ANALYSIS OF CURRENT CLOSE TO THE SURFACE OF NET STRUCTURES Mathias Paschen, University of Rostock, Germany

ANALYSIS OF CURRENT CLOSE TO THE SURFACE OF NET STRUCTURES Mathias Paschen, University of Rostock, Germany mathias.paschen@uni-rostock.de Abstract The development of the theory of (pelagic) trawls is stagnating for more than twenty years although numerical methods have been rapidly developed for a structure analysis of snowballing fishing gears. The reason for this aberration in theory is based on the assumption that all parts of the trawl are exposed to the same undisturbed water-current regarding value and direction. Koritzky already verified by numerical as well as experimental investigations in 1974 that fluid-structure interactions have to be considered in many cases of parametercombinations regarding angle of attack, d/a-ratio as well as mesh opening. This is very important especially for the aft part in the fishing gear. Therefore the author and his students have started to analyze systematically the physical phenomenon of hydro elasticity of net-like structures as a basis for a trustworthy prediction of hydrodynamic loads of fishing gears some years ago. Goal of the present paper is to present selected results of both numerical simulations and wind tunnel tests of current close to grids in case of a small angle of attack. Introduction The hydro-elasticity plays an important role in the computation of flow-loaded flexible net structures, whenever the current is significantly affected by the presence of the net. This becomes especially evident in the analysis of trawls. If the shape of net is assumed to be constant, then the water volume flowing into the trawl mouth needs the same time unit Δt to flow along the netting and out of the fishing gear. This correlation can be presented in a mathematical way as follows: v n A da = v n S ds, (1) with v being the speed vector, A the cross-sectional area of the net mouth, S the coat surface of the trawl and n A and n S the perpendiculars to the surfaces A and S. 307

DEMAT '09 EXPERIMENTAL INVESTIGATION OF FISHING GEAR In general, equation(1) is only fulfilled in case the speed v changes along the netting surface in relation to the original speed in the range of the net opening. The change of speed of a liquid particle on its way inside the net results from the presence of the fishing gear itself. As known it can be described as sum of local and convective acceleration (Gl. 2). dv dt = v t + (vt )v. (2) The degree of influence the fishing gear has on the current directly depends on the fishing gear s hydrodynamic transparency. The wider the mesh is opened, the smaller the d/a -ratio and the larger the local angle of attack with respect to the undisturbed inflow, the less the current is affected by the net and the higher is its hydrodynamic transparency. Nowadays modern methods of computer fluid dynamics already allow complex flow simulations for complicated tasks. It is possible to combine CFD methods with algorithms for the computation of trawls. Thus the relations from equations (1) and (2) are also considered. However, those simulation programs will be much more complex, more complicated and also more expensive than the ones we know so far. As a consequence thereof higher intellectual requirements will have to be fulfilled by their user. The author assumes, that apart from the methods predicted developers of fishing gears for various reasons want to use more simple methods in the future, too. One set-up could be for example to extend hitherto existing computation algorithms by special flow analyses to those areas of the trawl characterized by a particularly small hydrodynamic transparency. This refers to the tunnel and to the cod-end of trawls. This paper is meant to give a first setup in this matter. Because the currently most important simulation methods for computing trawls are reasonably restricted to steady motions, we want as well assume the motions of the fishing gear to be straight-lined and homogeneous in our following considerations. This assumption includes that the shape of the fishing gear can be regarded as temporally constant. The liquid regarded has the following characteristics: infinitely expanded, incompressible and 308

resting in the infinite. The assumption of the fishing gear s motion being straight-lined and homogeneous does not restrict the analysis of fluid-structure interactions. Anyhow it is useful, since the local acceleration disappears in this consideration. Only the convective acceleration (v T )v is left. It only depends on the location. The geometry and the surface properties of the structure in flow has to be known, of course. In the given paper the author concentrates on the experimental and theoretical flow analysis at a net grid. The experimental investigations have been conducted in the large subsonic wind tunnel of the Department of Mechanical Engineering and Marine Technology of the Rostock University. The grid analyzed is aligned parallel to an even base plate as well as parallel to the undisturbed incident flow. Its geometrical angle of attack is zero for this reason. The distance between base plate and grid was about h = 160mm. End plates contactlessly attached to the edges of the grid avoid a three-dimensional flow passing the grid. At the end of the grid an even plate is attached contactlessly transverse to the flow, see figure 1. The plate can be exchanged by plates of various transparency. It simulates for Fig. 1 Experimental rig [Ristow] example the presence of fish caught in a cod-end. The flow field measurements 309

DEMAT '09 EXPERIMENTAL INVESTIGATION OF FISHING GEAR near the grid take place contactlessly by means of Particle Image Velocimetry (PIV), see figure 2. The resulting flow forces as well as the pressure point at Fig. 2 Position of the PIV-system in the wind tunnel [Ristow] the grid are determined by means of a six-component balance. In this respect the present paper subsequently follows to the experimental investigations by Paschen, Knuths, Winkel, Ristow. The theoretical flow investigations with perpendicularly arranged plate are conducted applying the method of conformal mapping. It is the aim to examine whether a description of the experimentally determined flow field can also be represented by means of the potential theory. Finally a mathematical set-up is introduced, which permits a conversion of well-known hydrodynamic factors at grids with free flow around them to those at grids with blockage. The findings are presented and discussed. 310

Methodical procedure and first results Prediction of flow by means of conforming mapping The function ζ = z a2 (3) z where ζ = ξ + i η and z = x + i y allows to gain the flow past an even plate arranged transverse to the inflow velocity v 0 from the potential flow past a circular cylinder according to figure 3. One point Q(ξ, η) of the plane ζ corresponds to Fig. 3 Adoption of the image function ζ = z a2 z one point P(x, y) of the plane z. By definition the flow past the circular cylinder is shown in the z-plane. The plane ζ corresponds to the plane examined in the experiment. The experimentally analyzed area between the base plate and the horizontally arranged net grid with 0 <η h at different distances ξ to the perpendicularly arranged plate is of special interest. The calculation of the flow distribution in plane ζ is done in 4 steps: 1. Defining the point Q(ξ, η) in the ζ-plane for computing the flow velocity v ζ (ξ, η); 311

DEMAT '09 EXPERIMENTAL INVESTIGATION OF FISHING GEAR 2. Calculating the coordinates of the corresponding point P(x, y) in the z- plane; 3. Computing the velocity v z (x, y) in point P(x, y); 4. Counting the velocity v ζ (ξ, η) in point Q(ξ, η) on the basis of the known velocity v z (x, y). Below the necessary mathematical equations are given: The coordinates of the complex ζ-function, see equation (3), follow from ξ = x (1 η = y (1 + a2 x 2 + y2) (4) a2 x 2 + y2), (5) If the center of the circle coincides with the point of origin of the z - plane and R = a than we get an un-curved plate in the ζ - plane. Its length is l = 4 a according to figure 3. For this reason is R = a = 1 2h. The correlation between the points on the circle (x, y) and the points on the plate (0,η) is given according to the equations (4) and (5) by ξ = 0 und η = 2 y. (6) We locate a circular cylinder with an infinite length perpendicularly to the z - plane. The ambient fluid is assumed incompressible and inviscid. The fluid motion is irratotional. A velocity potential Φ can be used to describe the fluid velocity vector v(x, y) = (v x, v y ) at the point P in the z- Cartesian coordinate system. That means that v(x, y) = Φ x e x + Φ y e y (7) where e x and e y are unit vectors along the x- and y-axes, respectively. In case of a still circular cylinder in a uniform current the velocity potential (7) has to fulfil the two following random conditions: 1. The current of the fluid has to be undisturbed in a remoteness from the cylinder, i.e. v x = v and v y = 0. 2. The velocity of the fluid perpendicular to the circle s surface has to be zero in case of x 2 + y 2 = R 2. 312

The velocity potential according to equation (8) fulfils these conditions. ) Φ= v x (1 + R2 (8) x 2 + y 2 R is the radius of the cylinder and v is the velocity in an infinite distance in front of the body. Therefore the components of the velocity v x und v y are v x = Φ x v y = Φ y = v 1 + y2 x 2 ( x2 + y 2) 2 R2 (9) = v 2 x y R 2 ( x2 + y 2) 2. (10) The velocities v z and v ζ of corresponding points in both planes are inversely proportional to the segments dz und dζ. That means v ζ = dz dζ v z. (11) We have to note that at this juncture v z and v ζ are the conjugate-complex terms of the respective velocity vectors, see Betz, i.e. v ζ = v ξ i v η and v z = v x i v y. The absolute value of dz dζ results from equation (3). That means dz dζ = 1 1 + (. (12) ) a 2 2 x 2 +y + 2 a 2 (x2 y 2 ) 2 (x 2 +y 2 ) 2 The application of the conformal mapping for the parameter η 120 mm and η 160 mm used in the wind-tunnel tests leads to the curve progressions v ξ v (triangles) and v η v (diamonds) shown in figure 4. One can see that the plate at ξ = 0 influences significantly the velocity in its neighborhood. As expected the horizontal component of the velocity decreases whereas the vertical component of the velocity increases. To carry out advanced numerical analyses it is more feasible to approximate the velocity components v ξ as well as v η by means of polynomials like equations (13) and (14): 6 v ξ (ξ) = a i ξ i v (13) and v η (ξ) = i=0 6 b i ξ i v (14) i=0 313

DEMAT '09 EXPERIMENTAL INVESTIGATION OF FISHING GEAR Fig. 4 Calculated (standardized) velocity distribution v ξ v We found following coefficients: and v η v according equation (11) a 6 = 24.207 a 5 = 80.436 a 4 = 107.88 a 3 = 75.144 a 2 = 28.899 a 1 = 5.8768 a 0 = 0.4945 b 6 = 13.039 b 5 = 49.411 b 4 = 78.038 b 3 = 66.544 b 2 = 32.874 b 1 = 9.1657 b 0 = 1.1783 The ratio of v η and v ξ leads to the angle of attack α of the velocity vector v against the abscissa. tan α(ξ) = v η(ξ) v ξ (ξ). (15) We assume that the current will not be significantly influenced by the horizontally positioned grid during the wind-tunnel tests. When we do this then we nearly know the angle of attack α in the experimentally analyzed grid position of 120 [mm] η 160 [mm]. We get α(ξ) α grid (ξ) and α grid (ξ) arctan v η(ξ) v ξ (ξ). (16) 314

Equation (16) leads to the graph in figure 5. Fig. 5 Angle of attack α α grid in case of 120mm η 160mm Prediction of load Lift F l is theoretically described by Kutta and Joukowsky for many applications of slender bodies like wings, plates, rotating cylinders, etc. F l = ϱ b v Γ (17) Here ϱ, b, v and Γ mean respectively the density of the fluid, the extension (breadth) of the analyzed structure perpendicularly oriented to the direction of flow, the undisturbed current far from the structure and the circulation. The direction of the lift is well-defined by the unit vector e l as e l = v (e n v ). v (e n v ) e n is the normal unit vector of the structure surface in a Cartesian coordinate system fixed in the space. The formation of lift always necessitates the existence of circulation according to equation (17). The author and his collaborators analyzed the characteristic of current close to net panels, net grids as well as netting cones in a big number of different experimental investigations. But, the existence of a circulation had never been 315

DEMAT '09 EXPERIMENTAL INVESTIGATION OF FISHING GEAR observed along both a single mesh and the whole net structure. That means the hydrodynamic force of a net grid F η which is passed through by a fluid and perpendicularly oriented to the direction of the undisturbed velocity far from the net v is definitely no lift. The physical source of this load we can find in the existence of both convective inertia effects and loads due to viscosity and eddy generation of the fluid according to formula (20). df η = dv η dt F η = = (m ηη ) (m ηη ) dm ηη + c d90 ρ 2 n 0 d v 2 η dξ (18) dv η dt ( vη t dm h + c d90 ρ 2 n 0 d (l) + v η ξ v ξ + v ) η η v η dm ηη + v 2 η dξ (19) + c d90 ρ 2 n 0 d v 2 η dξ (20) (l) m ηη represent the added mass of the investigated grid. We assume that dm ηη = m ηη dξ whereas ξ m ηη n 0 π ξ 4 d2 ρ = const. c d90 represent the drag coefficient of a circular cylinder in case of transverse flow, n 0 and d mean the number and the diameter of transverse mesh bars of the grid, respectively. Because, the tests were carried out at v = const. the local accelerations v ξ t and v η t are zero v ξ t = v η t = 0. We may postulate an incompressible fluid. Because of that we find based on the equation of continuity v η η = v ξ ξ. Thus the load F η is given by equation (21) F η = m ηη ξ 0 l [ vη ξ v ξ v ξ ξ v η ] dξ + c d90 ρ 2 n 0 d 0 l v 2 η dξ. (21) 316

The position of the pressure point of the grid ξ cp relating to the rear edge of the grid is defined as ξ cp l = 1 l m ηη ξ 0 [ vη l m ηη 0 ξ l Here l is the length of the grid. ξ v ξ v ξ ξ v ] η ξ dξ + cd90 ρ 2 n 0 d 0 l v2 η ξ dξ [ vη ξ v ξ v ξ ξ v ] η dξ + cd90 ρ 2 n 0 d l v2 η dξ. (22) The hydrodynamic force F ξ of the grid is generated by viscosity and eddygeneration of the real fluid. In the following we only consider the reduction of velocity v ξ ξ < 0 due to the rotation of v according to figure 4. Then we can formally write the non-linear drag term as df ξ = c d0 1 2 ρ v ξ 2 b dξ. (23) c d0 is the drag coefficient of the grid in case of parallel inflow. We insert v ξ according to equation (13) into equation (23) and get F ξ = c d0 1 2 ρ b 0 v 2 ξ l dξ (24) F ξ = 0.949 c d0 1 2 ρ v 2 b l. (25) The result show that the drag of the grid is about five percent less in comparison to a grid in undisturbed current with v ξ = v. Results The grid used for the experimental investigations in the wind tunnel was made of aluminium. Its dimensions were 1450 mm in length and 750 mm in breadth. The mesh bars consisted of smooth tubes. Diameter and mesh size were d = 10 mm and a = 83 mm, respectively. The hanging ratio of the mesh was u 1 = 0.5. To compare the calculated angle of attack plotted in figure 5 with experimental data we are going to use the measured distribution of streamlines, see figure 6. A comparison shows a good qualitative correlation between experiment and theory. Unfortunately, the measurement of the velocity distribution below the mesh knot at the rear edge close to the vertical plane was disturbed by the shadow of the mesh. A measurement of the velocity was also impossible behind the vertical plane because of the measuring conditions. 317

DEMAT '09 EXPERIMENTAL INVESTIGATION OF FISHING GEAR Fig. 6 Streamlines close to the rear edge of the grid Figure 7 illustrates both calculated and measured dimensionless coefficients of vertical load. One can see two significant results: a remarkable difference between theory and measurement as well as a big gradient of c η ξ. Probably the definition of the drag of the grid by coefficients of smooth cylinders according to equation 20 is too rough. Beyond that it is possible that the computed velocity is too high because viscosity effects are neglected. Fig. 7 Dimensionless coefficient of vertical load c η, calculated as well as measured 318

Conclusions The theoretical considerations presented in this paper line up in basic investigations on fluid-structure interactions of trawls done during the last few years. The focus in this paper has been set on a simple mathematical model based on potential theory to analyze the effect of a fluid blockage on both the velocity distributions inside and load of a netting construction. The prospective aim is to introduce simple mathematical fluidic models into established computer programmes for quality-improving designs of fishing gears. First consolidated findings are available. References 1. A. Betz (1948): Konforme Abbildung.(Conforming mapping) Springer Verlag, Berlin etc. 2. H.H. Koritzky: (1973) Beitrag zur Bestimmung von Widerstand und Auftrieb ebener Netztücher. (Contribution to the determination of drag and lift of plane netting) Dr.-Ing.- Thesis, University of Rostock 3. M. Paschen, H. Knuths, H.-J. Winkel, E. Ristow (2007): Flow investigations of net panels for small angle of attack. In: Contributions on the Theory of Fishing Gears and Related Systems Vol. 5, Shaker Verlag, Aachen 4. E. Ristow (2010): Experimentelle Strömungsanalysen an einem Netzgitter bei bodenparalleler Anordnung und einstellbaren Strömungsleiteinrichtungen. B.Sc-Thesis, University of Rostock 319