_{1}

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Trawls and plankton nets are basically made up of conical and cylindrical net sections. In conical sections the flow will pass through the inclined net wall with a noticeable angle of attack, and then the flow, filtration and drag can be suitably modelled e.g. by a pressure drop approach [1]. In cylindrical and other non-tapered net sections, such as foreparts and extension pieces in trawls and plankton nets, the flow is directed along the net wall and is best considered in terms of a boundary layer. Boundary layer theory and turbulence models can be used to describe such flow, but this requires extensive numerical modelling and computational effort. Simplified approximate formulas providing a qualitative description of the flow with some quantitative accuracy are therefore also useful. This work presents simplified parametric expressions for boundary layer flow in cylindrical net sections, including the boundary layer thickness and growth rate along the net, the filtration velocity out of the net wall, the decrease in mass flux through the net due to the growing boundary layer, and the effect of twine thickness, flow (towing) velocity and the dimensions of the net. These expressions may be useful for assessing the existence and extension of a boundary layer, for appropriate scaling of boundary layer effects in model tests, for proper placement of velocity measurement probes, for assessing the influence on filtration and clogging of plankton net sections, and more.

When a cylindrical net is towed through water a boundary layer develops and grows in thickness along the inside and outside of the net wall. Mass conservation and pressure boundary conditions imply that a transverse velocity out of the net is induced and that the mass flux through the net decreases downstream. In practice, the boundary layer can be assumed to be turbulent all along the net, and classic turbulent boundary layer results may be used given that the velocity across the wall is small compared to the flow velocity outside the boundary layer, i.e. v/U < 0.01 [

Prandtl’s one-seventh power-law for the turbulent boundary layer along a smooth plate is given by Equations (1) and (2) [2,3]. Here x is the position along the wall (i.e. the distance from the net mouth), δ(x) is the boundary layer thickness at x, δ(x)/x is the boundary layer growth rate, y is the radial distance from the wall, u(x,y) is the boundary layer velocity profile, U is the undisturbed incident flow (towing) velocity, Re_{x} = Ux/υ and Re_{x} = UL/υ are the relevant Reynolds numbers, υ is the kinematic viscosity, L is the length of the wall, and c_{f} and C_{f} are the skin-friction and drag coefficients for (one side of) the wall, respectively.

Roughness primarily affects the skin friction coefficient c_{f}, which in turn affects the boundary layer thickness δ(x) and overall drag coefficient C_{f}. Considering the similarity between Prandtl’s expressions for δ(x)/x, c_{f} and C_{f} we now assume that a relative increase r_{ε} in c_{f} due to a roughness ε results in a corresponding relative increase in δ(x)/x and C_{f} also, cf. Equation (6). Roughness can be categorized in three regimes; smooth, intermediate and fully rough. Explicit relations for c_{f} exist for the smooth and fully rough regimes, cf. Equations (3) and (4) [2,3], while for the intermediate regime only a complex implicit relation exists (cf. Equation (6)-(82) in [_{ε} may now be estimated from Equations (3) and (4). The expression for c_{f}_{, smooth} is only slightly more accurate than c_{f}_{, Prandtl} in Equation (1), but more consistent to use in Equation (5) since c_{f}_{, Prandtl} could result in r_{ε}-values less than 1 in some cases. The expression for c_{f}_{,}_{ }_{fully}_{ }_{rough} applies to so-called sand-grain roughness, which is the most commonly used roughness model.

_{ε}(x) estimated from Equation (6) for some values of ε at U = 1 m/s, indicating that the boundary layer growth rate and filtration may be more than doubled for coarse netting compared to very fine netting. Note that while boundary layer thickness generally decreases with increasing velocity, the effect of roughness on the boundary layer thickness increases with increasing velocity, see

inserted directly into Equations (8)-(10).

Hence we assume that the boundary layer thickness along a cylindrical net can be approximated by δ_{ε}(x) in Equation (6), and that the velocity profile across the boundary layer can be approximated by u(x, y)/U = (y/δ_{ε}(x))^{1/7}. Under these assumptions expressions for the mass flux, radial filtration velocity and drag for a cylindrical net can be found analytically. The mass flux Q through a cross-section of the net is found by integrating the axial velocity over the cross-sectional area, assuming undisturbed flow outside the boundary layer (i.e. in the central core of the net), and using integration by parts for the power law expression, cf. Equation (9). Here R = D/2 is the radius of the net, r is the radial distance from the centreline of the net and y = R – r is the radial distance from the net wall. The radial velocity v(x) across the net wall is found by averaging the loss in Q from x to x + dx over the circumferential strip 2πRdx and making use of Equation (8), yielding Equation (10). The drag coefficient for the cylindrical net, normalized by the frontal area A_{0}, is derived in Equation (11), where L is the length of the net and τ_{w} = c_{f}∙ρU^{2}/2 is the wall shear stress. C_{f} typically lies in the range 0.001 - 0.010 [_{f} and C_{f} apply to one side of a wall, and that we must include both the inside and outside of a cylindrical net when calculating the drag.

The simplified expressions in Equations (9)-(11) are compared with experimental measurements from Vincent and Marichal (1996) [_{m} = 30 mm, for three undisturbed flow (towing) velocities; U = 0.29, 0.51 and 0.85 m/s. There are no noticeable differences between the measured boundary layers for the different velocities in [_{D}_{, A0} ≈ 0.65 for all three velocities (no drag forces or coefficients are given

in [

The simplified expressions presented here compare quite well with the measurements in [

The present approach may still provide useful estimates of the boundary layer flow in cylindrical net sections in a very simple manner. For fine-meshed netting such as in plankton nets the porosity is low and the roughness small, and then the present approach may be quite representative.