve shells tend to face downstream (Fig. 13). The authors made more than 500 orientation measurements of the bivalve shell Unio sp. (elliptical asymmetric shape) in convex-up positions in recent point-bar deposits. Although other data of elliptical bivalve shells in convex-up positions show that the long axis is parallel to the flow direction (Behrens and Watson 1969, Futterer 1982), the species Donax variabilis and Petricola pholadiforms have more stable positions with the posterior shell end pointing upstream. Elongated triangular shells (Mytilus edulis) also produce a long-axis orientation pattern parallel to the flow (convex-up transported), as seen by Kelling and Williams (1967) and Nagle (1967). These data demonstrate that the major axis is the main controller in the orientation of elongated shells: however, the bimodal pattern of umbo orientations in elliptical shell species reveals that the umbo produces a secondary effect in shell orientation.

Figure 13.
Shell orientation under unidirectional flow. (I) Final orientation of the shells transported in a convex-up initial position. (II) Final orientation of the shells transported in a concave-up initial position. (A) Elliptical shell shapes. (B) Circular shell shapes. (C) Triangular shell shapes. Question mark next to D. quadrisulcata refers to this species umbo orientation results pointing upstream which is in contrast with his more hydrodynamic position (umbo to downstream). The shell photographs are modified from Zenetos et al. (2009), Fassatoui et al. (2019) and World Register of Marine Species (2022).

The majority of shell orientation data comes from experiments conducted with the shell in a convex-up position because it is a hydrodynamically more stable position, as well as being the regular position of shells on rock surfaces or sediments in response to wave and current action (Bailey and Erickson 1973, Kreisa and Bambach 1982, Posenato et al. 2013). However, it is known that shells can be transported over short distances in concave-up positions until they flip over to adopt a more stable position (Brenchley and Newall 1970). Under such conditions, the transport orientation of the elongated shells differs distinctly from that of the convex-up shells, where the longer axis tends to stay perpendicular to the flow direction.

The literature on the orientation of circular shells (Rhipidomella sp., Andara ovalis, Chione cancellata, and Glycymeris glycymeris) moving in a convex-up position reveals that the umbo tends to point downstream (Nagle 1967, Behrens and Watson 1969, Futterer 1982). The Codakia orbicularis species’ results agree with these data, as its threshold of motion data showed larger critical shear stress values for downstream facing umbos, and their umbo final orientation measurements also pointed in the flow direction. Although Divaricella quadrisulcata is more hydrodynamic when its umbo faces downstream, the final orientation measurements showed that when this species started to move with the umbo pointing upstream, it could maintain this orientation until its movement stopped. Due to these distinct observations, it is difficult to infer how Divaricella quadrisulcata valves would be oriented in the paleocurrent analysis.

CONCLUDING REMARKS

Using laboratory flumes, this research investigated the threshold of motion and final orientation of the bivalve shells Anomalocardia brasiliana, Codakia orbicularis, and Divaricella quadrisulcata when transported by currents. The thresholds of motion results achieved led to the following conclusions:

• Valves in a convex-up position are entrained at a higher flow energy when the umbo is initially pointing downstream rather than upstream;

• The threshold of motion differences between species can be related to the shell shape with regard to the relationship between the shell mass and frontal shell area facing the current;

• Shells transported over a bed of low roughness have critical transport thresholds well below the Shields curve;

• Elliptical shells transported in a convex-up position tend to be oriented with the long axis parallel to the flow, with the umbos of opposite valves pointing toward opposite flow margins (bimodal pattern). This pattern was observed in both scenarios tested in this research: single-shell and shell-bed experiments;

• Circular shells transported in a convex-up position tend to be oriented with the umbo pointing downstream;

• Valves transported in a convex-up position with their umbo initially pointing downstream will continue to move. However, when the umbo faces upstream, the shells rotate on a horizontal plane until a hydrodynamically more stable position is attained and then stop moving.

ACKNOWLEDGMENTS

We thank the Instituto de Pesquisas Hidráulicas (IPH), the Núcleo de Estudos de Correntes de Densidade (NECOD), and the Programa de Pós-Graduação em Geociências from the Universidade Federal do Rio Grande do Sul. We are thankful to Pré-Sal Petróleo S.A. (PPSA) and the Libra Group, a consortium comprising Petrobras, Shell Brazil, TotalEnergies, CNODC and CNOOC Limited, in partnership with UFRGS (Project 826-IPH/PETROBRAS 5850.0105486.17.9 COQUINAS) for their financial support to the first author with special thanks to Maria José Resende Oliveira (PETROBRAS/LIBRA/AT) and Jian Zhao (CNODC BRASIL PETRÓLEO E GÁS LTDA/LIBRA/AT). We also appreciate the valuable comments of the reviewer Marcello G. Simões.

NOMENCLATURE

A = wet area (m²)

a = shell's longest axis (m)

b = shell's intermediated axis (m)

c = shell's inferior axis

D_n = nominal diameter (m)

D_sv = sieve diameter (m)

D/k_s = ratio between shell D_sv and mean bed D_sv

F_D = drag force (N)

F_L = lift force (N)

h = flow depth (m)

P = wet perimeter (m)

Q = discharge (m³ s^-1)

R = Pearson correlation coefficient

R² = coefficient of determination

Re* = grain's Reynolds number

S = flume slope

Sac = projected area on plan formed by axes a and c (m²).

u_* = shear velocity (m s^-1)

V = shell volume (m³)

W = submerged weight (N)

γ_s = specific weight of shell (N m^-3)

γ = specific weight of water (N m^-3)

θ_cr = critical Shields parameter

ρ = specific mass of water (Kg m^-3)

ρ_s = specific mass of shell (kg m^-3)

τ_b,cr = critical shear stress (Pa)

ψ = bed slope angle

ψ_r = repose angle

ARTICLE INFORMATION

• Manuscript ID: 20220080.
  How to cite this article: Silva F.R.S., Borges A.L.O., Toldo Jr. E.E., Fick C., Puhl E., Oliveira V.C.B., Cruz F.E.G. 2023. Threshold of motion and orientation of bivalve shells under current flow. Brazilian Journal of Geology, 53(1):e20220080. https://doi.org/10.1590/2317-4889202320220080

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Publication Dates

• Publication in this collection
  08 May 2023
• Date of issue
  2023

History

• Received
  16 May 2022
• Accepted
  15 Feb 2023