Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13, July 2002 Validation of Control-Surface Induced Submarine Maneuvering Simulations using UNCLE R. Parkajaksharl, M.G. Remoiigue, L.K. Taylor, M. Jiang, W.R. Briley, D.L. Whitfield (Computational Simulation and Design Center' Mississippi State University USA) ABSTRACT Physics based simulations of control surface induced maneuvers of a model submarine are compared with experimental results. The forces and moments acting on the body are computed using an UnRANS code (UN(~`F,) capable of handling moving control surfaces and rotating propulsors. A six degree of freedom code is used to calculate the trajectory and orientation of the model. Essential steady validations of drag and lateral force coefficients for the appended SUBOFF body and thrust and torque for the 5168 propeller show excellent agreement. Computed trajectories, orientations, velocities, force and moments are compared with experimental measurements for a vertical overshoot maneuver and three horizontal overshoot maneuvers of the ONR Body 1 Radio Controlled Model (RCM), a free running model submarine. As carefully as can be determined, the level of agreement with experiment is regarded as extremely good. INTRODUCTION Submarine maneuvering simulations are of great interest to the naval community from the standpoints of design, analysis and operational safety. The key requirement for maneuvering simulations is accurate prediction of the forces acting on the hull, propeller and control surfaces. Current techniques treat the forces and moments as functions of the motion state variables and their derivatives md approximate them using truncated Taylor series expansions often containing second, third and cross derivative terms (Boger, 1997~. The derivatives are determined using a combination of empirical and theoretical correlations with the empirical coefficients being computed from captive model tests or tests on individual components. Maneuvering codes based on scale models are not only expensive but also suffer from a serious limitation in that some of the phenomena of interest do not scale well to full scale Reynolds numbers. The current methodology also has major difficulties in dealing with extreme maneuvers, propeller effects and even small changes to the design (Boger, 19971. Thus a validated unsteady RANS based maneuvering code could lead to improved understanding of the complex physics that govern these problems and would be a valuable addition to extant techniques. Some of the thinking that went into the genesis of this effort to do full physics-based maneuvering simulations are detailed in McDonald (1991) and Boger (1997~. FLOW SOLVER The basic flow solver of Taylor (1991) and Whitfield (1991) is comprised of an iterative implicit finite-volume scheme, Roe/MUSCL fluxes, numerically computed state-vector flux linearizations, and an approximate-Newton iteration solved using LU/SGS relaxation. The cell-centered finite-volume scheme using artificial compressibility and time-dependent curvilinear coordinates can be written as aq = _[ditf -fv) +dj (g -gv )+ dk (h - ha)] = it(q) (1) Here q = J(p, u, v,w) is the solution vector, J is the Jacobian of the inverse coordinate transformation, p is pressure, u, v, and w are Cartesian velocity components, and ~ is time. The steady residual vector is R(q), the inviscid and viscous flux vectors are denoted fg,h and fv' gv ~ hv respectively. The central difference operators ~i ~ ~ j ~ ~k apply to the respective curvilinear ;, ~ and coordinate directions. The inviscid fluxes are approximated by Roe's (1981) scheme, with a third-order MUSCL extrapolation of left and right state vectors, qR and

A, as implemented by Anderson, Thomas and van A+ afi+l~2 A- ~i-l/2 Leer(1986). The flux approximation for the idirection Ai = a ~ ~ i = ~ R (7) f. 1 =f(q )+A (qR,qL).(qR_qL) (2) 1 1 2 Here, A_3f lOq is the inviscid flux Jacobian, A-is defined by A- = SASSY, where S is the matrix of right eigenvectors of A, A- is a diagonal matrix containing the negative eigenvalues of A as nonzero elements, and the overbar denotes a Roe average. Analogous definitions apply to the j and k directions, with B _ Og I ~q and C _ Oh I aq . Details and nonsingular eigensystems that use metric information from only one direction are given in Taylor (1991). An iterative implicit nonlinear scheme for solving (1) is given by i~qn,S+l /~ =—R(qn+i,S+~ ) (3) where /~( On = (.)n+~ _(.)n, and s = 0,1,... is an iteration index. The spatial residual is linearized as R(qn+l,s+l ~ = R(qn+l,s )+~3 n+l,s(? qn+l,s ~ (4) where /\s(~) _ Ads+ _~)s, and Fin+ s(~) is a linear spatial difference operator made up of flux derivatives to be defined subsequently. This leads to the following iterative linearized implicit scheme: t?t -I] +5n+l,s ( )] (a? qn+l~s )=RU (qn+l,s ~ (I) The physical unsteady residual R u is defined as R u (qn+i ~ = LAt-iIzZ76(qn+l _ qn' + R`< no+ '] (6) where Ip _ [O. l, l,l lT . The flux linearization matrices are computed using numerical state-vector flux linearizations, as proposed by Whitfield et al. (1994). These are defined as in A A Analogous definitions apply to B j and Ck . The linearized flux derivative operator i3n~ ~ can now be defined as 3n() = - Ai_l ( )i-l + (Ai - Ai ~ ( )i + Ai+l ( )i+l BJ-I()j-l +(`Bj -BJ ~ ()j+BJ+I()j+l Ck-l ( )k-1 + (`Ck Ck ~ ( )k + Ck +1 ( )k+l (8, The solution of (5) for /\sqn+~'s is obtained by Lower- Upper Symmetric Gauss-Seidel (LU/SGS) relaxation. Introducing a second subiteration index m and appropriate definitions for the matrix D and operators i, and ~ 2 ~ the LU/SGS scheme can be written as tDn+l~s + :3ln+l,s (.) Jo? qn+l,s )8 + :32n+l'S (?sqn+l~s )m =Ru (qn+l,s ) [LDn+l~ + 32n+l's ( ) ]( ~sQn+l'S In + :3ln+l~s (?sqn+l~s )* =Rii (qn+l,s ~ (9) The definitions of D, At, andS2 for the LU/SGS scheme are Do)=? I+(Ai-A~ + (,Bj—BJ \~+ j(`C k—Ck i)] ( )i, j, k 31() = A`-I ( )i-l BJ-I ( )j-l Ck-l ( )k-l 52() = Ai+1 ( )i+1 BJ+I ( )j+l Ck+l ( )k+1

The right hand side of (5) contains additional terms (Janus, 1989) to ensure that the Geometric Conservation Law (Thomas, et al, 1978) is satisfied in the regions with deforming grids. Further discussion of this algorithm can be found in Briley (2000~. PARALLEL SOLVER The parallel solution process consists of a scalable solution algorithm implemented to run efficiently on sub-domains distributed across multiple processes and communicating through MPI. The solution algorithm has multiple nested kernels viz. time step, FAS multigrid iteration, LU/SGS iteration etc. and the block- to-block coupling is at the innermost level i.e. in the solution of the linear system. A block-Jacobi type updating of the subdomain boundaries ensures efficient parallelization with a small incremental cost incurred in terms of sub-iterations required to recover the convergence rate of the sequential algorithm. Details about the parallel algorithm can be found in Pankajakshan (2002~. TURBIlLENCE MODELS Turbulence is modeled in UNCLE using one of two available 2-equation models viz. the modified Shih and Lumley he model (ShTh et al. 1993, Yang ~ al, 1995 and Liou et al., 1994) and the q-co model (Coakley, 19831. The convective terms and dissipative terms are computed in a manner analogous to the corresponding terms in the mean flow solver. The turbulence model is loosely coupled, implying that the mean flow and two-equation model are solved in sequence with each using the latest available flow or turbulence quantities. The parallel solution algorithm and the associated message passing routines used in the turbulence models are directly derived from the main solver. Typically the thin layer approximation is used and integration is performed up to the wall using tightly packed grids with off the wall spacings chosen to give yF values less than 1 for the first cell. CONTROL SURFACE DEFLECTIONS One of the basic requirements for this study was the ability to generate valid grids around 25 degree control surfaces deflections within the context of a parallel multi-block simulation. The first approach (Jiang, 2000) could be termed "deform and regenerate" and involved deforming the grid and then regenerating when the deformations became too large. The regeneration was done using reference surfaces corresponding to various intermediate deflection angles. The reference surfaces were generated and tested a priori and then read in during the actual simulation. Each block which contained any portion of the deforming sections of the grid read in these surfaces and used them for regeneration as and when required. Thus the grid regeneration scheme was scalable with only a small increase in memory required to store the reference surfaces. This method was used successfully for a number of simulation involving deflections of the sailplane and rudder (Pankajakshan, 2000~. However several deficiencies in this method were identified which precluded it from being used productively on a regular basis. Specifically, the generation and placement of the reference surfaces was ad hoc and the success of the method relied heavily on the knowledge of an experienced grid generator. This made it cumbersome and error-prone when used without this expertise. It was also found that there were minute variations in the geometry during the deformation phase and that the method could fail without extensive pre- simulation testing. Drawing on the experiences from the first method, a second control surface deflection module (Remotigue et al., 2002) was developed using a more flexible and general approach. The grids in the deformation zones are generated by interpolating between a sequence of volume grids that span the maximum deflection angle. A minimum of three grids is required, but more may be needed for larger deflection angles. The interpolation is performed in an appropriately chosen cylindrical coordinate system and yields grids that preserve the quality and relative spacings from the original volume grids. To prevent loss of geometry, faces of the deforming blocks which constitute the hull are formed to the hull surface assuming that it is a body of revolution. The volume grids which are the inputs to the interpolation routine must satisfy certain requirements in order to be used in the deflection module. The outer boundaries of the grids must be identical and must exactly mesh with the corresponding faces of the full grid. The generation of the volume grids can be done using any structured grid generation package and the restrictions are not onerous enough to exclude any of the commonly used packages. USS_UNCLE (Remotigue, 2002), which is a preprocessor for setting up UNCLE simulations, has a module for testing of the interpolation grids and then writing out the grids and other inputs in the format required by the flow solver. The interpolation can also be tested for the full range of deflection in successively decreasing increments. Since USS_UNCLE and UNCLE use the same interpolation routines, a successful test in the pre-processing phase guarantees that valid grids shall be generated during the actual simulation.

PROBLEM SETUP USS_UNCLE is a general-purpose tool with a GUI for setting up flow simulations using the UNCLE code. It is used for partitioning the grid, applying boundary conditions, checking the load balancing efficiency and also automatically determining the block connectivities. It is also a grid manipulation tool capable of simple operations such as rotation, translation and mirroring as well as extraction, smoothing, reversal and swapping of physical and computational space axes. It can be used to specify moving control surfaces as well as the blocks and faces that make up a sliding interface. During the application of boundary conditions, surfaces can be individually tagged in order to obtain the force and moment histories on particular components such as rudders, stern-planes or individual propeller blades. USS_UNCLE checks for a host of common setup errors such as undefined or over-defined boundaries, block boundaries with gaps, negative volumes etc. A snapshot of the RCM surface grid within the USS_UNCLE user interface is shown in Figure 1 Figure 1 Snapshot of USS-UNCLE with RCM geometry showing tagged components 6-DEGREE OF FREEDOM CODE The forces and moments computed by the RANS solver are used by the 6 Degree of Freedom (6-DOF) code to compute the instantaneous linear and angular velocities. A body-fitted non-inertial reference frame and a fixed inertial coordinate system are used for the angular and linear components of the computation respectively. In the body-fitted reference frame, the moments of inertia of the model are constants and this greatly simplifies the solution process. The use of a four-variable attitude propagation (Stevens et al., 1992) or quaternion representation eliminates the so-called "wraparound" problems that arise from singularities at certain orientations in a formulation based on Euler angles. The quaternion formulation has the added advantage that the quaternion rate equations are linked to the angular rates by a set of linear differential equations. The linear velocities and displacements are obtained by directly integrating the accelerations after addition of the buoyancy and gravity terms in the inertial reference frame. The angular rates and the attitude quaternion satisfy the relations = J QB Jo) + J T (11) O -R Q R O —P and -Q P O O P Q R _p o -R Q -Q R O _p -R -Q P O P. Q and R are the three components of the angular rate ~ J is the moments of inertia matrix, T is the total moment acting on the model while q is the attitude quaternion of the body-fixed reference system. The quaternion rate equation is solved using a Stage Runge-Kutta-Merson integration scheme with built in error estimation. The linear displacements and the attitude quaternion computed at each time step are used to translate and rotate the grid into position for the next time step. Verification and validation of the WOOF modules were carried out using simple problems from celestial mechanics and spinning tops involving elliptical orbits, . . nutat~on and precession. The GOOF module also allows simulations where the vehicle follows a prescribed path specified in the form of a time history of displacements and attitudes. The primary reason for the development of this module was to approximate to the degree possible, the initial

conditions of a vehicle prior to initiation of a maneuver simulation under 6-DOF control. The WOOF module also allows the automatic computation and addition of a "body force" term for allowing self-propulsion in simplified models without the actual rotating propulsors. The term is made equal to the total drag of the vehicle during a straight and level simulation and is appropriately vectored during the maneuvering simulation. A similarly computed moment term can be used to compensate for the rolling effects of the propeller without resorting to small deflections of the control surfaces to achieve the same effect. ROTATING PROPULSORS The initial propulsor module was a scalable parallel implementation of the method developed by Janus (1989) where rotating and stationary blocks deform and then re-connect within a pre-defined deformation zone. The connectivity between the blocks in relative motion is always one-to-one and information is exchanged across the interface through point-to-point messages in a complex but periodic pattern. While this technique is suitable for handling a large class of problems, it runs into some difficulties in cases where the interface needs to be positioned in between physically proximate geometry components. To overcome this limitation, the sliding interface technique (Chen at al., 2001) was implemented wherein the blocks slide across each other at the interface without any deformation or one-to-one connectivity. In this approach the blocks on the two sides of the interface can have different numbers of points in the circumferential direction but must match up in the radial direction. Thus the mismatch between cell faces is confined to the circumferential citrection and the use a one dimensional interpolation scheme based on arc length keeps errors to the minimum. VALIDATION STUDIES The flow solver has undergone a large number of validation studies on a wide range of flow problems ranging from accelerating cylinders to centrifugal compressors. The steady flow validations include inflected stern (Taylor et al., 1991), prolate spheroid (Taylor et al., 1995), wingbody junction (Sheng et al., 1994) and various SUBOFF configurations (Sheng et al., 1995 & Jonnalagadda et al., 1997) along with the P4119 (Sheng et al., 1996) and P5168 propeller flows. In validating unsteady flow, comparisons with experimental results have been made for vortex preservation/convection, maneuvering prolate spheroid (Taylor et al., 1995), flapping hydrofoil (Taylor et al., 1993) and a low speed centrifugal compressor (Sheng at al., 1997 & 1996~. The physical quantities compared include pressure, velocity, forces, moments, shear stress as well as the effect of grid resolution, time-step, turbulence and other algorithmic parameters on these variables. _~= _ -God , ~ . ~ . ~ ~ D:: --10~D ~0 18,0 - .~:~t ~~} · ,._. As. c - ~ _ _ ~^ us - ~: - - b(:- -9 ._~ ~ lo,, ever ~ . .~ A__ 5 —1~ _ ~ i .. o~ _ <~ e - ~0 _~- I ~ I --o .~ ~5 s o ... Pv an.. 5 5 x: Hi, a - Figure 2 Comparison of axial force coefficients computed using two turbulence models do: ~ o.~s _ 4 :: : ;, _ ~' _ hi: ~~ ~—E :~ coypu" q_~ sly is :~.+ :~ :~ ..~. a, . .~.. .^ .,. . at. -God -age ~ te~ ~~t .. ~ (:: 0~) Figure 3 Comparison of lateral force coefficients computed using two turbulence models Validation of force and moment computations performed on the SUB OFF geometry as well as the thrust and torque computations on the P5 168 are particularly significant in the context of maneuvering simulations with rotating propulsors since they are necessary precursors to any maneuvering simulations. In figures 34, the axial and lateral force coefficients and yawing moment coefficient computed using two

turbulence models for a SUBOFF bare hull with stern appendages are compared with experimental data (Huang et al, 1992) for 3 angles of drift. The results show that the q-co model tends to substantially overpredict resistance while the k-£ model is generally within 5%. The lateral force coefficient and the yawing moment coefficient show a drift in accuracy towards the higher yaw angles. Figure 5 shows that the computed thrust and torque coefficients (open water) for the 5168 propeller agree well with experiment (Chesnekas et al., 1998) for a range of advance ratios. ~~e ~ ~ ~ ~ ' 1 ~ . . hag Thou am: 0~0118 _ ~ h'~oltr a' =,~: ~ Hi _ '^ .~ . ; 8~: _ . ·. ~ o r, .o :~ ~ amp. -~3 ~.~ =~.^, at. ~ - .~ _ OC~d _ Bed t ~~ - t 85;~7 ' .~ - 004 1 ~ . 1 ~ , , 'I ~ ~ _~.0 0.0 ' 10 . —~~ ~ ~ | :d~ee) Figure 4 Comparison of yawing moment coefficients computed using two turbulence models. ?4 - ~ :~5 u o e" ~ 1 ~ =. e: a, ~ _ . ~ _~ ~ \ —lime like ec Up—~ ~~ c_~tct t~ . . ~ : - ·_-, \ 0~ 1 L5: Adhere Beet, 1 ~ ~ Figure 5 Comparison of computed thrust and torque coefficients for Propeller 5168. MANllEVERING SIMULATIONS An early step towards a full maneuvering simulation was the periodic flow solution shown in Figure 6 for a SUBOFF configuration with rotating propeller and control surfaces with gaps. The model is at a fixed attitude of 5 degrees roll and 10 degrees pitch and yaw. The 4.5 million point, Unblock grid was run at a Reynolds number of 12 million at a time step requiring 350 cycles per revolution of the propeller and took 160 hours to complete on 50 processors of a Cray T3E-900. Figure 6 Axial velocity contours for submarine drifting at fixed incidence. The same grid and blocking scheme was used for a rudder-induced maneuver by coupling the computed forces and moments with the GOOF code and using displacements and attitude quaternions to move and orient the grid at each time step. The rudder was deflected by 10 degrees in the time taken by the model to travel a quarter of its body length. The rudder was then left in that position till the end of the simulation. Since this was a notional submarine model, the mass and moments of inertia used were ad hoc and the simulation was a test of the elements that made up the maneuvering code and their proper integration. Contours of the X-component of the velocity at various stages of the maneuver are shown in Figure 7.

Figure 7 Contours of axial component of velocity at various lateral deflections for rudder induced maneuver The large simulation times associated with the previous runs were due to the small time step imposed on the simulation process by physics of the rotating propulsor. The use of a body force propulsor would allow the use of much larger time steps with some loss in the fidelity of the simulation. This was done in the form of a sailplane induced maneuver simulated using a modified version of the grid used in the previous simulations with the propeller replaced by the body force model. Due to the larger time step used, the total simulation time was 53.2 hours on 50 processors of a Cray T3E-900. The predicted trajectory and axial velocity cuts at various points during the simulation are shown in Figure 8 Figure 8 position and orientation along trajectory at selected times with axial velocity contours. ONR BODY1 RADIO CONTROLLED MODEL The ONR Body 1 Radio Controlled Model (RCMXFaller et al., 2001a) is a fully instrumented free- running model submarine with rudders, sternplanes and sail and is propelled by a centerline mounted 3-bladed propeller. The model has a mass of 2260 kilograms and is about 6 meters in length. Maneuvering experiments (Faller et al., 2001b) conducted at the Maneuvering and Seakeeping Basin at the Naval Surface Warfare Center, Carderock Division using the RCM include constant heading and depth runs, horizontal and vertical overshoots as well as controlled and fixed plane turns. ROM MANEUVERING SIMllIAIIONS RCM maneuvering simulations were done using both the body force model and a rotating propulsor. The body force simulation was done using a 57 block 6.06 million point grid. The rotating propulsor simulation was done using a 73 block, 7.1 million point grid. An initial straight and level solution was run with each grid until a converged solution was obtained. These solutions served as the initial conditions for the maneuvering simulations. Ideally the RCM should have pro angular rates, accelerations and non-axial velocities at the start of the experiment. In reality initial conditions include small but non-zero values for the three angular rates, angular accelerations as well as for the non-axial velocity and acceleration components. The attitude also tends to be slightly off the ideal with a roll angle of around 3 degrees. While the prescribed motion module can be used to some extent to match certain aspects of the initial conditions, a perfect match of the corresponding angular rates and accelerations is highly impractical. The simulations include two types of maneuvers viz. vertical overshoots (VOVR) and horizontal overshoots (HOVR). In a VOVR maneuver, the stern- plane is deflected to a maximum value and left at the value until the pitch angle of the submarine reaches a certain value known as the Execution Pitch Angle (EPA). At the EPA, the deflection is reversed until the maximum deflection angle is reached in the opposite direction. In the case of the HOVR, the rudder is deflected until it reaches a maximum and is then deflected in the reverse direction upon the model reaching the Execution Yaw Angle (EYA). All the cases discussed in this paper correspond to a model speed of 3.048m/s (lOft/s).

Simulation speed was 0.037 seconds/hour with a time step of 0.001 seconds on a Cray T3E-900. In Figure 11, the computed Z-component of the forces acting on the body show the right trends and inflection points with some error in magnitude. Some drift is seen towards the end of the simulation but this might be partly due to the maximum sternplane deflection not matching the experiment. The X- component is mainly influenced by the propeller and is not significant for this body-force based simulation. Since the roll is not modeled, the Y-component is zero. The velocities in Figure 12 show similar trends, but the positions shown in Figure 13 are predicted extremely well. The computed moment histories are compared with experiment in Figure 14 and show all the major trends in spite of the differences in control surface deflection histories near the maximum. The angular rates and orientations are compared in Figure 15 and in Figure 16 respectively and are in reasonable agreement. Both the experimental force and moment histories show a sinusoidal pattern of unknown origin imposed on the larger trends. It is speculated that this variation is related to the propeller. There was a reduction in the quality of the grid near the control surfaces because the interpolation was done using only three grids to span 50 degrees of deflection. The spikes in the computed force and moment histories were caused by this deterioration in grid quality. .,, Figure 9 RCM with direction and orientation of axes. RESULTS The computed results from four simulation runs will be presented using the nomenclature from the experiment. Four runs were made with the body-force propulsor while a fifth was run with an actual rotating propeller using the sliding interface module. The body force runs include 1 VOVR (Run 41) and 3 HOVR (Runs 13, 18 & 27) simulations. Run 13 is very similar to Run 18 and the results of this simulation are not presented here. The final run is a repeat of Run 18 with a rotating propulsor. All the quantities compared with experiment are in the body-fixed coordinate system except for the displacements and orientations. The directions of the various axes and the color coding used to refer to them are shown in Figure 9. All simulations presented here were run with 3 multigrid levels and using 3 multigrid cycles per timestep. The he two- equation model was used at a Reynolds number of 18.6 million based on model length. RUN41 RUN41 was a VOVR maneuver with a maximum stern-plane deflection of 26.6 degrees and an EPA of 5 degrees. The rudder was under automatic control but the deflections were of the order of 0.6 degrees. The initial 2.4 seconds of the experiment with no stern- plane deflection were not simulated in order to reduce the computational cost. The rudder deflection was kept at zero for the entire simulation and the initial roll angle of the model was not simulated. The maximum grid deflection in the experiment was 26.6 degrees but the simulated maximum stern-plane angle had to be restricted to 25 degrees since the grids were generated assuming a maximum deflection of 25 degrees. RUN 18 Run 18 was a HOVR maneuver with a maximum rudder deflection of 20 degrees and an EYA of 30 degrees with the stern-plane being fixed at the neutral angle of 0.54 degrees. This case was run with a time step of 0.004 seconds with a simulation speed of 0.25s/hr on 57 processors of an IBM-SP3. The prescribed motion module was used to give an initial roll to the model before starting the maneuvering simulation. The computed force histories in Figure 17 compare well with experiment except for the sinusoidal pattern seen earlier. The velocities and positions also match well in Figure 18 and Figure 19 respectively. The computed moment histories in Figure 20 capture all the major trends with small variations in magnitudes. In Figure 21 the roll rate is reasonable given the lack of the propeller while Q and R show trends in keeping with the moment histories. The orientation in Figure 22 shows good agreement with a gradual deterioration towards the end of the simulation. RUN 27 Run 27 was a HOVR maneuver with a maximum rudder deflection of 10 degrees and an EYA of 30 degrees with the stern-plane being fixed at around 0

degrees. Runtimes and time steps were similar to that of Run 18. This maneuver was executed at a slower rate and in a direction opposite to that of Run 18. The quality of the agreement with experiment is similar to that of Run 18 except for the roll rate Figure 27) and the roll (Figure 28) which compare extremely well. This could be due to the slow rate or the direction of this maneuver or both. Figure 10 Axial velocity cut showing effects of propeller and deflected rudder. RUN 18 W1MI PROPELLER This HOVR simulation includes the propeller rotating at a fixed speed of 480 RPM and was simulated using the sliding interface technique. A time step of .001 seconds was used for this simulation yielding simulation speeds ranging from 0.03 seconds/hr (Cray T3E-900) to 0.034 seconds/hour (I13M-SP) The Y-component of the force history was predicted more accurately by the propelled simulation (Figure 29) though the sinusoidal pattern was still absent. The prediction of the moment about the Taxis (Figure 32) was also better. As a result the predicted yaw rate (Figure 33) and the predicted yaw Figure 34) match extremely well with experiment. Even though the initial 3° roll of the RCM was not modeled for this case, the roll rate shows better agreement with the experiment than the body-force case. DISCUSSION One of key features of unsteady simulations of the kind presented here is that small errors tend to accumulate and get amplified and cause the simulation to drift. This implies that force and moments may have to be computed with a level of accuracy not usually expected of steady computations. To achieve this, particular attention has to be paid to transition, turbulence modeling, grid refinement and algorithmic issues. Several partial runs were made to study the effects of turbulence model, timestep, sub-iteration count, initial conditions on the trajectories and none of these factors were found to make a substantial impact on the trends presented here. The RCM experiments were carefully conducted to in minimize experimental uncertainty using instrumentation with very high accuracy (Failer et al., 2001b). Qualitatively, the repeatability of the experiments were also extremely high. However, in the absence of quantitative measures of imprecision, the uncertainty bounds on the experiment could not be computed. The effective moments of inertia of the RCM were measured with an accuracy of 1% and an estimate of the added mass term was then subtracted from the measured value. Thus there is a possibility that the moments of inertia have a high degree of uncertainty in comparison to the rest of the experimental data. The angular rates and accelerations in maneuvering simulations are strongly coupled with the linear accelerations and velocities through complex hydrodynamic interactions. This makes error tracing and analysis extremely difficult since it is not easy to distinguish between cause and effect. CONCLUSIONS Given the complexity and the myriad sources of errors that can impact the kind of unsteady simulations presented here, the agreement with experimental data can be considered to be extremely good. However, there is also a need to identify and reduce the errors such that longer simulations can be run without being overwhelmed by the gradual error accumulation. Further work is needed in the areas of turbulence modeling, grid refinement and vortex preservation before such simulations can be run with a high degree of confidence in the results. It is hoped that the comparison with the component force and moment histories will shed some light on any changes in solution methodology aid "ridding strategies that will have to be made to further improve accuracy. ACKNOWIEDGE~ENTS This work was supported by grant N00014011045501050428 from the Office of Naval Research. The grant monitor is Dr. L. Patrick Purtell. This support is greatly appreciated. This work was also supported in part by a grant of HPC time from the Arctic Region Supercomputing Center and MSRC-

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in Computational Field Simulations, Honolulu, Hawaii, 2002. [30] Taylor, L.K. and Whitfield, D.L., "Investigation of the Accuracy of an Unsteady Incompressible Euler [201 Roe, P.L., "Approximate Riemann Solvers, Equation Algorithm" Proceedings of the International Parameter Vector, and Difference Schemes," Journal of Computational Physics, Vol. 43, 1981, pp. 357-372. [21] Sheng, C., Chen, J.P., Taylor, L.K. Jiang, M., and Whitfield, D.L., "Unsteady Multigrid Method for Simulatiing 3-D Incompressible Navier-Stokes Flows on Dynamic Relative Motion Grid," AIM Paper No. 97-0446, Jan. 1997. [22] Sheng, C., Taylor, L.K., Chen, J.P., Jiang, M., and Whitfield, D.L., Multigrid Computations of 3-D Incompressible Internal and External Viscous Rotating Flows," MSSU-EIRS-ERC-9~1, Feb. 1996. [23] Sheng, C, Taylor, L K., and Whitfield, D.L., "An Efficient Multigrid Accleration for Solving the 3-D Incompressible Navier-Stokes Equations in Generalized Curvilinear Coordinates," AIM Paper No. 94-2335, 1994. [24]Sheng, C., Taylor, L.K., and Whitfield, D.L., "Multignd Algorithm for Three-Dimensional Incompressible High-Reynolds Number Turbulent Flows," AIAA Journal, Vol. 33, No. 11, Nov. 1995, pp. 2073-2079. [25] Shih, T.H., and Lumley, J.L., "Kolmogorov Behaviour of Near-Wall Turbulence and its Application in Turbulence Modelling," Computational Fluid DYnamics, Vol. 1, 1993, pp. 43-56. [26] Stevens, B.L, and Lewis, F.L., Aircraft Control and Simulation, John Wiley and Sons, Inc. New York, 1992, pp. 3940. [27]Taylor, L.K., Arabshahi, A., and Whitfield, D.L., "Unsteady Three-Dimensional Incompressible Navier- Stokes Computations for a 6:1 Prolate Spheroid Undergoing Time-Dependant Maneuvers," AIM Paper No. 95-0313, 1995. t28] Taylor, L.K. Busby, J.A., Jiang, M., Arabshahi, A., Sreenivas, K., and Whitfield, D.L., "Time Accurate Incompressible Navier-Stokes Simulation of the Flapping Foil Experiment, Proceedings of the Ah International Conference on Numerical Ship Hydrodynamics, Iowa City, IA, Aug. 1993. [29] Taylor, L.K. and Whitfield, D.L., "Unsteady Three-Dimensional Incompressible Euler and Navier- Stokes Solver for Stationary and Dynamic Grids," MM-Paper No. 91-1650, 1991. Conference on Scientific Computing and Modeling, Eastern Illinois University, Oct. 1995. [31] Thomas, P.D., and Lombard, C.K., "Geometric Conservation Law and its Application to Flow Computations on Moving Grids," AIAA Journal, Vol. 17, No. 10, 1978, pp. 1030-1037. [32]Whitfield, D.L, and Taylor, L.K., "Discretized Newton-Relaxation Solution of High Resolution Flux- Difference Split Schemes," AIM Paper No. 91-1539, 1991. [33] Whitfield, D.L, and Taylor, L.K., "Numerical Solution of the Two-Dimensional Time-Dependant Incompressible Euler Equations," MSSU-EIRS-ERC- 93-14, April 1994. [34] Yang, Z., Georgiadis, N., Zhu, J. and Shih, T.H., "Calculations of Inlet/Nozzle Flows Using a New ke Model," AIM Paper 95-2761, 1995.

2 -— C '~' : P 2~2 _ ~ ~ _ ~ 2~6 _ ~ _ -:16 _ ~ ~ r ~ id 45 - - — .~ - ~ ~ ~'-2- i, I_ ~2: ~ _ .''' ~ ''"2:~"~"~2 ~j~ 1 _ _ ~ ' A 2' " ~ _C:dmpu~ ~ J - A,,,- Explement : 2 ~ 2, :'S: '2 ^: :~.2.,. Figure 11 Comparison of forces for Run 41 ~[ ~ %_' :j^: - 'C! 4 _ ~-~Comp~a~ ~22~':: f _ l _ :0 _ '':2,., ~-_~ Exp~rlmant ~2''~'2.,, :2A:122221222 S—nPl~ ~''l''/ .... . ~ . . ~ If Figure 12 Comparison of velocities for Run 41 it. - . .~ . :8 - 1: -342 At: _ ;C:4mputation As. ~ . . ,-~~,4~, EXp~iflm~~= ~'.22....~ at) ~ :.. .,:. ,:: >4l. rime (.$) Figure 13 Comparison of position for Run 41 Figure 16 1ll" _ 5. _ ~ . £ 1} - ' E .5* "lO - - = C:omputation E ~ Expertrnerit ~ > Sternplane .. , ~ , TIE" t,$)f Figure 14 Comparison of moments for Run 41 ~ 14 .! :§ ._ ~ 10 _ a34f _ 7 ~ 2;0 ~- .x' .a s~ -'10 .e ._ ~ ~: . ~_o . .~ ~ ~. ~r . ~ -~; -,( _~ 1 -~. ~ _ ~. ~ ~: ~: ~. . —O~mp~:atior * ~ Exp"Im~t . ~v ii .S~rnp~ne~ L ~ I ~ I I ~ ~ ~ ' ~f ~ ~: Figure 15 Comparison of angular rates for Run 41 . _ 7 _ Computa~ - ~ ~ Exp~ment _~ To~- ~S~Fnp~r~ _f ~;.~ ~ ~ i ~ ~.~.i Jp,,, ,~ Pil~ ~—f . —~v i,~ ~ ~ ~k;~+ ~ ~—jl'—~—'~ -+ . . Ti~ 1~) Comparison of orientation for Run 41

my; IF :1 . I. - .. ~ ... .. ... = ~ 2 4: fi ~ ~~ . ~ _ _:~_ . ~ -^ _ ,~iN ~ .0,~ ~ :~2~' 74~ 3 I~ ~ .' '~4~t3:_— 3 i~f ; ~ ,`_ ;! ~ ~ ~ Exp~i~ment F 3" ~ '3~33 fords ': ~ i:3 :..' ~ i :.:.ii.ii.~_ ; _~:i 'I ~ —Inns A___ T.i~ ~ ~ Figure 17 Comparison of forces for Run 18 .4 Is ~ 2 3~ 8: ~ _ ' en ~ r -1 a - 'pupation ,~,~v ~ 3- S;~9S-::V3 ~ <<,^~ 3.o~~~56. p ,, 3 ~ .,, a: jj,,.j:ij,3 333.; '~c'e ~ ~ ~ '''"' I'd'' ~ ~'23~ ~ _ - _r~~—·~ —e~ REV—-r i,.. <.::-xs~ vet : ' 1 1 -, I Tamp Figure 18 Comparison of velocities for Run 18 lo. ,_. hi - . ~,4 ,3~ ,:3i~ _ 32, :' _ c0~ aft: i: —~—,, Experlment ~~: ~ ~ ~ Rough 2: 4: TI~ 1',): Figure 19 Comparison of position for Run 18 ~- _ Jl ~ u in. ~ ~0 :~0 . - _ Compmadon A,, Experiment ~;; ~P~IJd~ ~ 'i'.j, 4~.,OiO~.b . .,:: ::':i: ::*. :< . S 2 : A . :. ~ t _~ ~ air i' Tim t.~1 Figure 20 Comparison of moments for Run 18 1 _ :C~mp~6w · ~ ~ EXpof~lt~ i.:f.: :.: Rudder ;—7~ Fit ?~ I' lo i' ~ ~ ~ _ ~~ .,. * ~ >* <~ " I*** *<a *o*.~.:~> ^~— ~ _ ~ -18 _ _- _ a _ ~ r~ ; * ~ ::~ i: _ . 1 . I 3 Tim A) Figure 21 Comparison of angular rates for Run 18 IN _ :~ ._ I-= ~ - : _ Ida . , , . , ~ S. i.i i ... i....; ~: .= .. _ ~,~1 ~~ _ ~.~ —~.~ .. _~ '~A _ `,, .. ; i;. _ _ ~ T~ ~~ -I _ C>ompu~tion ~ .` ~ it:: ~~ -~ ~ Experiment it ;i. - Ever i: :: . A lo. _ : - . ~ .. i ; ~ ~ 2 ~ I Tim. 1~' Figure 22 Comparison of orientation for Run 18

~3 hi it i_ . . -at: - ' ~ ~ l _~' I: , - : gab 1411: ' - _ -:l. :aj' - . I: - 0' :~: Tie [.~) _ Compression -- -~ E - eriment ; :~:~: Rudder ' - Y fly \ · ~ 3~ .. A . ., i, bomb,, t$J~ '.: ~ ~ ~ Fii~ ~ :~ .. ,' :. T: :3'': :' :: :; ,.. .': 'A ' S~ And ·..C..ii"'.V---.::: v.:: ~ _ l. Figure 23 Comparison of forces for Run 27 . at. _ ._ ~ :' ., . -1 _ . _~omp~tion .? ~ ill . 3.~...,W,F,V,,,,,~ ~ ~ Expenme~ .. . i ~ . . , W . . .~ —Ed ~ 1 1 ~ :. 5: :~e Aid (a.) Figure 24 Comparison of velocities for Run 27 :69 1 :~ _. ~ 4 . ~ -__ ._._ ~ ,. _ _, . = pomp - .tlon ., .,. -~ -. E~rimen' ~ Rudder ._ ·: c; ~ ~ . . Tall .... - P-~ - = ~ L : :=_~ —~, __ far ' :` _ Roll Figure 25 Comparison of position for Run27 Figure 28 . _~. Yen ~ - ~_~. ~ ~ - - ~:~:~ .;.: :.,:.::jijj :::::j.~:.j:::j:~ al i: ~ . , ,, . —~ . i—3 22: ~ i,' ~ :2,~ ~ ,3 I'll Bi ', i, By' ~.j to' _ Cornputa~on -` .~ . S~] ~ ~ -: ~ E~'iiment :~' f A + - '> .. ~ ~ :: R. Our ~ ~ ~ i ~ ~ - al . . ~ . ' ~ ~ AH To ~~ Figure 26 Comparison of moment for Run 27 I: is. .16 a . - ~ IU . — ., .,., , .~ ~ , I ~ 1 _ . . —Worry - tent a,. ~ Experiment 8~ - . - ~~ ~ A. _~ i. ... I :~ .~ i: . . .. .: . :: i. ,. . a... Ti Tame ~) Figure 27 Comparison of angular rates for Run 27 w ~ - a Compumion - - -,. }A - Met ; --My Rawer I,, P4,~ Io ~ ~ ~ ' .,,;~ _ i .~ ~ ~ at, lo_ ~ . . . ~ .~ _ Ya. ash : ~~ ~ . ~ · 1 18 Tim ts' ::: .. i. . ~ :: Comparison of orientation for Run 27

- ~ is ·4 - ~ ~ - : ~ .1: - .1~1l Figure 29 Comparison of forces for Run 18 with propeller u ' At: ^,:~,s,~ >USED A 55J ~~ 2 ~ i' ~ :. 5,, 1 _~ _ S—UW,iOn 1p~lhF) · in, .> a,,,, ~ ~~ ~ F~) ° ~ f~d~ ~ ~ sit , ~ .__ _ , :W : 'it i. :~ ~ ha ,: .;: ~ i'~ Figure 30 Comparison of velocities for Run 18 with propeller ~ _ _ ~ At. >_a = ~trro~u - ' ~ +—Gbn~on(Body Force} ~ - ·* Expert ~? . q[~— 2:~ .— I: As ~~ ~ ~ ^ ~ ~ ~ , C~7 . .. <. I, _ : · n~ use Figure 31 Comparison of positions for Runl8 with propeller .~ _ I 1: ~ 1 1 _ a I I |, F ~~s,:s~. ~,s ~ ;, sass X..S<~xS {v ~ set ~ : so ii'.. .~ n ~ ~ —Ma {Propell~lr~ ~~ ~¢ __ ~, ^~ ~~ldli-` (~forc*:ll ~: ~ . ~~- 'A ,, . ~ _ , ~ ~ -_ ~ a ~~ ~ 81 s ~ ~ ~ U ;~;~^ it, '-I: ~ ~ ~~ ~ _ _~ ~.ai6, it. . ~ I; Id,. ~ .l *. a ~ I Menu———f Amp force) ~~.~ ~ ~ ~~~ y~f --e - ~ "p~i~ ' I <,< ~ few _ s;^ Ret ~ _ ~ ; ''1 ~ , 1 . ::' ~ -I 1 s' Ja al b ~ i: -I ~ ll~4s, Figure 32 Comparison of moments for Run 18 with propeller ear . ~:4 Or :l = ID em by .. ~~ :~ , ~ ~ ~ sad ~ ~~',,,6~*F ::: ~ :::: .. 2: ~ > _ I ~~ ! ~~ ~ —~ ~ _ —Oa my Exp~ii~niS :s,:s;.:::~ F—— ~ , ~ s, ~ use Figure 33 Comparison of angular rates for Run 18 with propeller. _w . ~ ~ ~ A ~ I'' ~e :. ~1 ,8 is ~ I, ~ i L . . ~ _ — .'" ~ ... ::: ~ : ~. -it :041E ~:~' user :' ~- :~ :~: At. hi: -.H _~on~pe~ at, - ~ S.r.~on {loopy For - t I, ~ '~per~ent ~ _ . ~~, ~~ w" ~ i'' ~ .: $[ 11 i'l ~ ' I ~ ~ ~ ~ b ~ it ABE.. Figure 34 Comparison of orientations for Run 18 with propeller

DISCUSSION In-Young Koh Naval Surface Warfare Center, Carderock Division, USA What is the benefit of using UNCLE as compared to Fluent (commercial code) for the Navy users? AUTHORS' REPLY I (D. Whitfield) am not that familiar with Fluent. However, I did not know that it had the capability to perform 6-DOF maneuvering calculations based on a high resolution solver with rotating propulsor and moving control surfaces on grids with y-plus of one at Reynolds numbers of 10**9.