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Velocity is an integral quantity of the instantaneous vorticity field. Specific geometries are represented using surface source and vortex panels whose strength is prescribed to satisfy the no-slip and no-flux boundary conditions. Vorticity is diffused from the vortex sheets onto the body surface to maintain a vorticity balance. Vorticity in the flow is specified at points and the vorticity at any other point in the field is obtained via linear interpolation. Interpolation is performed by constructing tetrahedra using Delaunay triangularization. Tetrahedra provide the control volume to integrate over to obtain the velocity and the connectivity of the control points provides a basis to construct derivatives. A Baldwin- Lomax eddy viscosity model was implemented into the solution algorithm to model turbulent flow effects. This method was then validated for two disparate flow cases—flow past an unmanned undersea vehicle (UUV) at Reynolds numbers of one million and unsteady flow development past a cone. Attached flow past the UUV was compared with empirical turbulent flat plate results. Quality of the flow past a cone was compared with data obtained with experimental data. Validation of this method allows for a subsequent simulation of a UUV recovery problem. INTRODUCTION: Undersea vehicle hydrodynamics pose significant challenges for the computation of complex, three-dimensional unsteady flow fields. Major examples include submarine maneuvering problems, low speed maneuvering and control and unmanned undersea vehicle (UUV) recovery. Many of these complex flows are characterized by the production of vorticity and its subsequent interaction with the vehicle. This is generally the case when a vehicle encounters the wake of another body. For cases where vorticity is dominant, a Lagrangian vorticity based method can be used to compute the complex unsteady two-body hydrodynamics. Here, incompressible, unsteady fluid flow can be characterized by the instantaneous vorticity field alone. For cases of flow past bodies, the vorticity distribution in the boundary layer and wake determines the characteristics of the entire flow field. Vorticity based methods would seem to be a natural fit to solving these types of problem. This technique utilizes the velocity-vorticity formulation of the Navier-Stokes equations to solve for these flow variables on a Lagrangian mesh (calculational points advected by the local flow). The solution methodology has distinct advantages over traditional methods that rely on fixed grid solutions. The Lagrangian vorticity method is essentially grid free. It does not rely on a grid (structured or unstructured) in the traditional sense to evolve the vorticity associated with the unsteady flow. Vorticity is continually generated at the surface and is represented by the computational points that are also continuously generated. This makes the method naturally adaptive to coherent vortex structures and since the vorticity is advected by the flow, it is subject to little numerical diffusion. The use of a Lagrangian mesh allows for the straightforward treatment of moving surfaces and does not require incorporation of additional terms due to non-inertial reference frames. In addition, multiple bodies are included in a straightforward manner. The traditional vortex method (Chorin, 1973) describes the vorticity field by means of isotropic elements or ‘blobs', which have a strength that depends only on distance from their center; a frequently used strength distribution is the Gaussian. Careful comparison with theoretical and experimental data of two-dimensional calculations using isotropic blobs of uniform size has been reported by Sethian and Ghoniem (2) for a backward-facing step. High-resolution 2-D computations for flow past a cylinder have been conducted by Koumoutsakos and Leonard (1995) and Subramaniam (1996) and represent the standard for using blob methods to compute unsteady flow past surfaces. The original method has the feature that the identity and location of neighboring blobs are not needed to compute the Biot-Savart integral (which determines the velocity), so the algorithm for this computation is simple. Later works (e.g., Koumoutsakos and Leonard, 1995) typically employ accelerated methods (Greengard and Rokhlin, 1987; Strickland and Baty, 1995) to avoid an order N2 calculation (where N is the number of elements). Even with these elaborations, the simplicity of an approach the authoritative version for attribution. based on the traditional method remains attractive. There remain significant difficulties when applying vortex blobs to flows past a surface, however. One is the fact that near the surface, the blob vorticity distribution (a Gaussian) actually penetrates the surface so a finite value of vorticity is on the inside of the surface. This can cause problems when computing the boundary conditions. Another is that without sufficient overlap of the blob radii, the computed velocity and associated vorticity field will be extremely noisy. In the current method, which will be presented later, the vorticity in the field is piecewise
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This overlap can be very difficult to maintain, especially near the surface as the blobs are advected by the flow. For this reason, most blob methods (e.g. Koumoutsakos and Leonard, 1995) employ periodic interpolation of the blob strength onto a regular grid to most efficiently maintain overlap to avoid noisy boundary conditions. This can lead to artificial numerical diffusion, however, and still does not solve the problem of vorticity penetration into the surface due to the blob function. Another difficulty with blob methods is the wide range of scales that emerge during the evolution of flows past a surface. An example is the anisotropy of the vorticity distribution in a thin boundary layer. For these cases, the gradient of vorticity in the normal direction is much greater than that in the tangential direction. This presents another compelling motivation for developing an algorithm not based on elements of uniform size, but one in which the size and shape of the elements adapt to the local spatial distribution of the vorticity field. Blobs, whatever their size, have a constant radius and are therefore isotropic in nature. For thin boundary layers, anisotropic elements are desired. As problems are extended to three-dimensions, the above problems are only exacerbated. For this reason, there are only minimal examples of vortex blob solutions for flow past surfaces (e.g. Gharakhani and Ghoniem, 1996) We have developed a novel technique directly solving the vorticity equation on a Lagrangian mesh. Vorticity is specified at points in the field and is linearly interpolated by constructing tetrahedra. This method has the advantage that the elements are connected allowing the vorticity to be locally approximated as a function of position. This approach was introduced by Russo and Strain (1994) who examined inviscid vorticity fields for two-dimensional flow. Huyer and Grant (2000) extended the method to examine viscous flow past streamlined and bluff bodies. The current approach treats fully three-dimensional flow past multiple bodies. The vorticity is determined on a number of points in the field. These points are then connected to form a set of tetrahedra via a Delaunay tetrahedralization algorithm. The vorticity at any other point in the region is then approximated by linearly interpolating the vorticity at the nodal points with given shape functions based on the geometry of the tetrahedra. First and second order derivatives are computed via a second order least squares formulation from the elements connected at a given node (Marshall and Grant, 1995). The vorticity field and the locations of the calculation points are updated at each time step. Since the formulation is a Lagrangian approach, the advection term is automatically included. Viscous diffusion is accomplished using both an effective diffusion velocity (i.e. Strickland and Baty, 1995) as well as the second order Laplacian. Integration of the Biot-Savart integral provides the velocities. Another difficulty in treating high Reynolds number flow is accounting for turbulence. Traditional vortex methods have relied on random walk and other stochastic methods to simulate the high Reynolds number turbulent flow. Thus far, most deterministic solutions for viscous flow have been limited to low Reynolds number laminar cases. In the present method, a deterministic approach utilizing the full viscous equations was desired. Therefore, a turbulence model has been introduced to account for the lack of spatial resolution required to properly treat fully turbulent flow directly. A Baldwin- Lomax turbulence model was implemented into the viscous solution methodology to better model the turbulent flow characteristics. To the author's knowledge, this has yet to be implemented in any vorticity based solution methodology. This paper includes unsteady computational data collected for flow past a UUV at a Reynolds number of one million and unsteady flow past a cone for a Reynolds number of 50,000. Attached turbulent boundary layer flow over the UUV is compared with turbulent velocity profiles obtained from empirical results by Spalding and Coles (see White, 1974). Time averaged wake velocity data for flow past a cone is compared with data presented by Calvert (1967). The docking cone and UUV are fixed in space and the flow field computed. These validation tests are performed to establish a high confidence level in the method. This is followed by a simulation of UUV recovery and discussion of the results. METHODOLOGY: Surface Definition: An example of an Unmanned Undersea Vehicle (UUV) surface mesh is shown in Figure 1. To construct the mesh, surface body points and unit normals are required. Pseudo points are placed just above the body points. A Delaunay tetrahedralization algorithm (Borouchaki and Lo, 1995) is then applied. This essentially constructs tetrahedra encompassing all points and leaves a convex surface. The surface then consists of the faces of the tetrahedra, which are exclusively made up of the original body points. The UUV surface was defined with 586 points and a total of 1150 surface panels. These panels are then used to define the surface source and vortex panels. Satisfaction of the Surface Boundary Condition: Each panel on the body surface carries two velocity generators: a surface vortex distribution lying in the plane of the panel and a potential source. The sources are needed, mathematically, to ensure the no-flux boundary condition is met the authoritative version for attribution. properly. Uhlman and Grant (1993) showed that as the number of panels increases to infinity, the source strength approaches zero since the surface vorticity distribution can satisfy the no-slip and no-flux boundary conditions simultaneously. Both distributions are taken to be uniform over an individual panel and lie in an infinitely thin sheet on the surface. Thus the vortex strength parameter characterizing a panel is the velocity jump across the panel. The velocity due to a potential source, σ, and vortex panel strength, on a surface S is:
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Equations 14 and 15 are then modified using the total viscosity defined as: (18) Vorticity Boundary Condition: After the surface source and vortex sheet strengths are computed it is necessary to transfer the vorticity in the infinitely thin sheets into the volume. This is accomplished by adjusting the surface nodal vorticity values to satisfy the no-slip boundary condition exclusively. To do this, surface vortex sheet values must be minimized thus requiring: (24) The left-hand side is the volume integral of the first vorticity layer at node m that must balance the vorticity in the infinitely thin layer expressed in the right hand side term. The vorticity is desired at body point m that can satisfy the no- slip boundary condition. An iterative scheme is then used to set the vorticity. The surface velocity boundary conditions are computed and the vorticity sheet strengths determined. The vorticity is then determined by equation 24. The surface velocity boundary conditions are then re-computed and vorticity sheet strengths computed again. This iterative procedure continues until the magnitude of the vortex sheet strength is below 0.01. This typically only requires 2–3 iterations and converges quite rapidly. The reason for this is that the layer of vorticity is very thin so there is little difference in the velocity generated by the infinitely thin sheet and the thin vorticity elements connected to the surface. Initial Volume Vorticity Distribution, Euler Layer and Point Creation: After the surface panels are defined, it is required to initialize the volume vorticity on a set of points. Nodal vorticity values located at the surface body points are used to define the geometry. Additional points are then placed in layers staggered over the body nodes and the surface panel centroids. To ensure that sufficient resolution of the boundary layer vorticity is maintained close to the surface, a thin layer of “Euler layer” of fixed points is used. Typically, the field points are located normal to the body nodal points in successive layers. This also allows for the computation of derivatives using standard finite difference formulas as an alternative to the least squares approach summarized in equations 6 and 7. Vorticity evolution for points in the Euler layer is performed using the Lagrangian form of 8–10 then interpolating the values back on to the original point positions. On the first time step, the Euler Layer consists of seven sub-layers and ten additional sub-layers of points are Lagrangian (advected by the local flow). Figure 2a shows a close-up view of the typical initial distribution of points and subsequent cross-section of the tetrahedral mesh for a cone. Figure 2-b shows the developed point distribution and mesh cross-section at a later time. Figure 2: a) initial point distribution and b) point distribution at t=3.0. On the first time step, an initial boundary layer thickness is assumed, which is 10% of a fully developed, turbulent boundary layer on a flat plate. The boundary layer is assumed to be attached and of uniform thickness modeling an impulsive start. The purpose of this is to more easily initialize the vorticity field with the vorticity remaining close to the wall. The vorticity on the surface is determined from the boundary conditions and the vorticity in the volume is assumed to exponentially decay. The vorticity values on the surface and in the boundary layer are then adjusted using an iterative scheme to satisfy the no-slip and no-flux boundary conditions. New Lagrangian points are continually created above the Euler layer. Extended panels are formed that are effectively the authoritative version for attribution. the outer faces of the tetrahedra formed in the Euler layer. Since the points in the Euler layer are not advected, the tetrahedra in this layer can be explicitly set for all times. These extended panels on the outer surface of the Euler layer have the same characteristics as the surface panel directly underneath. New Lagrangian points are formed by first determining the closest point to the extended panel that is above the panel. To determine if the point is above the panel, the centroid of the surface panel is determined and a radius constructed to include the three panel vertices. If a point lies within the radius, its normal component is computed. The closest normal point within the radius is then found. If this point is greater than a prescribed value, a new Lagrangian computational point is created. The new point is located in the tetrahedron formed by the closest point and the three points connected to the nodes of the extended panel. The vorticity is
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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SIMULATION OF UUV RECOVERY HYDRODYNAMICS 840 shots of the vector plots taken at x/l=0.7 for t=2.0, 2.1 and 3.0. The vectors are colored based on the vorticity magnitude with the scale shown to the right. Maximum vorticity magnitudes of 100.0 were chosen to illustrate the boundary layer flow. Maximum surface vorticity values exceeded 2,000. These plots appear very similar. Upon closer inspection, however, there are qualitatively small changes in the overall flow as the unsteady vorticity field is advected over the UUV. These plots illustrate the unsteady attached flow over the UUV for high Reynolds numbers. Boundary layer axial velocities over the UUV were time averaged between t=2.0 and t=4.0 to obtain mean velocity profiles and the turbulent intensities in the velocity components (u', v' and w'). Mean axial velocity profiles for Re=1,000,000 are plotted in Figure 8 for the current computations and for empirical results obtained from flat plate boundary layers for Spalding and Law of the wake corrections of Coles (see White, 10). As can be seen, the computed velocity profile agrees very well with empirical solutions from Spalding and Coles. Even in the inner layer, the agreement is good except for slightly lower velocities at y/δ between 0.01 and 0.04. Figure 9 shows the same mean velocity profile with the scales compressed to highlight the near wall behavior. At the wall, the computations agree very well. From y/δ=0.01−0.04, it appears that the computational solution bows out somewhat. Good agreement is seen thereafter from y/δ=0.04−0.1. Figure 10 shows plots of the turbulent intensities (u', v' and w') as a function of distance with comparisons with these statistics obtained by Klebanoff (see Hinze, 1975). Since a turbulence model is used, the turbulent intensity statistics were expected to be minimal. The computations, however, are showing fluctuations in all three velocity components due to the inherent unsteady computations of the present method. u' appears to show the best agreement although it does not reach the same maximum near the wall as Klebanoff's data. v' and w' show poorer agreement. This is likely due to the lack of spatial resolution of the surface. The fact that the turbulent intensities are non-zero and the fact that u' follows the same trends as the experimental results demonstrates the potential of the current code to represent turbulent flow without a turbulence model. Of course, the spatial resolution would need to be greatly increased to properly accomplish this. Unsteady flow past a 45° cone was also computed and results of the unsteady flow field can be seen in Figure 11. Here, vector plots, representing the instantaneous velocity field, are colored based on the z-vorticity component with the x- y plane displayed. The surface is shaded based on the vorticity magnitude. This plot compares the instantaneous unsteady flow at t=2.0, 3.0 and 4.0 for Re=1,000 and Re= 50,000. For Re=1,000, the flow was assumed laminar and for Re=50,000, the turbulence model was used. In both cases, a ring vortex is produced in the wake of the cone and can be seen to grow in size as time is increased. The laminar flow case shows a much more coherent, well-defined vortex compared with the turbulent flow case. Quantitative comparisons of this flow condition are shown in Figure 13 for laminar and turbulent flow computations. The z-vorticity component in the wake was computed at one cone diameter downstream of the stern of the cone (corresponding to the location of the vortex core). The computational data shows the instantaneous velocity field at t=4.0. As can be seen, data for Re=1,000 show significantly higher vorticity with peak values on the order of 20. Turbulent flow vorticity values for Re=50,000 are much more modest by comparison with values approaching 10. In addition, the vorticity for the laminar flow case appears to be Figure 12: Z-Vorticity distribution at t=4.0 in the wake of the cone at x=2.0 (relative to the cone apex) for Re=1,000 (laminar flow) and Re=50,000 (turbulent flow). the authoritative version for attribution. Figure 11: Unsteady flow development past a cone for the laminar flow case (Re=1,000) and the turbulent flow case (Re=50,000). Velocity vectors are colored based on the wake z-vorticity and the surfaces are colored based on the vorticity magnitude.
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As frequency is increased, the gains decrease significantly so that by 1 Hz, the gain is only on the order of 10−5 for the pitching moment. At even higher frequencies, the gain drops off sharply so that, in effect, the vehicle doesn't even respond to the forces. As could be seen from Figures 9–11, the high frequency vortex shedding resulted in force fluctuation periods on the order of 0.1 seconds for resulting frequencies of 10 Hz. The corresponding gain is on the order of 10−8 resulting in negligible change in UUV trajectory. Strouhal shedding frequencies on the order of 0.75 Hz were observed from the moment fluctuations in Figure 11. The corresponding gain in pitching moment is 1.356x10−5 and is 1.334x10−3 in vertical force. Assuming a forcing function that produces 1625 N-m (1200 ft-lbs.) of pitching moment and 280 N (63 lbs) of vertical force, the resulting change in trajectory is on the order of 9.1 mm (0.36 inches). This change should be corrected during a docking procedure and is easily within the window of error. Although the vortex shedding off of the cone will likely have minimal impact on the trajectory of the UUV, motion of the submarine or the presence of ocean currents will cause the cone wake to vary as well. A sinusoidal wave pattern could be established on the wake of the cone. For moderate ocean currents, and temporal variation with 10 second period superimposed on the cone wake would not be unreasonable. This wave pattern could result in pitch and moment fluctuations on the order of 0.1 Hz. The bode plots shows a gain in pitch moment of 3.8x10−3 and gain in vertical force of 8.58x10−2. For a 1625 N-m (1200 ft-lbs.) pitch moment amplitude and 280 N (63 lb) vertical force amplitude, this would result in changes in vehicle trajectory on the order of 1.37 meters (4.5 feet). This altered trajectory could be significant in terms of docking the UUV. Future flow calculations should focus on these types of phenomena. CONCLUSIONS: A novel Lagrangian vorticity method has been presented to compute the unsteady hydrodynamics associated with naval unmanned undersea vehicles. Since the flow is incompressible, the pressure term is not required for solution leaving a velocity-vorticity formulation. In fact, since the velocity is a direct integral quantity of the vorticity, this method demonstrates that the unsteady flow can be described by the vorticity alone. The Lagrangian nature of the calculation requires only that the diffusion term be solved explicitly. The advection term is automatically included since the points are moved with the local flow. The diffusion velocity concept is also used to move the points into regions of zero vorticity. This avoids the difficulty of establishing empty points to diffuse the vorticity onto. The diffusion equation was modified accordingly and solved at each time step. Effects of grid resolution and validation of the velocity calculation and diffusion algorithm were conducted by comparing computational results for analytical solutions for Hill's spherical vortex and a columnar vortex. The unsteady turbulent flow past a UUV at Re= 1,000,000 and the unsteady flow development in the wake of the cone were both investigated to demonstrate the effectiveness of the current method. Mean turbulent boundary layer velocity profiles agreed quite well with empirical results from Spalding and Cole. Turbulent fluctuations agreed surprisingly well for u' but not as well for v' and w'. This was expected due to lack of surface resolution. The fact that turbulent quantities are computed at all demonstrates the potential of this method to compute turbulence directly. Unsteady flow past a cone showed the vortex ring structure produced. Initially, the vortex ring was symmetric but developed asymmetries as the flow developed. The laminar flow case showed a much more coherent vortex compared with the turbulent vortex. This is exactly as intended and was the reason for implementing a turbulence model. Consequently, the turbulent flow results for the wake velocity profiles were in much better agreement with experimental test cases. Simulation of UUV recovery hydrodynamics displayed some interesting flow field phenomena. The unsteady flow development in the wake of the cone showed the vortex ring structure produced. Initially, the vortex ring was symmetric but developed asymmetries as the flow developed. As the asymmetry was manifested, shedding of vorticity was observed. The shed vorticity did not appear as well-formed vortex ring but as convoluted hairpin vortex structures. Proximity of the UUV affected the way in which the vorticity was shed. For cases where the UUV was close to the UUV, the separated vorticity region between the UUV and the cone maintained coherent vortex ring structures whose asymmetry varied with time. As the UUV was placed downstream, the vortex ring structure became elongated and elliptically shaped. Here, definite ‘chunks' of vorticity were shed which impacted the UUV. As the wake of the cone impacted the UUV, significant transients in local pressure were observed. There was a reduction in stagnation pressure at the nose of the UUV and increase in pressure at the point of maximum radius of the the authoritative version for attribution. UUV. Pressure fluctuations were observed as the vorticity dominated wake impacted the nose of the UUV. Calculations at mid-body showed fluctuations in pressure as well. Even though there was large pressure fluctuations, integrated force computations showed little fluctuations during initial wake impact. Only as the flow became developed were significant force fluctuations seen. Also, the magnitude of the force fluctuations rapidly diminished, as the UUV was placed downstream. Vertical placement of the UUV produced altered wake structure and unsteady loading on the UUV. Placing the UUV below the cone altered the vortex wake interaction with the UUV and also produced negative normal forces and nose down pitch moment. For this case, sinusoidal fluctuations in
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Representative terms from entire chapter: