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Viscous Flow Past a Ship In a Cross Current V.C. Patel, S. Ju, I.M. Lew (Iowa Institute of Hydraulic Research, The University of Iowa, USA) ABSTRACT A numerical method for the solution of the Reynolds- averaged Navier-Stokes equations is used to calculate the viscous flow over the stern of a ship in a cross current, i.e., a ship in yaw. The solutions are started with assumed initial conditions downstream of the bow. The numerical results are compared with the limited data that are avail- able. Although the calculations are successful in describ- ing the port-starboard flow asymmetry and vortex forma- tion, the solutions indicate a need for a better resolution of the bow flow. INTRODUCTION Patel, Chen and Ju (1988, 1990) have recently devel- oped a numerical method for the solution of the Reynolds- averaged Navier-Stokes (RANS) equations and applied it to study the flow around ship hulls under the assumptions that the ship is symmetric about the vertical centerplane, and is advancing in a straight course in a calm sea. To the authors' knowledge, all viscous-flow calculations for ship hulls performed over the years with different methods have been made with these restrictions (see Patel, 1988, for a review). In this paper, extension and application of the numerical method of Patel et al. to asymmetric flow around the stern of a symmetric ship advancing in a straight course in a cross current, as depicted in Figure 1, are considered. This problem is equivalent to a ship in a uniform stream at an angle of yaw. From a basic fluid-flow perspective, this situation is more general than that of a body of revolution at incidence, which has been studied experimentally and numerically in many investigations (see, for example, Patel and Back, 1985~. We now have a more complex body at incidence. The resulting flow also bears some resem- blance to the flow around a turning ship. A study of the flow around a ship in a cross current is thus of practical interest and also of value in further developing the capa- bilities of modern numerical methods. Before describing the extension of the method, it is use- ful to make two important observations with regard to the present work. First, as in the previous work, we will be concerned with the flow downstream of some section in the middle body of the ship. This implies that there is some uncertainty in the establishment of proper initial conditions which reflect the flow over the bow. Secondly, experimental information on the viscous flow past a ship in yaw is limited and, therefore, the success of the method cannot be ascertained with confidence. CALCULATION OF ASYMMETRIC FLOWS A detailed description of the basic numerical method and its applications to symmetric flows about double models, i.e., with the water and keel planes treated as planes of symmetry, is given in Patel, Chen and Ju (1988, 1990~. The modifications in the method and the associated com- puter programs to calculate the flow around a ship double model at an angle of yaw are relatively minor. The main changes are concerned with the size and shape of the com- putational domain, the initial conditions, and the boundary conditions. These are outlined below. As shown in Figure 1, a positive value of yaw angle or is used to represent a cross current from the port side. A cylindrical (x, r, 0) coordinate system in the physical plane Is used for the velocity components. For the symmetric- flow calculations presented in Patel et al. (1988, 1990), the solution domain in the transverse plane was (0° < ~ < 90°), i.e., it extended from the keel plane, ~ = 0°, to the water plane, ~ = 90°. Plane-of-symmetry conditions were applied on these planes. For the current application, the required solution domain is (-90° < ~ ~ 90°), and plane-of-symmetry conditions are applied on the water plane at ~ = -90° and ~ = 90°, on the port and starboard (or, windward and lee) sides, respec- tively. The outer boundary of the solution domain is located farther than before, at r = 2.0, i.e., it is a cylindri- cal boundary, two ship lengths in radius. The number of grid points in the radial direction is increased to 40 to accommodate the more severe variations of flow quantities in that direction. The locations of the transverse sections (x= constant) and the number of grid points in the axial direction are the same as before (= 501. The number of grid lines in the circumferential direction was increased to 27. Thus, the present calculations have been performed with a 50 x 40 x 27 grid, which is close to the finest that could be accommodated on a CRAY XMP/48 supercom- puter. At the outer boundary of the solution domain, uniform- stream conditions, i.e., Ux = UO cos a, Ur = UO sin a sin 9, US - UO sin a cos 0, are imposed. The calculations were started downstream from the bow, with the most upstream transverse section located at x = 0.3. The pro- f~les used at this section were generated in a similar manner to that described in Patel, Chen and Ju (1990~. Briefly, it involves prescription of a girthwise distribution of the boundary-layer thickness 8, the friction coefficient Cf (= ~ Presently Manager of Technical Research Center, Daewoo Shipbuilding & Heavy Machinery Ltd., Korea. 2 Visiting Professor from Chungnam National University, Daejeon, Korea. 721

GUM, where U.; is the friction velocity), and the velocity at the edge of the boundary layer Us. These are used, together with the law of the wall and the law of the wake, to generate the profiles of the longitudinal velocity Ux inside the boundary layer, and the reduction from Ups to unity in the inviscid flow is assumed to take place as r2. Uniform stream conditions are specified beyond a distance of five boundary-layer thickness from the hull. In the first instance, Ur and US components are determined by interpolation in the boundary-layer profiles and the known conditions at the wall and in the uniform stream, and k and £ are obtained from correlations for a flat-plate boundary layer. As the solution progresses, the values of Ur, Us, k and £ within the boundary layer are updated by scaling those calculated at the first downstream station. This process is continued only for the first 20 global sweeps, and then the initial profiles are fixed. This procedure for the generation of the initial conditions does not affect the principal quantity, i.e., the axial velocity profile. However, it ensures that the subsequent solution is carried out with initial profiles of transverse velocity components and turbulence parameters which are compatible with the governing equations. In the present calculations, the girthwise distribution of the boundary-layer thickness and friction velocity were assumed to be the same as those in the symmetric (zero yaw) case but the velocity at the edge of the boundary layer was prescribed from an inviscid solution. This is obviously an approximation and the resulting initial conditions do not properly reflect all of the flow phenomena that may occur over the bow, particularly at an angle of yaw. In the calculations at nonzero yaw angles, some diffi- culties were encountered in the application of the boundary conditions in the ship centerplane and at the exit plane. As noted in Patel, Chen and Ju (1988), for the symmetric- flow calculations with a= 0, the ship centerplane was extended as a false wake plane, and the condition of sym- metry, Us= 0, was applied on it. Also, along the wake centerline, r = 0, the conditions Ur = Ue = 0 were implic- itly imposed. In the present application, the false wake plane is no longer a plane of flow symmetry although it is a plane of symmetry for the numerical and. Also, Ur and US do not vanish along r = 0. In fact, Ur and Ue change rapidly near the singular point r = 0 in the grid. These points are no longer calculated directly in the extended method. Because we are now considering a half plane of -90° < ~ < 90°, there are no explicit boundary conditions to be satisfied at the false wake plane. However, because of the rapid changes occurring in the velocity components in cylindrical coordinates near the singular point r = 0, increased accuracy and stability of the solution procedure was obtained by interpolating in these components in Cartesian coordinates, and then transforming back to the cylindrical coordinates. Finally, at the downstream (exit) boundary, the condi- tions Px' = 0 and (Ux, Ur, Us, k, £)Xx' = 0 are imposed, where x' denotes the direction of the uniform stream. This requires complicated interpolations near the exit plane as x' is different from the ship axis x. All values required in the exit plane were extrapolated linearly in the x direction first and then shifted by the amount of (Xf - Xf 1) tan a in the r direction, where Xf and Xf 1 denote, respectively, the x coordinates at the last and the next to last axial stations. RESULTS With the modifications described above, the numerical method was employed to calculate the flow around double models of the HSVA Tanker (Wieghardt, 1982) and the SR107 Bulk Carrier (Okajima et al., 1985~. The hull shapes are shown in Figure 2. Comparisons between calculations and experiments for both hulls with symmetric flow, at zero yaw, were presented and discussed in Patel et al. (1988, 1990) There is no experimental information with asymmetric flow around the HSVA Tanker but some measurements at a = 5° and 10° have been reported by Nishio et al. (1988) for the SR107. In the presentation of the results all quantities are made dimensionless by the reference velocity UO and the ship length L. Figures 3 and 4 provide an overview of the effects of yaw angle on the HSVA Tanker at Re = 5 x 106. The limiting, or wall streamlines determined from the calculated friction vectors on the hull are shown in Figure 3. The port and starboard asymmetry is very clearly seen. The regions of strong streamline convergence are those where the pressure and the friction coefficients assume low values. These are also the regions which are readily iden- tified in surface flow-visualizations using such techniques as oil flow, wool tufts, and dye injection. Figure 4 shows the velocity field at a section near the stern, x/L = 0.9. In general, the effect of the cross current is to drive the flow from the port to the starboard side. This produces a thinning of the boundary layer on the port side and a thickening on the starboard side. The secondary motion at this section reveals reversals in direction that are characteristic of converging limiting streamlines and vortex formation. They are also associated with a local decrease in the wall shear stress and thickening of the viscous layer. These flow features are particularly difficult to handle in space-marching numerical methods usually employed for the solution of the three-dimensional boundary-layer equa- tions. Here they are predicted without any special treat ment. The general features of these and other results presented below are qualitatively similar to those observed on a body of revolution at angles of attack. However, subtle differ- ences arise in the shapes of the distributions and in the magnitudes of the pressure and friction coefficients due to the present three-dimensional geometry and, in particular, due to the rapid changes in this geometry at the stern. The remaining results pertain to the SR107 hull. In this case, comparisons are made between the calculations and experimental data at a = 10° and Re = 2.7 x 106. We examine the velocity field as well as the contours of the axial component of vorticity, fox, in the format presented by Nishio et al . (1988~. Figures 5, 6 and 7, show the results at three sections, x/L = 0.5, 0.7 and 0.9, respec- tively. From Figure 5 it is clear that the effect of the cross cur- rent is to thin the boundary layer on the port side and thicken it on the starboard side. A vortex is observed around the turn of the bilge, particularly in the experiment. In general, at this section the calculations reproduce the overall trends but differ in intensity. For example, the measured secondary motion and axial vorticity are both stronger than predicted, and Figure 5(c) indicates some difference in the location of the vortex. It is possible that better agreement between measurements and calculations can be secured, at least in some respects, by adjustment of the initial conditions at x/L = 0.3. An alternative, of course, is to include the bow in the calculations. We shall return to this later. 722

At the next axial station, x/L = 0.7, the results are similar to those at the previous section. Figure 6(b) again indicates that the secondary motion observed in the experiment was considerably stronger than is calculated although there is good correspondence in the extent over which axial vorticity is found. The calculations fail to reproduce a detached core of high axial vorticity. At the last section where measurements were made, x/L = 0.9, Figure 7 shows continued qualitative agreement between the calculations and data but somewhat greater differences in the details. The dual vortex structure indi- cated by the experiments is perplexing because it gives the appearance of vortex shedding and yet such a phe- nomenon, even if it were present, could not have been captured by the simple pressure probes employed for the measurements. The calculations also do not produce the high levels of axial vorticity observed in the experiments. Be that as it may, Figure 7(c) shows that the calculations are successful in identifying the region over which there is . . ~ . . . slgmilcant axle . vortlclty. Although the calculations were continued well into the wake, there is no data to gauge their success. An interest- ing feature of the calculated wake is that the longitudinal vorticity (secondary motion) is destroyed rather rapidly although a measurable axial velocity defect persists for large distances. DISCUSSION This first attempt to calculate the viscous flow over a ship in a cross current is regarded only as a partial success because it has raised a number of issues which need fur- ther investigation. Among these, the most important is the problem of specifying realistic initial conditions at the upstream station. Numerical experiments performed with different initial conditions during the course of this work indicated that the strength and location of the vortices on the lee side depended rather markedly on the initial condi- tions. For this reason, it was decided to present results obtained with the same initial conditions as those used ear- lier and found satisfactory for the zero-yaw case. Of course, calculations can be performed for the inviscid flow and the boundary layer over the bow to determine these conditions in a systematic way. But, it is possible that even these will fail if, as the results from the present calcu- lations indicate, a vortex is formed well ahead of midships. In fact, the likelihood of vortices arising near the bow is greatly increased for a full hull form, and with a bulbous bow, at angles of yaw. Thus, it appears that the problem of initial conditions may have to be addressed, in the final analysis, by extending the full Navier-Stokes type of solu- tions to include the complete ship. At model scale, this would also require a careful treatment of transition or trip- ping devices. The failure of the calculations to reproduce the observed high levels of axial vorticity in well defined cores is another feature brought forth by this investigation. The differences between calculations and experiment arise as early as midships and continue over the stern. It is possi- ble that this is due, at least in part, to the use of the wall- functions approach in the turbulence model. In this approach, the flow near the wall is not explicitly calcu- lated, and therefore, flow features arising from a roll up of the near-wall layer, such as vortex formation, are unlikely to be resolved in detail. A near-wall turbulence model, along with integration of the equations of motion up to the wall, are needed to obtain an adequate resolution of these features. Finally, it should be noted that the present calculations were carried out for a ship in a uniform stream (in unre- stricted waters) whereas the experiments on the SR107 were conducted on a double model in a wind tunnel. The effects of tunnel blockage, which increases when the model is mounted at an angle of yaw, remain to be explored. CONCLUDING REMARKS Calculations of the flow past a ship in a cross current, presented here, have served to illustrate not only the capabilities of the numerical method employed but also the additional complexities that arise in real-life situations. The asymmetries in the stern flow, resulting from asym- metric stern shapes, cross currents, or maneuvers, are obviously of interest in the design of propulsors and appendages, and in the prediction of hull vibration. The present calculations represent a first step in the development of numerical methods capable of addressing these issues in a comprehensive and realistic manner. ACKNOWLEDGEMENTS This research was partially sponsored by the Office of Naval Research, first under the Accelerated Research Initiative (Special Focus) Program in Ship Hydrodynamics, Contract N00014-83-K-0136, and then under Contract N00014-88-K-0001. The calculations reported here were performed on the CRAY X/MP-24 supercomputer of the Naval Research Laboratory, and on the CRAY X/MP-48 machine of the National Center for Supercomputing Applications at Urbana-Champaign. REFERENCES Nishio, S., Tanaka, I., and Ueda, H. (1988), "Study on Separated Flow around Ships at Incidence," J. Kansai Soc. Nav. Architects, Japan, Vol. 210, pp. 9-17. Okajima, R., Toda, Y., and Suzuki, T. (1985), "On a Stern Flow Field with Bildge Vortices," J. Kansai Soc. Nav. Architects, Japan, Vol. 197, pp. 87-95. Patel, V.C. (1988), "Ship Stern and Wake Flows: Status of Experiment and Theory," Proc. 17th ONR Sym. Naval Hydrodyn., The Hague, The Netherlands, pp. 217-240. Patel, V.C. and Baek, J.H. (1985), "Boundary Layers and Separation on a Spheroid at Incidence," AIAA Journal, Vol. 23, pp. 55-63. Patel, V.C., Chen, H.C. and Ju, S. (1988), "Ship Stern and Wake Flows: Solutions of the Fully-Elliptic Reynolds-Averaged Navier-Stokes Equations and Comparisons with Experiments," Iowa Inst. Hydraulic Research, Uni. Iowa, IIHR Report No. 323. Patel, V.C, Chen, ~G, and Ju, S. (1990), "Computa- tions of Ship Stern and Wake Flow and Comparisons with Experiment," J. Ship Research, to appear. Wieghardt, K. (1982), "Kinematics of Ship Wake Flow, The Seventh David W. Taylor Lecture," DTNSRDC- 81/093. See also Z. Flugwiss Weltraumforsch., Vol. 7, pp. 149-158. 723

z 43 =- 90° 11 ~ - 90° Port\ Starboard a e= 0 (a) transverse section A-A A ~ (' ~_. Ct:~ _ . / UOA ~ (b) waterplane (a) HSVA Tanker lLLL:~Sl (b) SR107 Bulk Carrier Figure 1. Ship in a cross current; notation Figure 2. Hull shapes a = 10° (a) port (by starboard Figure 3. Surface streamlines, HSVA Tanker 724

whoa -. 1_ .~2 _ no -~ od - or _ 17 -. od -. OB _ , ~ - n2 aft (a) Axial velocity (b) Transverse velocity Figure 4. Velocity field on HSVA tanker at transverse section x/L = 0.9 0.7 1~ '1 -- Experiment Calculation (~1 Axial velocity us (b) Transverse velocity 28 24 20 16 . (c) Axial vorticity \ \l1 6 Figure 5. Velocity field and axial vorticity on SR107 at x/L = 0.5 725

Experiment Calculation AIL = 0.7 (a) Axial velocity ~ ~o.s _ :~_ 0.809 0.6 GIL = 0.9 At 0.6 (a) Axial velocity (b) Transverse velocity TIC (c) Axial vorticity =~84 16 12 Figure 6. Velocity field and axial vorticity on SR107 at x/L = 0.7 (b) Transverse velocity (c) Axial vorticity 24 2016 Figure 7. Velocity field and axial vorticity on SR107 at x/L = 0.9 726