Software Design Document for a Six DOF Unsteady Simulation Capability in ANSYS-CFX

Transcription

1 Copy No. Defence Research and Development Canada Recherche et développement pour la défense Canada DEFENCE & DÉFENSE Software Design Document for a Si DOF Unsteady Simulation Capability in ANSYS-CFX ANSYS Canada Ltd. 554 Parkside Drive Waterloo, Ontario NL 5Z4 Phone: FAX: ansysinfo@ansys.com Contract Number: W Contract Scientific Authority: Dr. eorge Watt, george.watt@drdc-rddc.gc.ca, et 381 Defence R&D Canada Atlantic Contract Report DRDC Atlantic CR January 007

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3 Software Design Document for a Si DOF Unsteady Simulation Capability in ANSYS-CFX ANSYS Canada Ltd. 554 Parkside Drive Waterloo, Ontario NL 5Z4 Phone: (519) FAX: (519) ansysinfo@ansys.com Contract: W Scientific Authority: Dr. eorge D. Watt, george.watt@drdc-rddc.gc.ca, (90) Defence R&D Canada Atlantic Contractor Report DRDC Atlantic CR January 007

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5 Abstract This report presents the software design for a 6 degree-of-freedom submarine simulation capability in ANSYS-CFX ( CFX ). It documents the underlying theory, implementation in the CFX software system, verifies the algorithms, and presents a preliminary validation. Two main approaches to the problem are considered, both using a body fied mesh. In one, the mesh is rigid and moves in 6 DOF with the submarine so that apparent body forces for 6 DOF motion must be accounted for in the fluid equations of motion solved by CFX. In the other, the mesh translates but does not rotate with the submarine; it deforms to follow the submarine locally using the CFX Arbitrary-Langragian-Eulerian (moving mesh) formulation of the fluid equations and requires only the apparent body force terms for the linear accelerations. In either approach the equations of motion for the submarine (solid body model) are also solved to determine the apparent body forces and, if required, any mesh motion. The rigid mesh approach is chosen for initial evaluation. In this approach CFX solves the flow about the submarine, then passes the unsteady hydrodynamic forces on the submarine surface to its solid body model (which account for submarine inertia, buoyancy, propulsion, control forces, etc.), and receives back the solid body kinematic information needed to propagate the net coefficient update loop/time step. A second order scheme is used to integrate the fluid and solid body equations of motion in parallel. The method accurately predicts analytical potential flow predictions of ellipsoid added masses. Résumé Le présent rapport établit la conception de logiciel pour une capacité de simulation d un sousmarin selon 6 degrés de liberté (DDL) en ANSYS-CFX ( CFX ). Il documente la théorie sous-jacente et son intégration au système logiciel CFX, vérifie les algorithmes et contient une validation préliminaire. Deugrandesapproches au problème sont étudiées, les deu faisant appel à une maille fie de corps. Dans l une des deu approches, la maille est rigide et se déplace selon 6 DDL avec le sous-marin, de sorte qu il faille prendre en compte les forces apparentes de corps du mouvement selon 6 DDL dans les équations fluides du mouvement résolues au moyen du logiciel CFX. Dans l autre approche, la maille se déplace, mais ne tourne pas en même temps que le sous-marin; elle se déforme pour suivre le sous-marin localement au moyen des préparations arbitraires-langrangiennes-eulériennes CFX (de maille mobile) des équations fluides et requiert uniquement les termes des forces apparentes de corps pour les accélérations linéaires. Dans les deu approches, on calcule aussi la valeur des équations des mouvements applicables au sousmarin (modèle de corps solide) pour déterminer les forces apparentes de corps et, au besoin, tout mouvement de la maille. L approche aée sur une maille rigide est choisie en vue d une évaluation initiale. Dans cette approche, le logiciel CFX calcule la valeur de l écoulement autour du sous-marin, fait passer les forces hydrodynamiques instables eercées à la surface du sous-marin à son modèle de corps solide (pour tenir compte de l inertie du sous-marin, de sa flottabilité, de sa propulsion, des forces eercées sur ses commandes et d autres facteurs) et reoit les données cinématiques rétrospectives du corps solide nécessaires à la propagation de l intervalle de temps/de la boucle de mise à jour des coefficients qui suit. Un schéma de secondordre sert à l intégration des équations des mouvements en parallèle du corps solide et fluide. La méthode prédit avec précision les prévisions analytiques d écoulement potentiel des masses ajoutées ellipsoïdales. DRDC Atlantic CR i

6 Eecutive Summary Introduction DRDC Atlantic is collaborating with ANSYS Canada to develop an unsteady, 6 degree-offreedom (DOF), computational fluid dynamics (CFD) submarine maneuvering simulation capability. This is needed to validate fast, coefficient based simulations used to investigate maneuvering limitations and establish safe operating envelopes for underwater vehicles. Several countries (eg, the US, UK, France) validate their simulations using a free swimming scale model which is currently the best predictor of full scale submarine maneuvering performance. Such a facility is unaffordable by Canada. Validating with CFD is affordable and can provide better detail. The disadvantage to using CFD is that its predictions are not as reliable as eperimental measurements. But CFD technology is evolving quickly and it is worth evaluating the capability now. By collaborating with a successful commercial CFD vendor, there is the potential for commercialization which would minimize ongoing maintenance and development costs. This report describes the current ANSYS Canada implementation of the simulation capability using their commercial CFD code CFX. Preliminary validation work is also presented. Principle Results The current CFX implementation requires that the flow field be discretized with a rigid, body fied mesh etending from the surface of the vehicle out to the far field. The mesh and boat move together controlled by the same 6 DOF solid body equations of motion used by the DRDC Submarine Simulation Program (DSSP). CFX solves the flow about the submarine, passes the unsteady hydrodynamic forces to the solid body equations (which account for inertia, buoyancy, propulsion, control forces, etc.), and receives back the velocities for the net time step. Theoretically, any maneuver can be modelled in which the boat is deeply submerged and isolated from any other vehicle or boundary. Significance of Results Evaluation of the CFX simulation capability has begun. It is being used to investigate a submarine rising maneuver that generates a roll instability that can result in ecessive roll as the submarine surfaces, a maneuver operators have asked DRDC about in the recent past. Conventional quasi-steady coefficient based hydrodynamic models have difficulty modelling this maneuver. The submarine can be modelled as an isolated deeply submerged body throughout the maneuver because the free surface is unimportant in the development of the underwater roll instability. The evaluation is taking place at the University of New Brunswick using the DRDC generic submarine shape for which etensive eperimental data are available. Future Plans Preliminary work has shown that an alternative approach to the problem using a moving mesh formulation is feasible. This would allow the mesh to deform with time as the boat moves toward or away from a boundary or other vehicle, as would be the case for littoral or two-body problems. Minimal development work is required to implement moving mesh because the basic capability already eists in CFX. Finally, it is desirable to incorporate a generalized version of this capability in subsequent commercial versions of CFX. A commercial capability supported by all CFX users is an economical way to handle future maintenance and development costs. ANSYS Canada Ltd., 007, Software Design Document for a Si DOF Unsteady Simulation Capability in ANSYS-CFX, DRDC Atlantic CR ii DRDC Atlantic CR

7 Sommaire Introduction RDDC Atlantique s est associé à ANSYS Canada pour mettre au point une capacité instable de simulation de manœuvres de sous-marin de dynamique des fluides computationnelle (CFD) selon 6 degrés de liberté (DDL). Cette mise au point est nécessaire en vue de la validation de simulations rapides, fondées sur des coefficients, qui servent à l étude des restrictions des manœuvres et à l établissement d enveloppes de fonctionnement sûr pour véhicules sous-marins. Plusieurs pays (dont les États-Unis, le Royaume-Uni et la France) valident leurs simulations au moyen d un modèle-échelle autonome qui est actuellement le meilleur outil de prédiction du rendement de manœuvre d un sous-marin pleine grandeur. Le Canada n a pas les moyens de se doter d une telle installation. La validation à l aide de la CFD est cependant abordable et permet d obtenir de meilleurs détails. L inconvénient de la CFD, c est que les prédictions qu elle permet d obtenir ne sont pas aussi sûres que les mesures obtenues lors d essais. La technologie CFD évolue cependant rapidement, et il vaut la peine d évaluer la capacité maintenant. Une collaboration avec un fournisseur commercial prospère de CFD ayant réussi offre une possibilité de commercialisation, ce qui réduirait au minimum les frais permanents de développement et de maintenance. Le présent rapport décrit la mise en œuvre qu effectue actuellement ANSYS Canada de la capacité de simulation au moyen de son logiciel en code CFD commercial. Le travail de validation préliminaire est aussi présenté. Résultats Dans le cadre des travau en cours de mise en œuvre du logiciel CFX, le champ d écoulement doit être discrétisé au moyen d une maille fie de corps rigide qui s étend de la surface du véhicule jusqu au champ lointain. La maille et le navire se déplacent ensemble et sont commandés par les mêmes équations de mouvement de corps solide selon 6 DDL dont RDDC se sert dans le cadre de son programme de simulation de sous-marins (DSSP). Le logiciel CFX permet de solutionner l écoulement autour du sous-marin, fait passer les forces hydrodynamiques instables au équations de corps solide (pour tenir compte de l inertie, de la flottabilité, de la propulsion, des forces eercées sur les commandes et d autres facteurs) et reoit les vitesses de retour pour l intervalle de temps qui suit. En théorie, n importe quelle manœuvre peut être modélisée de manière à ce que le navire soit immergé à une grande profondeur et isolé de tout autre véhicule ou limite. Portée L évaluation de la capacité de simulation du logiciel CFX a commencé. Elle sert à l étude d une manœuvre de remontée d un sous-marin qui génère une instabilité susceptible d entraîner un roulis ecessif à mesure que le sous-marin remonte à la surface, manœuvre que les opérateurs ont demandé à RDDC d étudier récemment. Des modèles hydrodynamiques fondés sur des coefficients quasi stables classiques posent des difficultés pour la modélisation de cette manœuvre. Le sous-marin peut être modélisé comme corps isolé immergé à une grande profondeur dans toute la manœuvre parce que la surface libre n a pas d importance dans le développement de l instabilité du roulis sous l eau. L évaluation a lieu à l Université du Nouveau-Brunswick au moyen de la forme générique de sous-marin de RDDC, à l égard de laquelle on dispose d une grande quantité de données epérimentales. DRDC Atlantic CR iii

8 Recherches futures Les travau préliminaires ont montré qu une autre approche au problème, qui fait appel à une préparation de maille mobile, est possible. Cela permettrait une déformation de la maille à mesure que le navire se déplace en direction d une limite ou d un autre véhicule ou qu il s en éloigne, ce qui serait le cas des problèmes à proimité du littoral ou en présence de deu corps. Il faut mener des travau minimes de développement en vue de la mise en œuvre de la maille mobile, la capacité de base étant déjà prévue dans le logiciel CFX. Enfin, il est souhaitable d intégrer une version généralisée de la capacité à des versions commerciales subséquentes du logiciel CFX. Une capacité commerciale dont se serviraient tous les utilisateurs du logiciel CFX constitue un moyen économique de s occuper des frais ultérieurs de développement et de maintenance. ANSYS Canada Ltd., 007, document de conception de logiciel pour une capacité de simulation instable de siddl en ANSYS-CFX, document CR de RDDC Atlantique. iv DRDC Atlantic CR

9 Software Design Document for a Si DOF Unsteady Simulation Capability in ANSYSCFX Prepared for DRDC Atlantic CFX Report # 01 ANSYS Canada Ltd. 554 Parkside Drive Waterloo, Ontario NL 5Z4 Phone: (519) Fa: (519) ansysinfo@ansys.com December, 006

10 Disclaimer ANSYS Canada Ltd. ( ANSYS ) makes no representation with respect to the adequacy of the results contained in this report, for any particular purpose or with respect to its adequacy to produce any particular result. Defence Research and Development Canada ( DRDC ) assumes all professional engineering responsibility connected with the use of these results. DRDC acknowledges that the results contain modelling assumptions and that the results are not warranted to be free of error. ANSYS shall not be responsible for any use made of this Report or any other report, materials, equipment, or information arising from, or related to, the work described herein either by DRDC or any third party. ANSYS Canada Ltd. Reference: Project Number: 073 ANSYS Contract Number: W /001/HAL Report Number: 01 Date: December 1, 006 This report written by: Stephen Dajka Manager, Directed Development Dr. Philippe odin CFD Integration Developer Dr. Andrew erber CFD Consultant, University of New Brunswick Page of 55

11 Contents 1 Introduction... 4 Theory... 5 Nomenclature...5 Kinematic Relationships...7 Rigid Body Equations of Motion...11 Fluid Equations of Motion...18 Tracking Body in Inertial Frame...0 Boundary Conditions... 3 ANSYSCFX Implementation... 6 Solution Procedure...6 Variable Time Steps...8 Internal Data Arrays...9 Sequencing of CFD and Solid Body EOM Solution...30 Solid Body Solver Organization Validation... 3 Validation Using Ellipsoid Shapes...33 Validation with Fully Appended Submarine Shapes...48 Full NavierStokes SiDOF Submarine Simulations Recommendations for Future Work Page 3 of 55

12 1 Introduction This report presents the detailed software design of a sidegree of freedom (DOF) submarine simulation capability in ANSYSCFX. The intent of this document is to provide a fully documented manual of the underlying theory, validation and verification process, and the resulting implementation into the ANSYSCFX software system. The report serves as a means for communication with DRDC on all aspects of the new model, and provides the historical continuity for future development activities. The approach taken in the model development has been to eplore and document alternative approaches, amenable to ANSYSCFX, for the solving sidof submarine motions. It is possible that all approaches will ultimately be available for use. This work is outlined in Chapter. This is followed by a description of the software implementation into ANSYSCFX in Chapter 3. The software implementation is on going and therefore this section is subject to change, for eample as the ANSYSCFX version level changes (currently at V10) aspects of the implementation details will need to be updated. The net chapter in the report covers validation (Chapter 4) which up to the present time has focused on clearly defined analytical or semianalytical benchmarks. This methodology, using derived added mass coefficients with viscous corrections and highly accurate eplicit solver solutions, is to provide a base level of verification and validation (and means for efficient debugging) before considering full coupling with the Navier Stokes equations. Detecting subtle implementation details or mistakes are much easier to detect using this approach. In Chapter 5 the remaining future work is described, which is focused primarily on si DOF simulations fully coupled to the governing fluid equations. Page 4 of 55

13 Theory This section describes the rigid body equations of motion (EOM) as generally used by DRDC in their underwater vehicle simulations. Since the ANSYSCFX implementation is eploring two alternatives to si DOF simulations it is important that the steps in the derivation of the submarine EOM be described carefully. As will be shown, the two alternative approaches require different forms of the submarine EOM. Furthermore the fluid EOM are also developed for general motion in a translatingrotating frame of reference. Here again different forms are required depending on the approach taken for simulating the submarine motion. Following presentation of the EOM theory, boundary condition implementation is described considering alternative simulation approaches. A number of auiliary relationships are required for overall implementation of the model. These are particularly important for relating quantities across frames of reference. These relationships are described in detail. Nomenclature As much as possible the nomenclature followed is that already used by DRDC in its submarine simulation work. In Table I the basic nomenclature is organized. The body fied coordinate system for the submarine is placed along the hull ais (chosen during the CFD mesh generation process) with the ais aligned forward, the yais to starboard, and the zais through the keel. Since the submarine coordinate system is not located at either the center of mass () or buoyancy (CB) the derivation of the EOM for the submarine must take this into account. In Figure 1 is shown the body fied coordinate system in relation to the as well as the linear (u,v,w) and angular (p,q,r) velocities. O 1 r,u,p Hull ais y,v,q z,w,r Figure 1 Body fied coordinate system Page 5 of 55

14 BρVg Buoyancy g ravitational constant K,M,N Body ais moments I ij Moments and products of inertia for submarine (about the center of gravity) l Length of submarine m Total mass including ballast p,q,r Body ais submarine angular velocities r Position vector t Time u,v,w Body ais submarine velocities U Submarine speed V Volume of eternal hydrodynamic envelope W Submarine weight including ballast,y,z Body ais coordinates,y,z Submarine center of gravity in body coordinates B,y B,z B Submarine center of buoyancy in body coordinates o,y o,z o Inertial coordinates X,Y,Z Body ais forces v v v A,α pi & + qj & + r& k φ Roll, rotation about body ais θ Pitch, rotation about body yais ψ Yaw, rotation about body zais ρ Fluid density v v v,ω pi + qj + rk Subscripts/Superscripts Indicates submarine center of gravity in body coordinates B Indicates submarine center of buoyancy in body coordinates S Static stability forces o Indicates inertial coordinate system CFD Forces/moments derived from CFD solution P Forces/moments derived from propulsor Time derivative Table I Submarine nomenclature Page 6 of 55

15 Kinematic Relationships.1.1 Particle Motion in a TranslatingRotating Frame of Reference In the development of the si DOF motion equations, a number of kinematic relationships are needed for describing general particle motion in a translatingrotating frame of reference, and rigid body motion in translation and rotation. We will start with the most general motion situation first. z y z o O 1 R y o r o r 1 A O o Figure Particle displacement vectors for A with O 1 in translation and rotation To describe an accelerating fluid particle in a moving (translating and rotating) frame of reference consider an inertial coordinate system with origin O, and a moving coordinate system with origin O 1. We want the fluid particle position, velocity and acceleration relative to the aes at O 1. The inertial coordinate system by definition is non accelerating. The position of A, at an instant in time, can be obtained by vector summation as: r o R+ (.1) r 1 and the velocity by differentiation with respect to the origin O: d r dt dor dor + (.) dt dt o o 1 For the velocity of point A it is better to replace the last term with one differentiated with respect to origin O 1 (shown by a dt), which results in: d 1 d o r dt o d d r dt o r1 (.3) dt R Page 7 of 55

16 where: d o r1 d1r1 + dt r1 (.4) dt The angular rate of rotation of the moving aes,, now appears in Eq..3 with the ais of rotation acting through O 1, and a direction defined relative to the inertial coordinates. Now by further differentiation the acceleration appears: d r dt dor do d1r1 do dor + r (.5) dt dt dt dt dt o o 1 applying the rule and result of Eq..4 to the third and fifth terms respectively and combining like terms results in: d dt r dor d1r1 do d1r + + r1 + ( r1 ) + (.6) dt dt dt dt o o 1 Equation.6 describes the acceleration of a particle at position A at an instant in time. All of the terms, in order from left to right, represent the following accelerating motion: The acceleration of point A in the inertial coordinate system The acceleration of the origin O 1 of the moving aes relative to the inertial aes The acceleration of point A relative to the moving aes The angular acceleration of point A due to the moving frame The centripetal acceleration of point A due to the moving frame The Coriolis acceleration Note that the last three accelerations appear as a result of the rotation of the moving aes. To simulate flow in the moving frame in ANSYSCFX the user would be epected to define the following quantities at any instant in time: The angular velocity and acceleration and d o dt respectively. Note that this includes defining the rotation vector relative to the inertial frame. The velocity and acceleration of O 1, namely R d o dtand d o R dt respectively. The position of A relative to the origin O 1, i.e. r 1, which is available from the coordinates of each node in the CFD mesh. Page 8 of 55

17 The quantities appearing in Eq..6 that are not predefined are the acceleration, d1 r1 dt d1u1 dt, and velocity, d 1r1 dt u1 (that appears in the Coriolis term), of position A relative to O 1 which can be obtained from solving the fluid EOM. The forces acting on a fluid volume are required in completing the fluid EOM, of which some of them are apparent body forces resulting from the rotating frame reference. Development of the fluid EOM is the subject of a subsequent section and for that purpose we finish by writing Eq..6 in a manner emphasizing the unknown velocity components u 1 : d o u dt o d d u d o 1 1 o + + r1 + ( r1) + u 1 (.7) dt R dt dt.1. Rigid Body Kinematics Kinematic equations are also needed to describe the rigid body motion of the submarine. In developing these equations it should be noted that the rotational state of the submarine is defined as ω, which can be different than the rotation of the translatingrotating coordinate system,, fied to the submarine at O 1. For rigid body motion around a fied point O 1, the velocity and acceleration of point A located on the body at a distance r 1 from O 1 is obtained by: d r dt d dt r ω r (.8) 1 1 ( r ) ω& r + ω ω (.9) 1 However, for general rigid body motion relative to a fied (inertial) frame of reference at O we have, using Fig. as a reference: r R+ (.10) 0 r 1 d0r0 d0r d0r1 + (.11) dt dt dt where the third term from the left is replaced using Eq..4 while noting that for a rigid body d 1 r 1 /dt0. This gives for velocity and acceleration: Page 9 of 55

18 d d 0 r dt dt r d 0 + ω r1 (.1) dt R d0r + ω& r1 + ω ( ω r1 ) (.13) dt or writing in terms of the velocity u 0 : d 0 u dt 0 d0r + ω& r1 + ω ( ω r1) (.14) dt giving the acceleration of point A on the rigid body relative to an absolute coordinate system. This equation can be compared to the equation for a fluid particle (Eq..7), where differences relate to the fluid particles relative velocity to O 1. What remains is to relate Eq..14 to the situation where a local coordinate system is fied at point O 1 on the submarine body. Consider the case where the coordinate system at O 1 is not rotating but does translate so that 0. This situation is relevant to the CFD modelling approach where a moving mesh is utilized to handle the submarine rotational motion. In such a model the coordinate system orientation is not fied to the submarine body, although the aes origin is fied at O 1. The ais over time does not remain aligned with the hull ais and occurs because ω. The other case relevant to submarine motion is the situation where the submarine rotational state defines the O 1 coordinate system orientation such that ω. This case maintains the O 1 ais aligned with the hull ais, the yais oriented starboard and the zais toward the keel at all times (based on the DRDC conventions). These two cases are identified as it is intended in this document to identify the strengths and limitations of CFD models based on either approach. Finally it should be noted that in the situation where ω it can be shown (using the transformation implied by Eq..4) that the angular acceleration of the rigid body in the O 1 reference frame is the same in the inertial frame O, and therefore & ω&. Page 10 of 55

19 Rigid Body Equations of Motion.1.3 EOM with TranslatingRotating Coordinate System The equations of motion for the submarine are formulated around a body fied coordinate system, with the coordinate system on the ais of the submarine hull (located at O), but not necessarily coincident with the center of gravity or buoyancy of the vessel. For the linear motion of the submarine its acceleration is computed by evaluating the sum of all applied forces, the net effect of which act through the submarines center of gravity. The appropriate equation of motion here is: F mu& (.15) where u& is the acceleration of the center of gravity. In general rigid body motion, the acceleration of origin O on the body (which is moving relative to an inertial reference frame) is related to the center of gravity acceleration based on Eq..14: O ( r ) u & u& + ω& r + ω ω (.16) The acceleration, u& O, is more convenient, for analysis purposes, to have in the frame of the rotating body which leads to: ( u& ) + ω u + & ω r + ω ( r ) u & ω (.17) O yz O where ω in this case is the angular velocity of the body and fied coordinate system so that ω ( is the angular velocity of the coordinate system as outlined in section.). The result is that ( u& O ) yz and u O are relative to the body fied coordinate system yz. Substituting this equation into Eq. (.15) results in three equations for each component of acceleration as presented in full in Eqs. (.4). The angular momentum of the submarine must also be considered. For applied moments about a point O on the body the equation of motion applies: M O H& O (.18) However we are interested in using moments of inertia at the center of gravity, which is obtained by the relation: H & H& + r mu& (.19) O Page 11 of 55

20 Substituting Eq. (.19) into Eq. (.18) and noting that this only applies to a stationary frame. For a rotating frame the time derivative of H must be epanded to give: where and ( H& ) + H + r mu& M O H & + r mu& ω (.0) yz H & [ I]α (.1) H [ I]ω (.) I Iy Iz I Iy Iyy Iyz (.3) Iz Izy Izz [ ] The moments of inertia, [I], are evaluated at the center of gravity of the submarine. On this basis substituting Eqs. (.17) and (.1) thru (.3) into Eq. (.0), and neglecting small terms involving the square of coordinates (e.g. ), results in the three equations for angular acceleration as presented in Eq. (.5). Alternatively the parallel ais theorem can be applied to transform the moments of inertia to a parallel ais acting through body fied location O. In this case it can be shown that terms involving the square of the coordinates cancel out to eactly zero. In either treatment of [I], the form of Eq. (.5) is the same, however it would be preferable to have [I] supplied relative to parallel ais O to avoid any approimation..1.4 System of Equations for Solution Aial (ais), lateral (yais) and normal (zais) forces: [ u& vr+ wq ( q + r ) + y( pq r& ) + z( pr + q& )] XCFD + X S X P m + [ v& wp+ ur y( r + p ) + z( qr p& ) + ( qp+ r& )] YCFD + YS YP m + [ w& uq+ vp z( p + q ) + ( rp q& ) + y( rq+ p& )] ZCFD + ZS ZP m + (.4a) (.4b) (.4c) Rolling (ais), pitching (yais) and yawing (zais) moments: Page 1 of 55

21 y I p& + I q& + I r& + z ( I I ) qr ( r& + pq) I + ( r q ) I + ( pr q& ) z y z yz [ ( w& uq+ vp) z( v& wp+ ur) ] KCFD + KS + KP m y ( I I ) rp ( p& + qr) I + ( p r ) I + ( qp r& ) z y z [ ( u& vr + wq) ( w& uq+ vp) ] MCFD + M S + M P m z ( I I ) pq ( q& + rp) I + ( q p ) I + ( rq p& ) y yz y [ ( v& wp+ ur) y( u& vr + wq) ] NCFD + NS + NP m I I yz I y + z + + (.5a) (.5b) (.5c) Applied forces: X CFD, Y CFD, Z CFD X S Y S Z S X P, Y P, Z P ( W B) sinθ ( W B) cosθ sinφ ( W B) cosθ cosφ Force vector obtained from integrated solid body surface forces (pressure and shear) Weight and buoyancy component based on direction of gravity vector Weight and buoyancy component based on direction of gravity vector Weight and buoyancy component based on direction of gravity vector Thrust vector obtained from a model of the propulsion system for the submarine Applied moments: K CFD, M CFD, N CFD K M N S S S ( y W y B) cosθcosφ ( z W z B) cosθsinφ B ( W B) cosθcosφ ( z W z B) sinθ B ( W B) cosθsinφ + ( y W y B) sinθ K P, M P, N P B B B B Moment vector obtained from integrated solid body surface forces (pressure and shear) Weight and buoyancy component based on direction of gravity vector Weight and buoyancy component based on direction of gravity vector Weight and buoyancy component based on direction of gravity vector Torque/moment component obtained from a model of the propulsion system for the submarine. Page 13 of 55

22 Auiliary relations: Submarine velocity in inertial coordinates & o y& o ( sinφsinθcosψ cosφsinψ) + ( sinφsinψ cosφsinθcosψ) ucos θcosψ + v w + ( cosφcosψ + sinφsinθsinψ) + ( cosφsinθsinψ sinφcosψ) ucosθsinψ + v w z& o usinθ + vcosθsinφ + wcosθcosφ Submarine acceleration in inertial coordinates & u o & v o ( sinφsinθcosψ cosφsinψ) + &( sinφsinψ cosφsinθcosψ) u& cos θcosψ + v& w + ( cosφcosψ + sinφsinθsinψ) + &( cosφsinθsinψ sinφcosψ) u& cosθsinψ + v& w & u& sinθ + v& cosθsinφ + w& cosθcosφ w o Body fied angular velocities & φ ( rcosφ qsinφ) tanθ p + + & θ qcosφ rsinφ ψ& r cosφ + qsinφ cosθ System of equations: m mz my 0 m 0 mz 0 m 0 0 m my m 0 0 mz my I I I y z mz 0 m I I y I y yz my m 0 I I I z z yz u& X v& Y w& Z p& K q& M r& N RHS RHS RHS RHS RHS RHS X Y Z K M N LHS LHS LHS LHS LHS LHS (.6) Page 14 of 55

23 The righthand side matri is comprised of the applied forces and moments (designated by the subscript RHS), and the nonacceleration terms on the lefthand side, of Eqs..4 and.5. The nonacceleration terms are identified with the subscript LHS. The solution of Eq..6 results in estimates at the new time level for accelerations: u &, v&, w&, p&, q&, r& which in turn, with the time step know, allow for calculation of velocities: u, v, w, p, q, r Finally the calculation of auiliary derivative quantities (in the inertial frame) is then possible: & y&, z&, u&, v&, w&, & φ, & θ, ψ& O, O O O O O and from these the integrated quantities for position and angular movement: O, yo, zo, φ, θ, ψ.1.5 EOM with a Translating Coordinate System F mu& (.7) ( u& ) + ω& r + ω ( r ) u & ω (.8) O yz Noting that u & O u O (& ) yz when the coordinate system at O is not rotating (i.e. 0). Noting again that M O H & O H& + r mu& (.9) ( H& ) + r mu& M O H & + r mu& (.30) yz H & ( & ) yz H when the coordinate system at O is not rotating. Since the ais of the submarine hull no longer remains aligned with the translating coordinate system the moments of inertia now become functions of time and must be reevaluated at each new time interval. This can be seen from the definition of H where: [ I] ( ω ) [ I& ] ω [ I ] & ω [ I ] ω [ I ]α d H & + & + (.31) dt Page 15 of 55

24 Therefore the equation for rotational motion should be stated as: M O ([ I& ] + [ I] α) yz + r mu& H & + r mu& ω (.3) It is now possible to formulate the system of equations to be solved to track the motion of the submarine. Note that in this case the equations are presented in terms of [I] relative to the center of gravity (C). If [I] is supplied relative to a body fied ais acting through O, then Eq. (.34) should be modified by substituting [I] based on the parallel ais theorem. The parallel ais theorem relates moments of inertia about a parallel ais at O to that acting through the C..1.6 System of Equations for Solution Aial (ais), lateral (yais) and normal (zais) forces: [ u& ( q + r ) + y( pq r& ) + z( pr + q& )] XCFD + XS XP m + [ v& y( r + p ) + z( qr p& ) + ( qp+ r& )] YCFD + YS YP m + [ w& z( p + q ) + ( rp q& ) + y( rq+ p& )] ZCFD + ZS ZP m + (.33a) (.33b) (.33c) Rolling (ais), pitching (yais) and yawing (zais) moments: z y z y [ w& zv& ] KCFD + KS KP I p& I r& I q& + I& p I& r I& q+ m y + (.34a) y y yz y y yz [ u& w& ] MCFD + MS MP I q& I p& I r& + I& q I& p I& r + m z + (.34b) z yz z z yz z [ v& yu& ] NCFD + NS NP I r& I q& I p& + I& r I& q I& p+ m + (.34c) This results in a solution matri ordered as follows: Page 16 of 55

25 Page 17 of 55 LHS RHS LHS RHS LHS RHS LHS RHS LHS RHS LHS RHS z yz z yz y y z y N N M M K K Z Z Y Y X X r q p w v u I I I m my I I I m mz I I I my mz m my m m mz m my mz m & & & & & & (.35) This matri is the same as the matri for the EOM with a rotating translating coordinate system, however the LHS contributions to the X,Y,Z forces and K,M,N moments are now changed. In particular the time variation in the moments of inertia must be calculated. The solution of Eq..35 results in estimates at the new time level for accelerations: r q p w v u & & & & & &,,,,, which in turn, with the time step know, allow for calculation of velocities: r q p w v u,,,,, Finally the calculation of auiliary derivative quantities (in the inertial frame) is then possible: ψ θ φ & & & & & & & & &,,,,,,,, O O O O O O w v u z y and from these the integrated quantities for position and angular movement: ψ θ φ,,,,, O O O z y Note that in the case when the inertial frame is not rotating, and the ALE moving mesh option is used to incorporate the rotational motion of the submarine, the finite angular movements (φ, θ, ψ) are used to calculate the new location of the submarine boundary.

26 Fluid Equations of Motion The fluid equations of motion, as for the solid body equations of motions, are derived on the basis of velocities with respect to the inertial coordinate system. For the fluid EOM Eq..7 is substituted in order to obtain transport equations for mass and momentum on the basis of local velocities. This process results in apparent body forces applied to the RHS of the transport equations..1.7 Fluid EOM for TranslatingRotating Coordinate System In this section the fluid EOM for the case of a translatingrotating coordinate system is presented. Since the coordinate system is free to move in all si degrees of freedom, the apparent body force contribution (F b ) contains four terms. This is different then with a translating coordinate system where the result is only one term in F b, however the EOM must then be cast in an ArbitraryLagrangianEulerian (ALE) form. These equations are shown in the net section. The conservation equations for mass and momentum are as follows where u i represents the velocity field relative to a local coordinate system: Mass Conservation: Momentum: ρ ρu + t j j 0 (.36) ρui t ρuu i + j j F S + F B (.37) where P ui F S + (.38) i j j and the apparent body force d or do F b ρ + r1 + i (.39) dt dt ( r ) + u 1 Page 18 of 55

27 Page 19 of 55 For turbulent simulations the kω based Shear Stress Transport (SST) is used and remain unmodified from their standard form. kequation: j k t j k j j k k P ku t k σ ω ρ β ρ ρ * (.40) ωequation: j t j k j j P k u t ω σ βρω ω α ρω ρω ω (.41) For details on the turbulence model source terms the ANSYSCFX documentation can be consulted..1.8 Fluid EOM in ALE Form for Translating Coordinate System An alternative form of the fluid EOM is to consider one based on a translating coordinate system fied to the submarine body. This however requires using an ALE form of the conservation equations to accommodate a deforming mesh. The essential aspect of this is the calculation of a mesh velocity u mj, calculated on the basis of domain boundary motion. The domain motion in this application is the submarine movement. The mass momentum equations in ALE form are as follows: Mass Conservation: 0 ) ( + j mj j u u t ρ ρ (.4) Momentum: B S j mj j i i F F u u u t u + + ) ( ρ ρ (.43) where + j i j i S u P F (.44)

28 Page 0 of 55 and dt R d F o b ρ (.45) Note that as previously described the apparent body force, for a translating coordinate system, has only one term. To account for the moving grid, the turbulence equations are also cast in ALE form: kequation: j k t j k j mj j k k P u u k t k σ ω ρ β ρ ρ * ) ( (.46) ωequation: j t j k j mj j P k u u t ω σ βρω ω α ρω ρω ω ) ( (.47) One additional set of equations must be solved to support the ALE application, and that is mesh displacement Laplace solutions which diffuse boundary motion into the interior of the domain. The resulting solution, over the time interval integrated, allows for the etraction of the mesh velocity u mj. The mesh displacement equations have the form: Mesh displacement equations: 0 ' Γ j i j (.48) where o i i i ' (.49) Note that the displacement diffusion coefficient, Γ, in Eq. (.48) can be a function of near wall distance, or mesh volume size. Tracking Body in Inertial Frame During the simulation of the submarine motion, tracking the position and orientation (in time) of the body w.r.t to the inertial coordinates is important.

29 When the body ais is oriented relative to the inertial ais through angles yaw (ψ), pitch (θ) and roll (φ) the order of finite angular rotations are important (yaw about z, pitch about y and roll about ). This order is accounted for through an appropriate transformation matri, starting from the calculated (from the EOM) rotation rates p, q and r and the current state angular state ψ,θ, and φ. This transformation is embedded in the auiliary equations previously described for the angular motion (body fied angular velocities): & φ ( rcosφ qsinφ) tanθ p + + & θ qcosφ rsinφ ψ& r cosφ + qsinφ cosθ and translational motion (submarine velocity in inertial coordinates): & o y& o ( sinφsinθcosψ cosφsinψ) + ( sinφsinψ cosφsinθcosψ) ucos θcosψ + v w + ( cosφcosψ + sinφsinθsinψ) + ( cosφsinθsinψ sinφcosψ) ucosθsinψ + v w z& o usinθ + vcosθsinφ + wcosθcosφ These inertial quantities can subsequently be integrated in time to give the cumulative displacement and rotation of the submarine over the simulation period. Note that when the fluid EOM solution involves only a translating frame of reference then the calculated angular displacements are used to modify the submarine boundary position, which activates the ALE form of the fluid EOM. Similarly the submarine acceleration in inertial coordinates is needed in order to supply one term in the apparent body force F b in the fluid EOM. In this case the auiliary equations are (submarine acceleration in inertial coordinates): & u o & v o ( sinφsinθcosψ cosφsinψ) + &( sinφsinψ cosφsinθcosψ) u& cos θcosψ + v& w + ( cosφcosψ + sinφsinθsinψ) + &( cosφsinθsinψ sinφcosψ) u& cosθsinψ + v& w & u& sinθ + v& cosθsinφ + w& cosθcosφ w o Page 1 of 55

30 Boundary Conditions The boundary conditions employed in the simulation depend on the state of the submarine. How to implement this in a manner that provides stable outer loop convergence in time is an area open for investigation. In the following discussion it should be noted that an opening refers to a boundary condition that can allow both mass into or out of a domain boundary. For flow out of a boundary, conditions upstream are employed to close equations along the boundary. For flow into the boundary, information must be supplied by the user. In cases where the boundary condition is set as inlet or outflow, then flow is known to be into or out of the domain respectively and the user supplies appropriate information to close equations. Note that boundary conditions cannot change during coefficient loop iterations within a time step, but can be changed when moving to the net time level. Two options are suggested for treating boundary conditions as follows. Option 1 It is proposed in the first option that the boundary conditions (see Fig. 4a) be treated as an opening with a total pressure inflow condition calculated based on the submarine state values u,v,w,p,q and r determined from the previous time step. This total pressure condition would be applied to surfaces +,, +y, y, +z and z when flow is entering the domain. When flow is eiting, upstream conditions are employed for closing equations. Flow direction is determined based on the flow field from the previous time step. The applied total pressure at the boundary uses relative frame velocities as follows: o Vrel P rel Ps + (.50) with the static pressure at the boundary, P s, taken to be zero (i.e. the reference pressure for the solution) on the premise of quiescent flow, and is independent of the fameof reference of the solution. The velocity V rel is calculated on the basis of Eq..3 and state information which is described more fully for Option in the net section. For eiting flow velocity and direction is computed as part of the flow solution. In this boundary condition treatment, adjacent controlvolumes at the boundary have source terms, the apparent body force terms derived previously, which establishes the flow speed and direction adjacent to an opening. The boundary condition total pressure of course uses a velocity in Eq..50 based on the solution of the solid body EOM rather than controlvolume values. The remaining boundary condition is the submarine surface which can be treated as a no slip wall. Page of 55

31 Figure 4a Opening boundary conditions employed at all faces relative to the coordinate system. Option Another boundary condition possibility is one where the + and z boundary conditions (see Fig. 4b boundaries highlighted in blue) are treated as inlets with a specified velocity profile based on the submarine state values of u,v,w,p,q and r determined from the previous time step. In this case it would be assumed that the submarine will always be moving forward and upward with possible sidetoside motion. The submarine surface is treated as a noslip wall while the remaining boundaries (, +y, y, +z) are Page 3 of 55

32 taken as openings. The openings require information in the case of inflow, therefore a total pressure condition is used similar to the treatment given in Option 1. As described in Option 1, the controlvolumes at the boundary have source terms based on the apparent body force terms, which establish the flow speed and direction along any opening boundaries. The body force sources are also present along the + and z inlets, possible problems with this are described later. Equation.3 is used to compute the applied velocity components (magnitude and direction) at the inlet boundaries for Option, and is repeated here for clarity: dor dt o dor d1r1 + + r1 (.3) dt dt or rewriting in a more familiar nomenclature as: u O u+ u 1 + r 1 (.51) where u O is the velocity of a fluid particle at the boundary of the domain, u the velocity of the submarine origin (obtain from the solid body solution) and u 1 is the relative velocity of the fluid particle at the boundary. The angular velocity can be either zero or fied at the angular velocity of the submarine. In either case the applied boundary condition velocity (and direction) must be relative to the bodyfied coordinate system and so Eq..51 is rearranged to solve for u 1 : u uo u (.5) 1 r 1 In solving this equation r 1 is the distance between the boundary mesh point and the coordinate system origin located on the ais of the submarine (chosen at the time of mesh creation). The absolute velocity of the fluid particle is taken as zero (a quiescent ocean) far away from the submarine. It is interesting to note that a constant ocean current velocity could be applied here by setting u O to some value. The velocity u is obtained from the solid body solution, as is (ω) when the coordinate system rotates with the submarine. The resulting equations then become for the case when 0: and for ω: u u 1 O u (.53) u uo u ω (.53) 1 r 1 Since the state of the submarine is always changing with time the boundary conditions are continuously changing also. In the first case the profile is constant spatially at a given boundary however for the second case the profile varies spatially due to the rotation component. Page 4 of 55

33 This second boundary condition is less desirable since inconsistencies can arise at the inlets between the calculated controlvolume based velocities (magnitude and direction), set primarily by the apparent body force terms, and the boundary conditions velocity magnitude and direction set based on the state of the submarine. For this reason Option 1 is tested first. Figure 4b Boundary conditions employed relative to the coordinate system. Page 5 of 55

34 3 ANSYSCFX Implementation This section includes a description of how the theory has been translated into a ANSYS CFX sidof simulation. A primary aim of the section is to outline in detail the approach in resolving discretely the highly nonlinear force/moment coupling between the submarine and fluid EOM. Solution Procedure The system of equations governing the solid body motion can be considered as a functional relationship between the accelerations (linear and rotational) of the submarine k k+1 ( y& ) and the time step (t k ), current velocity state (y k ) of the submarine, and the applied forces (F CFD ) and moments (M CFD ) as summarized in: where and k k k k+ 1 k k+ 1 ( t, y, F, M ) k k+ 1 y& f CFD CFD (3.1) k k+ 1 k k+ 1 FCFD + FCFD F CFD (3.a) k k+ 1 k k+ 1 MCFD + MCFD M CFD (3.b) k k 1 [ u& v& w& p& q& r& ] k k+ 1 + y & (3.c) [ u v w p q r] k y k (3.d) The solution procedure involves iterating within a time step to obtain the average force /moment( F CFD and M CFD) conditions over the time step that results in a new predicted velocity state (y k+1 u k+1,v k+1,w k+1,p k+1,q k+1 and r k+1 ) at the net time level (k+1). The k k+1 computed submarine accelerations ( y& ) are always the average for the time step and are used as follows: y y t + y k+ 1 k k+ 1 k k &. (3.3) Within a time step, at each coefficient loop, the hydrodynamic variables u, v, w and p are recomputed providing a new set of forces and moments ( F and M ) to use in the k+1 CFD k+1 CFD Page 6 of 55

35 solid body solution. The repetition of coefficient loops, and reevaluation of the submarine state, is continued until the RMS residuals on the fluid equations are reduced to below a specified limit. The fluid forces F CFD evolve with time and can be viewed, when integrated over time step t k as an impulse applied to the submarine. For a functional relationship F(t), and assuming a linear profile for F between t k and t k+1 noting that t k t k+1 t k, the trapezoidal rule can be applied to compute the integral yielding: F k k+ 1 t k F k + F k+ 1 t k (3.4) The average force over the time step is then: k k+ 1 k k+ 1 F F F (3.5) + and gives the form used in Eq. 3.a. Therefore the integration of fluid forces F CFD is treated with a first order approimation consistent with the trapezoidal rule. A higher order scheme could be employed by using information on F available at time t<t k. During the validation eercise solutions were obtained using Eqs. 3.a and b to represent the F CFD and M CFD influence, and yielded ecellent results as long as abrupt changes due to blowing ballast were not present. With blowing ballast, as the rate of change in V ti /V i (volume of blown air to initial volume of ballast) increased, divergent behavior was found. The only remedy for this was to modify Eqs. 3.a and b by applying a zero order approimation of the form: F k 1 k CFD F k 1 CFD + F k CFD (3.6a) M k 1 k CFD M k 1 CFD + M k CFD (3.6b) This modification enabled solutions to be integrated with very good accuracy up to the point of full ballast being blown and beyond. These results are visible in Simulation Cases 1 through 8 in the section Validation with Fully Appended Submarine Shapes to be discussed subsequently. From the validation eercise it is clear that some of the additional submarine models such as blowing of ballast, propulsion and deflection of appendages can induce divergence when using a trapezoidal treatment of the CFD based forces. These initial studies suggest that in certain situations, such as blowing ballast and spiral movements, a very strong influence is introduced. It is recommended that a technique using first order with zero order blending be incorporated. Page 7 of 55

36 Variable Time Steps In anticipation of the lengthy computation times that will be required in the full 3D CFD calculations a variable time step capability was added within the source code that calls the solid body EOM. The latest version of ANSYSCFX allows control of time step from the CCL, however in order to allow for the development for more sophisticated time step control particular to submarines a source code addition was made. The present algorithm follows closely that enabled in ANSYSCFX by default, but likely will be refined as research evolves on the submarine simulations. The present algorithm seeks to reduce computation time by using an optimal time step and is not based on any error estimation technique. The essence of the algorithm is control of the number of coefficient loops eecuted within a time step. Reducing time step, for a properly posed problem, will normally reduce the coefficient loops required. Similarly increasing the size of the time step increases the number of coefficient loops. From eperience it has been shown that the most efficient time step is one that uses approimately three coefficient loops per time step. This takes advantage of the implicit formulation of the time integration in ANSYS CFX which places no limit on the time step used, but also ensures that too much computational effort is not epended within a time step. Further to the variable time step nominally secondorder time integration is employed with ANSYSCFX as well. This second order method uses information from the previous two time levels to reduce errors in the calculation at the net time level. A RungeKutta highorder time integration is not practical in implicit solvers such as employed in ANSYSCFX. The details of the timestep algorithm is now described. In this algorithm N represents the number of coefficient loops required in the previous time step, which is desired to be held within the limits N min <N <N ma. The time step T is to be bounded by T min and T ma. The new time step is incremented or decremented based on scaling factors calculated using functions: F inc 10 1/inc F dec 0.1 1/dec (3.7a) (3.7b) Where inc and dec are real parameters supplied by the user with defaults set to 10 in both cases. The new time step is then computed based on N min N or NN ma being nonzero. In such cases: Page 8 of 55

37 N min N > 0 t MIN(F inc (NminN) *t,t ma ) NN ma > 0 t MAX(F dec (NNma) *t,t min ) (3.8a) (3.8b) In the above calculation the new time step is accelerated to larger/smaller values the further N is outside of the optimal range. Obviously other methods can be applied for time step control, one of these being that the user directly specifies the scaling factors as: F inc inc F dec dec (3.9a) (3.9b) This latter approach is also provided as an option but is not the default method. Internal Data Arrays State variables: Two state arrays are maintained of the submarine motion, denoted by y and yp, and are organized as follows: y(1) U yp(1) u& y() V yp() v& y(3) W yp(3) w& y(4) p yp(4) p& y(5) q yp(5) q& y(6) r yp(6) r& y(7) o yp(7) & o y(8) y o yp(8) y& o y(9) z o yp(9) z& o y(10) φ yp(10) & φ y(11) θ yp(11) & θ y(1) ψ yp(1) ψ& Page 9 of 55

38 Sequencing of CFD and Solid Body EOM Solution Page 30 of 55

39 Solid Body Solver Organization Page 31 of 55

40 4 Validation The validation of the sidof simulation capability is suggested to be in phases. Initial validation should begin with available data based on simple ellipsoid geometries similar to that show in Figure 5. Following this validation should pursue realistic geometries. Figure 5 Ellipsoid geometry Analytical solutions for added mass forces are available based on potential flow theory. The solutions for ellipsoid shapes have been worked and thoroughly presented in the DRDC document Estimates for the Added Mass of a MultiComponent, Deeply Submerged Vehicle Part I: Theory and Program Description by eorge Watt. From this report the si added mass forces are calculated as: X Y ( w& + uq) + X q& Y vr Y rp Y r + Z wq Z q X u& + X + 4.1a u & w& q& v& p& r& w& ( wr up) + X qr Z wp Z pq Yv& + Y p& + Yr& + X ur + X 4.1b v & p& r& u& w& q& w& q& q& Z X w & ( u& wq) + Z w& + Z q& X uq X q + Yvp+ Y p + Yrp w& q& u& q& v& p& r& 4.1c Page 3 of 55

41 K M ( v& wp) + K p& + K ( r& + pq + X uv X ur + ( Z Y ) Y ) p& + X ( Y + Z )( vq wr) ( M N )qr q& r& q& p& r& q& ( u& + wq) + Zq& ( w& uq) + Mq& q& + ( Xu& Zw& ) uw+ Xw& ( w u ) + ( K N ) pr+ K ( p r ) Yvp r& p& r& r& r& w& q& w& v& vw + Y vr p& 4.1d 4.1e ( v& + ur) + K ( p& qr) + N r& ( X Y ) uv X vw+ ( X + Y )( up vq) N Yr& r& + Z wp ( K M )pq q& p& q& r& u& v& w& q& p& 4.1f The coefficients X, X etc. are calculated from potential flow theory for u& Xv&, Xw&, X p&, Xq&, r& a fied ellipsoid geometry. These coefficients remain constant regardless of the state of the submarine (u, v, w, p q, r etc.) or its orientation. The coefficients used in the ANSYS CFX solid body solver are nondimensionalized with the following: X X K K u&, Xv&, X w& Y u&, Yv&, Yw& Z u&, Zv&, Zw& Coefficient for non dimensionalizing 1 ρ L 3 1 ρ L 1 ρ L 1 ρ L 4 4 p&, X q&, Xr& Y p&, Yq&, Yr& Z p&, Zq&, Zr& 5 p&, Kq&, Kr& M p&, Mq&, Mr& N p&, M q&, Nr& u&, Kv&, Kw& M u&, Mv&, M w& N u&, Nv&, Nw& The density ρ is that of the fluid medium and L is the length of the ellipsoid along its primary ais. Validation Using Ellipsoid Shapes Using the above equations is quite straightforward. The calculation of the coefficients is done using DRDC s program, which outputs a 66 matri of coefficients of which all are zero ecept the diagonal ones in the case of an ellipsoid geometry. These coefficients are read into ANSYSCFX at the start of a solution and used by multiplying the coefficients with the current state of the submarine to obtain the added mass forces. For validation it is proposed to use the added mass terms based on a single ellipse shape in two ways. 1. Create a dummy submarine mesh, which will run very fast in a transient solution mode. The solid body solver will be given the force/moment state as calculated by the potential flow solution and calculates in return a new state for Page 33 of 55

42 the submarine. This new state will subsequently be used to compute new potential flow added mass forces. The solid body solver is then solved again in a repetitive manner. In this mode of operation the coupling of fluid force data and the solid body solver solution can be tested, including convergence within a transient coefficient loop sequence. The submarine model would have buoyancy and gravity forces active, and some sort of propulsion force to move the submarine in a stable trajectory. Ellipsoid shapes in general are not stable underwater without appendages. Finally, in this testing the apparent body forces and dynamic boundary conditions do not need to be active.. In this mode of testing an ellipsoid mesh is used with a transient CFD solution. Full coupling between the CFD solution and solid body solver is used. The potential flow solution, coefficient based, is output at the end of every time step for comparison with the converged forces from the CFD solution. In this mode of testing the CFD solution is obtained with no viscosity and slip conditions over the ellipsoid hull surface so that a potential flow solution is approimated. As in approach 1 above, the ellipsoid would operate under applied forces. In addition the CFD model would need to have the apparent body forces active and boundary conditions that adapt with the state of the ellipsoid. A basic ellipsoid mesh has already been created for these calculations. In the above two testing scenarios a relatively short trajectory period (i.e. period of transient simulation) can be studied. The goal is to validate coefficient loop convergence of the fluid force and solid body solver coupling, as well as the force/moment predictions obtained from the CFD solution. Ellipsoid eometry To start a 6:1 ellipsoid shape can be chosen. The added mass coefficients for this shape have been provided by DRDC as follows: X u & Y v & Z w & K & 0.0 p M q & N r & E E E E E3 Page 34 of 55

43 Note that the first three coefficient have been nondimensionalized by C1/ρL 3, and the last three by B1/ρL 5. Here L is the ellipsoid length equal to a, and the density ρ is that of the fluid. The geometric shape of the ellipsoid is governed by: a y + b z + c 1 (4.) with the ais chosen as the principle ais with ma 0.5 m and min 0.5 m. For a 6:1 ellipsoid then a 0.5 m, b m and c m. The moments of inertia are: ( b c ) 4 I ρπ abc + (4.3a) 15 ( a c ) 4 I yy ρπ abc + (4.3b) 15 ( a b ) 4 I zz ρπ abc + (4.3c) 15 where ρ is the density of the material here taken as 997 kg/m 3. The volume of the ellipsoid is given as: 4 V πabc (4.4) 3 so that the mass of the ellipsoid is calculated by: m ρv (4.5) The center of gravity is taken as the origin (0,0,0) and the center of buoyancy CB at the same location (0,0,0). For the dimensions given above the moments of inertia for the 6:1 ellipsoid are: I kgm I yy kgm I zz kgm Page 35 of 55

44 and the mass M kg. These quantities are requires as input to the ANSYS CFX solid body equation of motion model, including the locations of CB and. Page 36 of 55

45 Validation Cases: Method 1 Case 1a Linear acceleration of the 6:1 ellipsoid forward in the direction by applying a propulsion force with neutral buoyancy. The acceleration of the ellipsoid can be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if an applied propulsive force of 100 N in the positive direction is applied (X CFD 100 N), and using the added mass equation 4.1a for the added mass force, the acceleration can be calculated as: u& XCFD m CX u& The ANSYSCFX prediction for this case is an acceleration of m/s, which compares to the analytical result of m/s. This is an error of 0.0%. Case 1b Linear acceleration of the 6:1 ellipsoid forward in the ydirection by applying a propulsive force with neutral buoyancy. The acceleration of the ellipsoid can be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if an applied propulsive force of 100N in the positive ydirection is applied (Y CFD 100 N), and using the added mass equation 4.1b for the added mass force, the acceleration can be calculated as: YCFD v& m CY The ANSYSCFX prediction for this case is an acceleration of m/s, which compares to the analytical result of m/s. This is an error of %. Case 1c Linear acceleration of the 6:1 ellipsoid forward in the zdirection by applying a propulsive force with neutral buoyancy. The acceleration of the ellipsoid can be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if an applied propulsive force of 100N in the positive zdirection is applied (Z CFD 100 N), and using the added mass equation 4.1c for the added mass force, the acceleration can be calculated as: ZCFD w& m CZ The ANSYSCFX prediction for this case is an acceleration of m/s, which compares to the analytical result of m/s. This is an error of %. v& w& The last two digits represent the average of a small oscillation (+/ ) that was observed on the output. Page 37 of 55

46 Case 1d Angular acceleration of the 6:1 ellipsoid in the ϕdirection by applying a propulsive moment with neutral buoyancy. The acceleration of the ellipsoid can be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if an applied propulsive moment of 100Nm around the ais is applied (K CFD 100 N), and using the added mass equation 4.1d for the added mass force, the acceleration can be calculated as: MCFD p& m BK The ANSYSCFX prediction for this case is an acceleration of m/s, which compares to the analytical result of m/s. This is an error of %. Case 1e Angular acceleration of the 6:1 ellipsoid in the θdirection by applying a propulsive moment with neutral buoyancy. The acceleration of the ellipsoid can be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if an applied propulsive moment of 100Nm around the yais is applied (M CFD 100 Nm), and using the added mass equation 4.1e for the added mass force, the acceleration can be calculated as: MCFD q& m BM The ANSYSCFX prediction for this case is an acceleration of m/s, which compares to the analytical result of m/s. This is an error of %. Case 1f Angular acceleration of the 6:1 ellipsoid in the ψdirection by applying a propulsive moment with neutral buoyancy. The acceleration of the ellipsoid can be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if an applied propulsive moment of 100Nm around the zais is applied (N CFD 100 Nm), and using the added mass equation 4.1f for the added mass force, the acceleration can be calculated as: NCFD r& m BN The ANSYSCFX prediction for this case is an acceleration of m/s, which compares to the analytical result of m/s. This is an error of %. p& q& r& The last two digits represent the average of a small oscillation (+/ ) that was observed on the output. Page 38 of 55

47 Validation against Octave results The solid body equations of motion using apparent mass terms simplified for an ellipsoid are presented in Eq. 4.. These are solved using Octave and compared to ANSYSCFX results. The algorithm used for the integration in Octave was DASSL 1 which implements Backward Differentiation Formulas (BDF) of orders one through five to solve an IDE for y and y`. This enables a comparison of more comple cases when analytical results are not available. X u& Y v& Z w& N r& CFD CFD K p& M q& CFD CFD CFD CFD Yvr+ Z wq ( W B)sinθ m( vr+ wq ( q + r ) + y( pq r& ) + z ( pr+ q& )) v& u& u& g u& u& v& w& w& v& w& yy b mz ( g( u& vr+ wq) g( w& uq+ vp)) I M m ( g( v& wp+ ur) yg( u& vr+ wq)) I N zz q& r& q& r& g g g b b I I m X yy ( X Y) uv M pq+ ( W B)cosθsinϑ ( yw yb)sinθ ( I zz b + ( X Z ) uw N pr ( W B)cosθcosϑ ( zw z B)sinθ ( I I N u& + X ur Z wp+ ( W B)cosθsinϑ m( wp+ ur y( r + p) + z ( qr p& ) + ( qp+ r& )) m Y M q& r& v& X uq+ Yvp+ ( W B)cosθcosϑ m( uq+ vp z ( p + q) + ( rp q& ) + y( rq+ p& )) m Z w& ( yw yb)cosθcosϑ ( zw z B)cosθsinϑ m( y ( w& uq+ vp) z ( v& wp+ ur)) K p& g g g g g g b b g g g yy g g I I zz g g ) pr ) pq K. E. Brenan, et al., Numerical Solution of InitialValue Problems in DifferentialAlgebraic Equations, NorthHolland (1989) for more information about the implementation of DASSL. Page 39 of 55

48 Case 1g This test case uses a sphere with dimensions abc0.5m. Linear acceleration of the sphere upward in the zdirection as a result of buoyancy which is equal to twice its weight. The acceleration of the sphere can be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if the buoyant force is twice its weight and the added mass coefficient is calculated to be Z & π/ 6 the acceleration can be calculated as: w W w& m CZ w & The ANSYSCFX prediction for this case is an acceleration of m/s, which compares to the analytical result of m/s. This is an error of %. Integrating this twice to obtain the z displacement gives: 1 W z( t) t m CZw& Using ANSYSCFX to predict the value of the zdisplacement at t0.1s using a time step of 0.001s gives z(0.1) em while the analytical result gives z(0.01) Em. This is an error of 1% over the integration of 100 timesteps. Octave predicted z(0.1) em for this test case. This is an error of %. Validation Cases: Method These test cases replicate those undertaken for Method 1 ecept now a 6:1 ellipsoid mesh is used and the CFD solution provides the forces for input into the solid body solver. Considering the apparent body force (Eq..39) applied as a source term to each control volume in the solution: dor do F b ρ + r + 1 dt dt ( r ) + u 1 i each term is tested in the following si cases although not in isolation. Cases a, b and c test the first term on the RHS in the three coordinate directions. Cases d, e and f test the last three terms on the RHS about each of the three coordinate aes. The final case (Case g) considers the first term applied in two directions simultaneously. using equations presented in the DRDC document: Estimates for the Added Mass of a MultiComponent, Deeply Submerged Vehicle Part I: Theory and Program Description by eorge Watt The last digit represents the average of a small oscillation (+/ ) that was observed on the output. Page 40 of 55

49 Case a Linear acceleration of the 6:1 ellipsoid forward in the direction (See Fig. 4) by applying a propulsion force with neutral buoyancy. The acceleration of the ellipsoid can be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if an applied propulsive force of 100 N in the positive direction is applied (X CFD 100 N), and using the added mass equation 4.1a for the added mass force, the acceleration can be calculated as: XCFD u& m CX u& The ANSYSCFX prediction for this case is an acceleration of 6.58 m/s, which compares to the analytical result of m/s. This is an error of 0.046%. In the Figure below is shown the velocity field predicted in this case, which is relative to the bodyfied coordinate system. The velocity field is opposite to the acceleration of the body in the positive direction. Page 41 of 55

50 Case b Linear acceleration of the 6:1 ellipsoid in the ydirection (see Fig. 4) by applying a propulsion force with neutral buoyancy. The acceleration of the ellipsoid can also be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if an applied propulsive force of 100 N in the positive ydirection is applied (Y CFD 100 N), and using the added mass equation 4.1b for the added mass force, the acceleration can be calculated as: YCFD v& m CY v& The ANSYSCFX prediction for this case is an acceleration of 3.60 m/s, which compares to the analytical result of m/s. This is an error of 0.306% which is much larger than for Case a, however motion in the yais involves displacing much more liquid and therefore an increase in associated error. The main error source is likely a result of the discrete ellipsoid surface. In the Figure below is shown the velocity field predicted in this case, which is relative to the bodyfied coordinate system. Again, the velocity field is in the opposite sense to the motion of the solidbody. Page 4 of 55

51 Case c Linear acceleration of the 6:1 ellipsoid in the zdirection (see Fig. 4) by applying a propulsion force with neutral buoyancy. The acceleration of the ellipsoid can also be obtained analytically and compared to the ANSYSCFX solid body prediction. From Newton s nd law, if an applied propulsive force of 100 N in the positive zdirection is applied (Z CFD 100 N), and using the added mass equation 4.1c for the added mass force, the acceleration can be calculated as: ZCFD w& m CZ w& The ANSYSCFX prediction for this case is an acceleration of 3.60 m/s, which compares to the analytical result of m/s. This is an error of 0.306% the same as for Case b which involves the same displacement of the liquid around the ellipsoid body. The causes for increased error relative to Case a would be the same as for Case b. In the Figure below is shown the velocity field predicted in this case for body motion in the positive zdirection. Page 43 of 55

52 Case d This test case applies a fied propulsive moment to the vehicle of 100 Nm around the ais of Fig. 4. The solution is now coupled to the CFD solver, which calculates the forces eperienced in turn by the submarine. In the CFD model walls are treated as slip and viscosity is set to zero, so that the moments calculated by the CFD solution are a results of the apparent mass forces. The moments induced by the apparent mass forces can be calculated analytically as well, and a comparison made to assess accuracy. Comparison can also be based on the angular accelerations of the solid body, which is what is done below. The analytical solution for this case is based on the solid body equations of motion for angular momentum with the apparent body force based on the application of Eq. 4.1(d). For the applied moment above and the reduced equation of motion: KCFD p& m BK p& the analytical solution is rad/s. The CFD prediction for this case is rad/s with a percentage error of 0.001%. The velocity field associated with this prediction is given in the Figure below, where the counterrotating flow about the yais is as epected with the frameofreference fied to the ellipsoid body. Note that the opening boundary conditions treat both the inflow and outflows correctly around the entire flow domain. Page 44 of 55

53 Case e This test case applies a fied propulsive moment to the vehicle of 100 Nm around the y ais of Fig. 4. The solution follows as for Case d. The analytical solution for this case is based on the solid body equations of motion for angular momentum with the apparent body force based on the application of Eq. 4.1(e). For the applied moment above and the reduced equation of motion: MCFD q& m BM q& the analytical solution is rad/s. The CFD prediction for this case is rad/s with a percentage error of 0.36%. This is considered good keeping in mind that the ellipsoid surface is not perfectly smooth and that this motion, as opposed to Case d, involves a much larger displacement of the surrounding fluid. The velocity field associated with this prediction is given in the Figure below, where the counterrotating flow about the yais is as epected with the frameofreference fied to the ellipsoid body. It is clear that much more fluid is displaced than when the body rotates about its ais. Note that the opening boundary conditions treat both the inflow and outflows correctly around the entire flow domain. Page 45 of 55

54 Case f This test case applies a fied propulsive moment to the vehicle of 100 Nm around the z ais of Fig. 4. The solution follows as for Case d. The analytical solution for this case is based on the solid body equations of motion for angular momentum with the apparent body force based on the application of Eq. 4.1(f). For the applied moment above and the reduced equation of motion: NCFD r& m BN r& the analytical solution is rad/s. The CFD prediction for this case is 76.9 rad/s with a percentage error of 0.377%. Again this is considered good keeping in mind that the ellipsoid mesh surface is not perfectly smooth. Since motion about the zais involves the same displacement of fluid as that for motion about the yais the result should be nearly identical to that for Case e. The velocity field associated with this prediction is given in the Figure below where the counterrotating flow about the zais is as epected with the frameofreference fied to the ellipsoid body. Note that the opening boundary conditions again treat both the inflow and outflows correctly around the entire flow domain. Page 46 of 55