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Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

Chapter: Validation of the Flow Around a Turning Submarine

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Suggested Citation:"Validation of the Flow Around a Turning Submarine." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Paper submitted to TheTwenty-Fourth Symposium on Naval Hydrodynamics July 8-13, 2002, Fukuoka, Japan VALIDATION OFTHE FLOW AROUND ATURNING SUBMARINE Chao-Ho Sung, Ming-Yee Jiang, Bong Rhee, Scott Percival,Paisan Atsavapranee, and In-Young Koh David Taylor Model Basin NSWCCD, MD 20817-5700, U.S.A. ABSTRACT A numerical procedure based on solving the incompressible Reynolds-averaged Navier-Stokes equations in a steady rotating coordinate system has been developed for the prediction of the flow around a turning submarine. Computed results are com- pared with the measured data obtained from an unclassified sub- marine called ONR Body-l. The performance of the standard k—w turbulence model and a realizable k—~ turbulence model is compared and discussed. Several improvements in numerical methods including higher-order spatially accurate schemes, lo- cal refinements, and new wall function are suggested for more accurate predictions. INTRODUCTION Before full scale sea trials are performed, experimental data obtained by straight-line captive-model rotating arm experi- ments(RAE) and radio-controlled models(RCM) provide the cru- cial information for the design of maneuvering submarines. From the point of view of the validation of Computational Fluid Dy- namics(CFD) techniques, the validation of CFD with RAE data is most fundamental. This is because RAE can be treated as a time independent problem in a steady rotating coordinate system thus avoiding complications by time dependency and other oscil- latory flows. Yet RAE maintains most of the major flow physics unique to a maneuvering submarine such as vertical flow interac- tions with hull boundary layer and flow separations. Simplicity of the flow problems in RAE makes it easier to identify CFD difficulties and then to make necessary improvements. The pur- pose of this paper is to present the current predictive capability of a RAE model through validation with available experimental data. Improvements to increase the prediction accuracy will also be discussed. The outline of the paper is the following. The govern- ing equations of the incompressible Reynolds-averaged Navier- Stokes(RANS) equations and the turbulence models in a steady rotating coordinate system will be described. As expected, turbu- lence models play an important role in the accurate prediction of the flow around a maneuvering submarine. The numerical method used will be discussed next. Only some special features of the nu- merical schemes will be highlighted omitting most of the details. The experimental performed on an unclassified submarine-like body with a sail and four stern appendages (named ONR Body-1) in the Rotating Arm Basin at the David Taylor Model Basin will be discussed. The six components of hydrodynamic forces and moments were measured for only one combination of roll, pitch and yaw angle, rudder angle, sternplane angle, yawing angular velocity and ahead speed. This is defined as an experimental grid point. The six components of forces and moments are the most crucial information for the design of maneuvering submarines. The cross flows on the vertical planes at several locations down- stream of the sail were also measured using the Particle Image Velocimetry(PIV) technique. The measured data just described will then be compared with the computed results and discussed. In the conclusion, current status on the accuracy on the prediction of the six components of forces and moments will be discussed and future efforts to improve accuracy of prediction will also be described. GOVERNING EQUATIONS The incompressible RANS equations in a steady rotating co- ordinate system are 1 . .

= o ~ + ~[ninj +p*5ij - 2Q2(X2 +y2)5i +2(Q x r)iui] = ~ (I,~—~ij ) (1) (2) ~ * ~ * ~ * As = ~coS[3 cos 1 (~ij i:)jk ;'ki )] L U* = j(S*)2 + (Q*)27 S* = ~S`j*SiJ*' Q* = ~Qij*Qij* where x; and Uj are the Cartesian coordinate and velocity S * _ 1 ( dui + Bu componentrespectively,p*isthepressurepdividedbyaconstant t} —2 dxj density p, r is a position vector, Q is steady rotation rate in the z- direction, dij is the Kronecker delta, z' is the kinematic viscosity and ~rij is the Reynolds stress tensor. Qij* = 2(~ - ~) - Both the k—w and the k—~ turbulence models are consid- ered. The k—~ model by Wilcox[2] is 0dk + ~ (Uj k)—~ [(~ + ak ~t ) ~] = _7ij ~ - p*wk (3) bt + ~ (Uj~)—~ [(~ + ~ut) ~] = —O' k rij ~ _ >~2 (4) and the k—~ model by Shih[3] is + ~ (0jk)—~ [(U + aklJt) 7~] = —rij ~—£ (5) ~ + ~ (Uj£)—7~ [(~ + ~t) ~] = ClflS£—C2f2 k+~ + ~Vt(~)(~) (6) tJt = C~f~k(k + ~) (7y C —, 1 ~ 4.0+As e As = ~cos ¢, ~ = 3 cos-1 (~W*) Cl=max(0.43,5+77), 9=Sk W* Sij Sjk Ski U* = ;(S*) + (Q ) S* = ~Sij*St,*, Q* = ~Qij*Qij* (9) - 36ijk Qk ( 1 O) / Sl~f~ * ~ * = ~ ~ij Qij (8) In the k—w model, k is the turbulent kinetic energy, w is the specific dissipation rate and z/~ is the eddy viscosity given as k. The values of model parameters are given by ~ * = 0.09, ~ = 4i, 0l = g and (Jk = ~W = 2. For the definitions of the additional model parameters used in the k—£ model, readers are referred to the paper by Shiht33. For practical applications, it is important to point out that both models can be integrated right down to the wall without using the normal distances to the wall. It is also important to point out that the Shih's k—~ model is realizable in the sense that negative turbulent kinetic energy components can not be producedt4] as in the k—w model. The forms of the two-equations in the rotating coordinate system are the same as in the inertial coordinate system. This is expected since it can be proved that turbulence models must be form invariant under arbitrary translational acceleration of the ref- erence frame and should only be affected by rotation through the mean vorticity tensor[Si. The rotational effect is thus introduced explicitly only in non-linear models through the mean vorticity tensor defined in (10) in the expression for rij. For linear two- equation models, rij is a function of mean strain tensor defined in (9) only and rotation is not explicitly introduced. However, rotational effect can still be introduced to the linear two-equation models through the definitions of model parameters. Since all the model parameters in the standard k—~ model given by equations (3~-~4) are constant, the rotational effect is not introduced in this model. However, the rotational effect is introduced to the linear k—~ model, defined by equations (5~-~10), through C~ defined by equation (8) which is a function of the mean vorticity tensor. NUMERICAL METHOD The incompressible RANS equations are solved by the arti- ficial compressibility approach first proposed by Chorin t6] and subsequently generalized and improved by Turkel [71. A finite volume method is used. The mean flow (i.e., Eqns (1) and (2~) is spatially discretized by a second-order accurate central difference 2

method with fourth-order accurate dissipation terms. The logic for using the fourth-order accurate dissipation terms is twofold. During the early stage of time stepping to reach convergence, the fourth-order accurate dissipation terms act to suppress spurious oscillations. But once the convergence is achieved, the contri- bution from them to the solution is negligible because they are fourth order accurate compared to the second-order accurate spa- tial discretization scheme. The turbulent flow (i.e., Eqns (3~-~4) and (5~-~10~) is discretized by one of the several upwind schemes suggested by Yee [83. The reason for using an upwind scheme, not a central difference scheme, to solve the turbulent flow equa- tions is that the flux matrix is already diagonal; therefore there is no additional cost in doing characteristic formulation. The time stepping is based on an explicit one-step multi-stage Runge-Kutta method to reach a steady-state solution. This approach is not just applicable to the steady state solution but can also be extended to solve the time dependent equations in a very simple manner. Some discussion of this extension can be found in the papers by Jameson [9] and Liu et al.Ll03. Several convergence acceleration techniques including multigrid, local time step, implicit resid- ual smoothing, preconditioning and bulk viscosity damping have been implemented. To handle complex geometry, the multi-block grid structure is adopted. These numerical techniques have been implemented in a code named IFLOW, the acronym for Incom- pressible FLOW. IFLOW is intended to be a general- purpose pro- duction code for solving 2D, 3D, steady and unsteady problems. The code is highly modular in structure so different turbulence models and newly developed numerical schemes such as higher order schemes can be easily implemented. Some special features of the numerical schemes used will be highlighted. However, detailed derivations will be mostly omitted. PRECONDITIONED METHOD The preconditioned method is developed based on a system p-1 = of hyperbolic equations, but the idea goes back to the effort to reduce the condition number of a matrix in linear algebra. For by- . perbolic equations, the objective is to make the various speeds of different wave modes more or less the same so that convergence can be significantly accelerated. This is particularly important when the artificial compressibility approach is adopted to solve incompressible flows. The reason is that the sound speed, which is one of the wave modes in the incompressible flow, propagates much faster than the fluid speed. The result is a very slow conver- gence as often encountered in the attempts to compute low Mach number flows using any compressible flow code. The preconditioned mean flow (i.e., Eqns (1) and (2~) can be written in the conservative form as To 1qt + FX + Gy + Hz = 0 (11) where the preconditioned matrix PO and the three compo- nents of fluxes F. G. and H are defined as t1 + 7~3 2 yp-2u 73-2v 73-2W p_l _ t1 + ~ + y'>-2U 1 + y3-2U2 )~-2UV yp-2uw o - . (l+~+~-2V )~-2VU 1+~-2V2 y3-2vw t1 + ~ + y)~-2W )~-2WU )~-2WV 1 + :~-2w2 (12) q= G= IU F= V W v UV—Tyx V2 +p* _ ~ , VW—Tyz U U2 +p* _ UV—[Xy uw—[xz ?., UW—[ZX VW - [Zy w +p —[zz where Ct,3-2 and ~ are preconditioning parameters, rij,i, j = x,y,z, are Reynolds stresses. For mathematical anal- ysis, it is easier to write Eqnfll) in a non-conservative form. Neglecting the viscous terms, it can be derived as: p lQt + Aqx + Bqy + Cqz = 0 (14) The explicit forms of matrices A, B. C are omitted here. The preconditioning matrix P-t is different from the previous one Po ~ and is given by (1 + ~y)3-2 ~y>-2n 3-2v ,JB-2W- a>-2U oL3-2V oL>-2W 1 0 0 n 0 1 The condition or = ~y ensures the system of partial differential equations is well posed. However, the implication of well posed- ness in the case of numerical solution is not clear. In this paper, itissetor=~=Oand ,3-2 = nlaX(|u|2, 6), ~ = 0.7. (16) Through rather tedious algebraic manipulations, the eigen- values and left and right eigenfunctions can be found. The max- imum of eigenvalues is used to define a local time step. The eigenfunctions are of no use to central difference schemes except for establishing a nonrehecting boundary condition at far field. The final system of equations to be solved in the conservative 3

form is Po 1qt + FX + Gy + Hz = (po 1IPAIqxxx)x+ (po 1 IPBlQyyy)y + (pO 1 1PCIqzzz)z Since the formulation is based on hyperbolic equations only, viscous terms should be added to the fluxes F. G and H given by Eqn (13~. The right-hand side of Eqn (17) is the fourth-orderma- trix dissipation terms. The matrix dissipation gives the most accu- rate solution but is less stable because a smaller amount of dissipa- tion is added. As a compromise between accuracy and robustness, vector dissipation is adopted in this paper. For vector dissipation, matrices PA, PB and PC are replaced by their corresponding radius spectra. In the curvilinear coordinates, (i,i = 1,2,3, the maximum eigenvalue in the i-direction is given by )\maX = - (|uiI + ~/ui2 +4,32|ai12) ui=ue~, Pi MULTIGRID METHOD (18) Multigrid is one of the most effective methods to accelerate the rate of convergence and should be used routinely in every production code. The approach in IFLOW follows mostly the ideas of Brandt t11] and Jameson [12~. Several variations in- cluding V-, W- and F-cycles have been implemented. In general, the W- and F-cycles are more efficient but not significantly so. More levels of multigrid are more efficient but cost a little more. For simplicity, most computations performed with IFLOW use three levels of multigrid in V-cycle. The multigrid method is used routinely in IFLOW. For large scale computations on com- plex geometries, the starting of computation is often jumpy. To obtain a smoother start, a multigrid starting procedure is used. Consider a 3-level multigrid computation. A 2-level multigrid consisting of the medium and coarse grids is run for about 50 cycles and then the solution is extrapolated to the fine grid to start the 3-level multigrid computations. In general, a solution ade- quate for engineering applications can be achieved in 300-500 multigrid cycles. This level of efficiency is at least as good as the best Computational Fluid Dynamics (CFD) codes available but is far from the Textbook Multigrid Efficiency (TME) advocated by Achie Brandt [133. The goal of TME is to obtain an engineer- ing solution in about 10 multigrid cycles. Achieving TME is a noble endeavor and will no doubt have a significant impact on engineering applications of CI;D. INITIAL CONDITIONS The velocity relative to the fixed inertial coordinate system, vi, is related to the velocity relative to the rotating coordinate system, vr, by (17) vi = vr + Q x r = (u + Q-y)ex + (v—Qx)ey + wez (19) \ t:7, _ -l, , ~ _ where, as before, Q in the z-direction is the steady rotation and r = (x, y, z) is the position vector with respect to the rotating coordinate system, and ~ ex, ey, ez) are unit vectors . Since the fluid is stationary in the fixed coordinate system, vi = 0. This provides the following initial conditions for the mean flow in the rotating coordinate system as u = -Qy v=Qx w =0 (20) Other initial conditions are defined as the following. The pressure is assumed 1 everywhere. The turbulence kinetic energy k is set somewhere between (0.01~2 and (0.0001~2. For the k— model, w is set approximately 100 and then obtain the initial eddy viscosity according to the relation zJ~ = ~ . For the k— model, the initial eddy viscosity z'~ is set somewhere between 0.01 and 0.001 and the dissipation rate ~ is obtained from the relation 1~ = 0.09 ke . BOUNDARY CONDITIONS Only the solid wall and the farfield boundary conditions need to be discussed. At the wall, the three components of velocity and the turbulent kinetic energy k are set equal to zero, the pressure p is derived by taking the the component normal to the wall of the momentum equation (21. By neglecting the viscous term, this component gives ~ vp* = Q2x(ex ri) + Q2y(ey ri) (21) where n is the unit vector normal to the wall. It is noted that in the absence of rotation, the usual boundary condition for the pressure is recovered. The wall boundary condition of the specific dissipation rate ~ originally given by Wilcox (p. 148 in [21) is modified as I, _ aOo Ww— ~ ~W ,3 = 40 (22) where Qw is the vorticity at the wall and aO is a constant during calculation that can be varied from a value of 6 to 20 given by Wilcox. The choice of aO may vary the convergence rate slightly but once the convergence is achieved the solution is about the same. The motivation in deriving the modified wall bound- 4

ary condition (22) is to get rid of the requirement that the first grid normal distance must be given. The presence of the normal distance creates some difficulty in the coarser grids in multigrid cycle. In general, two-equation turbulence models require that the non-dimensional first grid normal distance y + be in the order of 1. If the finest grid satisfies this requirement of a value of 1, then the y+ of the coarser grids will be greater than 4. The consequence is that a proper turbulent boundary layer may fail to develop. With Eqn (22), the normal distance does not appear and the y+ of the first grid normal distance of the finest mesh should be in the order of 1 to 3. The wall boundary condition for the dissipation rate ~ in the k—~ model follows a suggestion by Shiht4] as ,:_ 0.251u4 - where u~ is the skin friction velocity. (23) In the absence of rotation, the gradients of the three compo- nents of the Cartesian velocity are set to zero. In the presence of rotation, there is a background flow described by equation (20~. Thus the gradients of the perturbed flow field are set to zero. The gradients of the turbulence quantities k, ~ and ~ are set equal to zero. The pressure is obtained by a non-reflecting condition discussed by Hedstrom [141, Rudy and Strikverda [151 and Sung [163. This is one of the most important boundary conditions for external flows and it can significantly affect both accuracy and convergence. The idea is based on the characteristic formulation of hyperbolic equations. The condition is formulated such that the outgoing solution modes will not be reflected back into the computational domain to corrupt the solution. For details readers are referred to the mentioned references. TWO-EQUATIONTURBULENCE MODELS Several two-equation turbulence models have been imple- mented in a general-purpose code IFLOW mentioned earlier. It is well known that the convergence of the two-equation turbu- lence models is rather temperamental. Two techniques have been used in IFLOW and as a result the same rate of convergence as using the Baldwin-Lomax turbulence model has been achieved. The first technique is the point-implicit method for source terms. Here, the positive part of the source term on the right-hand side is treated explicitly and the negative part is treated implicitly. This technique is, in fact, widely used. The second technique is to establish a lower bound for the specific dissipation rate ~ and ~ by applying the Schwartz inequality. To illustrate the method, it is sufficient to consider a linear Reynolds stress model lid = Ui'Uj' = 3 kdij—C~f~*w (2Sij) (24) where Sij is defined in (9). By Schwartz inequality, it can be shown that 'Ui/uj/2 ~ 4k2 Taking square of the both sides of Eqn (24) gives Fiji = 43 k2 + 2(C~fy Sky ) po' Po _ 25ijSij (25) (26) A lower bound for ~ is then obtained by combining Eqns (25) and (26) as A> (27) The proportionality factor in Eqn (27) can be taken as a value in the neighborhood of 2. Different values for this factor can affect the convergence rate. But once the convergence is achieved, they all give again about the same answer. The value used in this paper is 2.1. The lower bound of ~ for the k—~ model is derived in a similar manner and is given by r ~ > C~f~ky/-Po DESCRIPTION OF EXPERIMENT (28) A Rotating arm experiment was performed on an unclassified generic submarine model called ONR BODY-1 in the Rotating Arm Basin at NSWCCD as reported in Reference [11. Figure 1 shows a sting-mounted submarine for rotating arm testing. The model was attached to the sting through two sets of block gages which measured the longitudinal, lateral, and normal forces. The center of gravity of the model is located at L, = 0.4646 for the case investigated. The model is an axisymmetric body with a sail and four stern appendages. The shapes and locations of the four ap- pendages were deliberately made to differ from real submarines. The sail has a NACA 0014 section with an aspect ratio of 0.27. The four stern appendages are identical and have a NACA 0018 section with an aspect ratio of 1.2. The total length of the model is 17.0ft and the diameter is l.55ft. The radius to the model reference point at the center of gravity (CG) is 32.206ft. The angular velocity of the rotating arm is 0.163 radios correspond- ing to a linear speed of 5.26ft/s or 3.1 1 knots. Using the model length of 17.0ft and the linear speed of 5.26ft/s as the units for length and speed respectively, the non-dimensional angular velocity is r' =0.53 and the Reynolds number is 8.2 x 106. The model was oriented such that it was pitched at 2.0 °(bow up), rolled 2.1 °(to starboard) and yawed 9.5 °(to starboard). The rudder was s

Figure 1. Sting-mounted submarine for rotating arm testing (Ref.[1]) maintained at a fixed angle of -20° for a starboard turn and the sternplanes were set to -1.0° for bow up. Three components of forces and three components of moments were measured in the body-fixed coordinate system. A total of 20 laser sheets were also taken as the model passed through a location fixed in the inertial system using Particle Image Velocimetry(PIV). These laser sheets provides instantaneous cross flows on 20 cross sections along the model. Details of the experiment appear in [11. DISCUSSION OF RESULTS The size of the computational domain is severely restricted in the simulation of the rotating arm experiment. The restriction is particularly severe in the outer flow boundary. This is because the computational domain should not cross over the center of rotation implying that the outer flow boundary should not exceed the ro- tating arm length, which is 1.88 ship lengths. The computational domain used here is about one ship length in outer flow boundary and about 3 ship lengths downstream from the tail of the model. Three grids were used. The fine, medium and coarse grids having a total of 2.89, 0.36 and 0.075 million grid cells, respectively. For computer runs in this paper, the non-dimensional normal distance to the wall from the first grid cell center, y +, is approximately 1 for the fine grid, 3 for the medium grid and 6 for the coarse grid. A total of 12 blocks were used. The grid about the body is C-type and the grid about the appendages are H-type. A typical grid is shown in Figure 2. Notice that a small gap exists between body and rudder to allow for a rudder deflection of 20 °. IFLOW is run on Cray SV machines provided by The De- partment of Defense High Performance Computing Moderniza- tion Office(DOD-HPCMC) at NAVO. The code is parallelized by Open-MP. Using 4 CPUs, the efficiency varies from 2.0 to near perfection of 3.95 depending on how busy the machine is being used. A typical production run takes about 500-600 multigrid cycles. For an efficiency of 3.5 in 4 CPUS, the elapsed time is 3 hours and CPU time is 10 hours for 600 multigrid cycles using the medium grid. For the fine grid, the run time is approximately Figure 2. A typical grid used in the computation of flow around a ONR BODY-1 fully appended model. (Notice a gap between body and rudder to allow for a rudder deflection of 20°.) Figure 3. Convergence history of root-mean-square of pressure coarse grid ~ medium gric —--- fire grid 100 200 300 400 500 multigrid cycles multiplied by a factor of 8. Convergence histories of the root- mean-square of pressure residues for three grids are shown in Figure 3. The residues drop only about 3 orders of magnitude and rather bumpy. The drop is smaller than the usual practice with IFLOW. 6

Figure 4. Convergence history of force X' o.o -1 .o -2.0 -3.0 do X ~4 0 X -5.0 -6.0 -7.0 ~ = -A,— ' ;~ - t; _ ~ test data A coarse grid a medium grid - fine grid . _- , _ -80 F.,,, 1,,,, 1,,,, 1 1,,,, L . . 0 1 00 200 300 400 500 multigrid cycles _~ Thus the quality of the grids used here need to be improved. The convergence histories of the six components of forces(X ', Y' and Z') and moments(K', M' and N') together with the mean measured data are shown in Figures 4 through 9. Here, X',Y' and Z' are the forces in the x, y and z directions of a coordinate system fixed at the model and they are non-dimensionalized by 2 pUoo2 L2. K', M' and N' are the corresponding moments about the center of gravity of the model at at I;, = 0.4646 and non- dimensionalizedby 2pUoo2L3. Ingeneral,theconvergences are quite satisfactory and the forces and moments converged to steady values after about 200 multigrid cycles. The normal force Z ' and the pitching moment M' are very sensitive to the interaction between the vortex shed by the sail and the cross flow in the hull boundary layer caused by the rotation. The interaction is one of the most important and fascinating flow physics in the flow about a turning submarine and will be further discussed later. Since there is only one experimental data grid point (an ex- perimental grid point is defined in introduction), comparisons between computations and the measured X ', Y', Z', K', M' and N' are presented in tabular form in Table 1. At a first glance, the comparison looks disappointing. But after further considera- tion of experimental difficulties and uncertainties, the computed results appear encouraging. For example, a small misalignment in the sternplane pitch angle or rudder deflection by less than 1 ° can easily change the values of M' and 1\7' significantly. Should this happen, the predicted M' and N' will be within experimental error. To compensate for this uncertainty, multiple experimental data grid point should be taken so that the trend of measured data as angles of attack vary can be observed. Another uncertainty is Figure 5. Convergence history of force Y' 30.0 20.0 15.0 lo x ~ 10.0 0.0 2.0 1.0 0.0 -1 .0 lo X -2.0 - N -3.0 -4.0 -5.0 1 25.0 ~ _ _ , 5.0 it_ t_- - --- 1 . ~ 1 ~ -,,,, .,,, I L -5.u ~ ~ 100 · test data coarse grid 0 medium grid fine grid ~ , 1 - 1 — ~ 1 1,,,, 1 1 1 1 1 1 1 1 1 ~ I I 200 300 400 500 multigrid cycles Figure 6. Convergence history of force Z' t---- 1 1 L - -6.0 ~ ~ 100 of · test data coarse grid 0 - - medium grid fine grid 1 1 1 1 1 1 1 1 1 1 1 ~ I I I I , . , , 200 300 400 500 multigrid cycles the time dependency ofthe flow at such a high value of r ' = 0.53. A small value of r' represents a mild turn with a large turning radius and a large value of r' represents a sharp turn. For a sharp turn, not only the possibility of instrument misalignments increases but there is always a possibility that the flow may become unsteady. This is a potential problem since both the experiment and com- putation assume that the flow is steady in the rotating coordinate 7

Figure 7. Convergence history of force K' o.s n7 O.6 0.5 O.2 0.1 0.0 -0.1 10 n 6.0 an on o.o -en -4.n -6.0 a,0 . ~ ~ ~ _ · test dat _- coarse grid o medium grid _— fine grid ._— _- , . 1,,,,1 1,,,,1 ,,~l_- 0 100 200 300 400 500 multigr~d cycles Figure 8. Convergence history of force M' . _ · test data ~ coarse grid _ a medium end ..... ;~ fine grid ~~ 0 100 200 300 multigrid cycles 400 500 system. For these reasons, the historically measured data for r' in the range between 0.5 and 0.6 contain very large scatters even for a much simpler geometry like a body of revolution. More reliable data have been obtained for r' in the range between 0.2 and 0.4. In Table 1, the mean values of the measured X',Y',Z',K',M' and N' are presented in the first row, the corresponding standard deviations are presented in the Figure 9. Convergence history of force N' 10.0 O.O -on lo x -20 0 at -30.0 -40.0 Ann -60 0 0 100 200 300 multigrid cycles . 5 _ _ '1 _ ~ test data _ -- I-- coarse grid 0 - - medium grid _ ............ fine grid _ 1 ~ 1 ~ 1 1 1 1 1 ~ 1 1 1 1 1_ ~ 1 1 1 1 400 500 second row and the standard deviations in percentage deviations from the mean values are presented in the third row. The predicted values with the coarse grid, medium grid and fine grid using the realizable k—~ turbulence model are then presented. The percentage deviations from the mean measured values are listed in parentheses to give some idea of the agreement between data and prediction. The use of percentage deviation could create a false impression of excessive error when the mean value is nearly zero as in the case of M' here. Finally, the predicted results using the k—w turbulence model in medium grid are presented. Looking at the measured data, the percentage deviations in general fall within 10 to 20% with the exception of Z' which is about 90% and M' nearly 700% . These large percentage deviations are due to the small mean values of Z' and M' at that specific experimental grid point. From historically measured data of similar configurations, it is known that for a slightly different experimental grid point, say yaw angle of 8° instead of 9.5°, the magnitudes of Z' and M' can increase significantly to the orders of 10-3 and 10-4, respectively. This is almost one order of magnitude increase in Z' and two orders of magnitude increase in M'. This is the reason that more than one experimental grid point should be taken as discussed earlier. This is also the reason that the predicted values of Z' and M' are rather sensitive to the change of grid size. With these considerations in mind, the prediction is good. In particular, the predictions of X', Y', Z' and K' are quite good. It is expected that turbulence models will play an important role in the flow around a turning submarine. From Table 1, it can be seen that the prediction using the standard k—w model is con- 8 . . . .

Table 1. Comparison of calculated and measured Hydrodynamic Forces and Moments of Fully Appended ONR Body on Rotating Arm Captive Model Experiment (r' = 0.53, V = 3.11 knots, L = 17 ft. Re = 83 x106, o = -2.1°, ~ = 2°, ~ = 9.5°, sternplane rise 1°, rudder deflection 20° (x10-3) 1 (xlO-3) 1 (xlO-3) 1 (xlO-4) 1 (xlO-4) 1 (xlO-4) :~:Mean~ ~ |~ ~ IO Std. Dev. | 0.04 Std. Dev. in % | ~ 7 calc. k-e -1.53 coarse (44 DO ) calc. k-e -1.21 medium (10 % ) calc. k-e -1.29 fine and (17%) _ 5.29 0.14 5 3.28 (-38 % ) 5.06 (4%) 5.85 (11%) _ _ -0.15 .- 0.07 .- +91 .- -0.39 (160%) -0.33 (120%) -0.11 _ (-27%) _ 0.38 0.037 20 0.07 (-82 % 0.48 (26 0.53 (40 % -0.05 0.176 690 2.32 (4700% ) 0.85 (-1800% ) 1.54 (-3180% ) . - 1.50 0.103 14 -1.74 (16%) -1.57 (5%) -2.90 (93%) calc. k-~ -1.57 3.73 0.13 0.29 2.44 2.61 medium (43 % ) (-29 % ) (-187 % ) (-24 % ) (4980 % ) (274 % sistently not as good as the prediction using the realizable k—£ model. To understand the problem, a simple body of revolution called SUBOFF at incidence is considered. It is clear from Figure 10 for the prediction of Z' that all models predict well at angles of attack less than 4° and then deviate from the measured data at higher angles. It can be seen that the realizable k—£ model performs much better at higher angles of attack. The prediction of the moment M' is shown in Figure 11. In general, all models perform well but the realizable k—£ model still performs bet- ter. Better prediction of M' by the realizable k—£ model is not surprising since M' is essentially derived from the distribution of Z'. A better prediction of Z' gives a better prediction of M'. A conclusion can be drawn from this investigation. For a turbu- lence model to perform well in the prediction of the flow about a turning submarine it is essential that it can predict accurately the force and moment of a bare hull at incidence. One of the most important flow physics about a turning sub- marine is the interaction between the vertical flow shed from the sail and the cross flow on the hull boundary layer caused by ro- tation. This interaction results in a bow pitch-up moment of the submarine. For a barehull without a sail, there is no vertical flow shed and the flow about the body is symmetric. As a result, there is no normal force Z' nor pitch moment M'. In the presence of a sail, a vertical flow is generated. The flow direction of the cross Figure 10. Prediction of normal force Z' vs. incidence angles (94x48x64, Re= 1 4x 1 o6 ) · SUBOFF TEST: +a o SUBOFF TEST: -a 8 00 ~ — SUBOFF IFLOW(k-co) _ <~ . - SUBOFF IFLOW (k-£) 5.00 To 4-00 AX N 3.00 2.00 1.00 0.00 ~ no = . ~I,, ~,,,1 i/ ,) / / (/ ~ id' I/ / - ..vvt ~ 1 2 16 20 a (deg.) Figure 11. Prediction of pitch moment M' vs. incidence angles (94x48x64, Re= 1 4x1 o6 ) .~ an 3.00 2.50 2.00' o _ ~ X ~ ~ . 1.50t 1.00 O.50 0.006 · SUBOFF TEST: +a a SUBOFF TEST: -a SUBOFF IFLOW(k-~) I ~ 1 SUBOFF~IFLOW (ken ~ 1 1 1 ~ turf'' ~ ~ i' At'' 1~, · , , , 1 , , , 1 1 1 1 4 ~ 1 ~~ 1 8 12 16 20 a (deg.) flow of the shed vortex is opposite to that of the cross flow in the hull boundary layer. Cancellation of these two cross flows in opposite directions results in a lower velocity consquently a higher pressure on the upper hull in comparison to the lower hull. This downward normal force in the aft body causes the bow of 9

Figure 12. Longitudinal distribution of normal force Z',r',=0.4, ,3=9° for both bare hull and bare hull with sail 0.3 0.1 _ n lo, __ X v.v~ _ . -0.1 _ -0.2 _ I I I 1 1 1 1 ~ I ~ . . ~ . ~ . . . ;~- ,... _ ~ or— or. .-1 it., . rid ~ Cal wr ~ 1 Or Its ~ _ 1~ 1 1~ O barehull - sail ~ 1 -on ,,,, I I,,,, ,,,, I,,, ,,,,,, I,,,, I I,,,, I,,,, I I,,,, I, ~ ~ . t.0 0.1 . 0 3 0.4 0 5 0 6 0 7 0 ~ 0.9 ~ 0 the submarine to pitch up. This interesting flow physics is shown in Figures 12 and 13 where the longitudinal distributions of Z' and M' are shown respectively for a barehull and a barehull with a sail in rotation with A' = 0.4 and ,B = 9°. For a barehull, both Z' and M' are zero along the body. In the presence of a sail, a downward force(+Z') is generated on the hull aft of the sail as shown in Figure 12 resulting in a bow pitch-up moment as shown in Figure 13. For the ONR Body-1 fully appended submarine, the sail is located between I;, = 0.2 and 0.3. Figure 14 compares the mea- sured and computed crossbow velocities generated by the vortex shed by the sail at A, = 0.34. The location of I, = 0.34 is approx- imate since the laser sheets of the PIV techniques were taken in the fixed inertial frame while computation is done in the rotating frame. Thus the computed flow field needs to be projected onto the laser sheet plane for the comparison. It can be seen that the vortex shed by the sail is predicted well although the predicted location of the center of vortex is slightly lower and further out- board than the measured one and the strength is not as strong. The interaction between the vortex shed by the sail and the hull boundary layer resulting in a downward force producing a sub- marine nose-up phenomenon is shown in Figure 1 Sa. At A, = 0.34 where the vortex has just been formed, the upper hull is domi- nated by the sail-root high pressure field. As the vortex travels downstream, the interaction becomes stronger and a downward normal force is generated as shown at I;, = 0.48 in Fig 15b. This observation is consistent with the longitudinal distribution of Z' shown in Figure 12. Figure 13. Longitudinal distribution of pitch moment M',r',=0.4, p=9° for both bare hull and bare hull with sail 0.5 0.3 0.2 _ 0.1 O Or ~ _ ,,.= , . ~ ~a, -0.1 _^ -0.2 - _ -0.3 -0.4 -o.5o 0 0 1 . == _ ix _~_ _ _ ~ Jam Illl~llr! —~ barehull ~ sail ~ 1 the 1 t1111111111!111~1 I.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CONCLUSIONS A numerical procedure for the prediction of the flow around a turning submarine has been developed. The procedure is based on solving the incompressible Reynolds-averaged Navier-Stokes equations coupled with a standard k—~ turbulence model and a realizable k—~ turbulence model in a steady rotating coordi- nate system. Computed results are compared with the measured data obtained from an unclassified submarine called ONR Body- 1. ONR Body-1 consists of an unclassified submarine-like body with a sail and four stern appendages. The non-dimensional an- gular velocity is r' = 0.53 and the Reynolds number is 8.2 x 10 6. The model was oriented such that it was pitched at 2.0 ° (bow up), rolled 2.1°(to starboard) and yawed 9.5°(to starboard). The rud- der was maintained at a fixed angle of -20 ° for a starboard turn and the sternplanes were set to -1.0° for bow up. Only one data grid point listed above was taken. Three components of forces(X ', Y' and Z') and three components of moments (K', M' and N') were measured. Laser sheets using Particle Image Velocimetry(PIV) techniques were also taken to track the vortex shed by the sail. Comparing with the measured data, the prediction of the forces is in much better agreement with data than the prediction of the moments. The reason for the poor prediction of moments can not be ascertained since only one experimental data point was taken as explained in the discussion. The predicted crossbow of the vortex shed by the sail at L, = 0~34 compared well with the PIV laser sheet. The interaction between the vortex shed by the sail and the hull boundary layer results in a downward force in the aft hull and consequently a pitch-up of the submarine nose. This 10

Figure 14. Crossflow Velocities at I, = 0.34 (right behind the trailing edge of sail) -0.6 - .2 0 0.2 0.4 0.6 0.8 1 Y(~) Computed Crossflow Velocities for Plane #7 (x/L = .341 ) Figure 15. Calculated Pressure Coefficient (a). at =~ = 0.34(right behind the trailing edge of sail) (b). at L. = 0.48 - - - -0.4 t ~ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . . . . _ _ _ _ _ _ _ _ _ _ _ _ _ _ . a, . . . _ _ _ _ _ _ _ _ _ _ _ _ _ . in, . . _ _ _ _ _ _ _ _ _ _ _ _ . x. . _ _ _ _ _ _ _ _ _ _ _ _ . 5.26 fats Reference vector: W:: ~ - - - -: ~ \. :::::~:::::: ~ . . . . `... . . ~ · , ~ ~ , · , ~ , , , ~ , , , ~ , , , lo-,--,,, 1,-.- -0.2 0 0.2 0.4 0.6 0.8 1 y (ft.) important flow physics has been illustrated and explained with the computed flow field. As expected, turbulence models play an important role in the accurate prediction of the flow around an maneuvering submarine. A standard k—~ model and a realizable k—~ model were used in this investigation. For a simple body of revolution at incidence, the realizable k—~ model gives a much more accurate predictions of lift and pitch moment than the standard k—~ model. It is not surprising that the realizable k—~ model also gives a consistently more accurate prediction of the flow around a turning submarine. Since this investigation is the first time that the prediction of the flow around a fully appended submarine in rotation has been attempted based on the RANS solutions and the results compared with the measured data, several improvements in numerical meth- ods are needed for more accurate prediction. Since the accurate tracking of the vortex shed by the sail is important for accurate prediction of Z' and M', higher-order spatially accurate schemes should be used. The conventional second-order spatially accurate - - - - -2 y (ft.) Cp 0.04 0.03 0.02 o.ol Coo -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 .os -0.10 schemes used in the present investigation produce too much nu- merical dissipation which rapidly diffuses the shed vortex. Higher orcer schemes have smaller numerical dissipation therefore vor- tices will not be diffused as rapidly as second-order schemes. Local refinement technique is also very effective in tracking the vertical flows. It is also clear from this investigation that a grid with 3x 1 o6 cells is far from sufficient for accurate prediction. The majority of the grid cells are distributed near the wall because con- ventional turbulence models require the first y + be approximately 1. There will be a great saving of grid cells if a new wall function method can be developed so that a y+ in the range from 30 to 100 can be used. The new numerical methods just mentioned including higher-order spatially accurate schemes, local refinem- nent and new wall function method will be used for the prediction of the flow around a turning submarine in the next attempt. 11

ACKNOWLEDGMENT This work was partially funded by the Office of Naval Re- search, Code 333, monitored by Dr. Patrick Purtel and Dr. Ki- Han Kim and partially funded by the ILIR program at David Taylor Model Basin monitored by Dr. John Barkyoumb. Com- puter resources provided by The Department of Defense High Performance Computing Modernization Office(DOD-HPCMC) at NAVO is gratefully acknowledged. Useful discussions with Mr. Tom Moran and Dr. Jerry Feldman are gratefully acknowl- edged. REFERENCES 1. Fu, Thomas, C., Paisan Atsavapranee and David E. Hess, "PIV Measurements of a Turning Submarine Model (ONR BODY 1) Part 1: Experimental Setup", NSWCCD-50-TR-2002/019, 2002, Hydromechanics Directorate, Research and Development Report. 2. Wilcox, D. C., Turbulence Modeling for CFD, DCW In- dustries, Inc. CA, 1993 3. Shih, Tsan-Hsing, Jiang Zhu, William Liou, Kuo-Huey Chen, Nan-Suey Liu and John Lumley, "Modeling of Turbu- lent Swirling Flows", NASA Technical Memorandum 113112, ICOMP-97-08; CMOTT-97-03, August 1997. 4. Shih, Tsan-Hsing, Jiang Zhu and John Lumley, "A New Reynolds Stress Algebraic Equation model", NASA Technical Memorandum 106644, ICOMP-94 -15; CMOTT-94-08, August 1994. 5. Speziale, Charles G.,"Turbulence Modeling in Noniner- tial Frames of Reference", Theoretical and Computational Fluid Dynamics 1, pp 3-19, 1989. 6. Chorin, A. J., "A Numerical Method for Solving In- compressible Viscous Flow Problem", Journal of Computational Physics, vol. 2, 275, 1967. 7. Turkel, E., "Preconditioned Methods for Solving the In- compressible and Low Speed Compressible Equations", Journal of Computational Physics, vol. 72, 277, 1987. 8. Yee, H. C., "A Class of High-Resolution Explicit and Im- plicit Shock- Capturing Methods", NASA Technical Memoran- dum 101088, February, 1989. 9. Jameson, A., "Time Dependent Calculations Using Multi- grid with Applications to Unsteady Flows Past Airfoils and Wings", AIAA 91-1596, June 1991. 10. Liu, C., X. Zheng and C. H. Sung, "Preconditioned Multi- grid Methods for Unsteady Incompressible Flows", Journal of Computational Physics, vol. 139, 35-57, 1998. 11. Brandt, A., "Multigrid Techniques: 1984 Guide, withAp- plications to Fluid Dynamics", 1984,191 pages, ISBN-3-88457- 081 - 1; GMD-Studien Nr 85; Available from GMD-AIW, Postfach 1316, D-53731, St. Augustin 1, Germany, 1984. 12. Jameson, A., "Multigrid Algorithms for Compressible Flow Calculations", Lecture Notes in Mathematics, No. 1228, Proceedings of the Second European Conference on Multigrid Methods, Cologne, pp. 166-201, October 1-4, 1985. 13. Brandt, A., "Barriers to Achieving Textbook Multigrid Efficiency (TME) in CFD", NASA/CR-1998-207647, ICASE In- terim Report No. 32, April 1998. 14. Hedstrom, G. W., "Nonreflecting Boundary Conditions for Nonlinear Hyperbolic System", Journal of Computational Physics, vol. 30, pp. 222-237, 1979. 15. Rudy, D. H., and J. C. Strikwerda, "Boundary Conditions for Subsonic Compressible Navier-Stokes Equations", Comput- ers and Fluids, vol. 9, pp.327-338, 1981. 16. Sung, C. H.,"An Explicit Runge-Kutta Method for 3D Incompressible TurbulentFlows", DTNSRDC/SH -1244-01, July 1987. 17. Jameson, A. and L. Martinelli, "Mesh Refinement and Modeling Errors in Fluid Simulation", AIAA Journal vol.36, No. 5, May 1998. 12

DISCUSSION Steven Turnock University of Southampton, United Kingdom Would the authors like to comment on whether the impressive calculations they have carried out yet have the grid resolution necessary to achieve reliable results that can be used for design? Experience at Southampton suggests that typically four or five million cells on a multi- block structured mesh are required to accurately capture lift and drag on a single low-aspect ratio control surface. A simplistic analysis would therefore suggest at least an order more cells are necessary. Have the group considered moving to an unstructured mesh (hybrid cells) which should significantly reduce the computational effort. AUTHORS' REPLY The number of grid cells required depends on the degree of solution resolution demanded. Due to complexity of the experiment, we set the goal of achieving prediction within 10 to 20 % of the measured data in this first attempt. Apparently the 3 million grid cells used in this investigation is insufficient to achieve grid independent solution. We agree with the discusser that a far greater number of grid cells is needed. Our next attempt will still be based on the structured grid but with more sophisticated numerical techniques including local refinement, higher- order spatial discretization schemes and a new wall function method for the turbulence models. Use of these techniques should make it feasible to achieve high solution resolution with much less grid cells. Also the hybrid MPI/Open-MP technique will be used to maintain optimum parallelization efficiency even with a large number of processors in shared memory computers. After the code based on the structure grid has become standard production code, we will consider the approach based on unstructured grid.

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