Software Design Document for a Six DOF Unsteady...

Defence R&D Canada – Atlantic

DEFENCE DÉFENSE&

Contract Report

DRDC Atlantic CR 2007-020

January 2007

Copy No. _____

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

Software Design Document for a Six DOF

Unsteady Simulation Capability in ANSYS-CFX

ANSYS Canada Ltd.554 Parkside DriveWaterloo, Ontario N2L 5Z4

Phone: 519-886-8435FAX: 519-886-7580

www.ansys.com/[email protected]

Contract Number: W7707-3-2219Contract Scientific Authority: Dr. George Watt, [email protected], 902-426-3100 ext 381

This page intentionally left blank.

Software Design Document for a Six DOF

Unsteady Simulation Capability in ANSYS-CFX

ANSYS Canada Ltd.554 Parkside DriveWaterloo, Ontario N2L 5Z4

Phone: (519) 886-8435FAX: (519) 886-7580

www.ansys.com/[email protected]

Contract: W7707-3-2219Scientific Authority: Dr. George D. Watt, [email protected], (902) 426-3100 x381

Defence R& D Canada – AtlanticContractor ReportDRDC Atlantic CR 2007-020January 2007

Abstract

This report presents the software design for a 6 degree-of-freedom submarine simulation capa-bility in ANSYS-CFX (“CFX”). It documents the underlying theory, implementation in theCFX software system, verifies the algorithms, and presents a preliminary validation. Two mainapproaches to the problem are considered, both using a body fixed mesh.

In one, the mesh is rigid and moves in 6 DOF with the submarine so that apparent bodyforces for 6 DOF motion must be accounted for in the fluid equations of motion solved byCFX. In the other, the mesh translates but does not rotate with the submarine; it deformsto 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 linearaccelerations. 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 theflow about the submarine, then passes the unsteady hydrodynamic forces on the submarinesurface 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 propagatethe next coefficient update loop/time step. A second order scheme is used to integrate the fluidand solid body equations of motion in parallel. The method accurately predicts analyticalpotential flow predictions of ellipsoid added masses.

Resume

Le present rapport etablit la conception de logiciel pour une capacite de simulation d’un sous-marin selon 6 degres de liberte (DDL) en ANSYS-CFX (“CFX”). Il documente la theoriesous-jacente et son integration au systeme logiciel CFX, verifie les algorithmes et contient unevalidation preliminaire. Deuxgrandesapproches au probleme sont etudiees, les deux faisantappel a une maille fixe de corps.

Dans l’une des deux approches, la maille est rigide et se deplace selon 6 DDL avec lesous-marin, de sorte qu’il faille prendre en compte les forces apparentes de corps du mou-vement selon 6 DDL dans les equations fluides du mouvement resolues au moyen du logicielCFX. Dans l’autre approche, la maille se deplace, mais ne tourne pas en meme temps que lesous-marin; elle se deforme pour suivre le sous-marin localement au moyen des preparationsarbitraires-langrangiennes-euleriennes CFX (de maille mobile) des equations fluides et requiertuniquement les termes des forces apparentes de corps pour les accelerations lineaires. Dans lesdeux approches, on calcule aussi la valeur des equations des mouvements applicables au sous-marin (modele de corps solide) pour determiner les forces apparentes de corps et, au besoin,tout mouvement de la maille.

L’approche axee sur une maille rigide est choisie en vue d’une evaluation initiale. Danscette approche, le logiciel CFX calcule la valeur de l’ecoulement autour du sous-marin, faitpasser les forces hydrodynamiques instables exercees a la surface du sous-marin a son modelede corps solide (pour tenir compte de l’inertie du sous-marin, de sa flottabilite, de sa propulsion,des forces exercees sur ses commandes et d’autres facteurs) et reoit les donnees cinematiquesretrospectives du corps solide necessaires a la propagation de l’intervalle de temps/de la bouclede mise a jour des coefficients qui suit. Un schema de secondordre sert a l’integration desequations des mouvements en parallele du corps solide et fluide. La methode predit avecprecision les previsions analytiques d’ecoulement potentiel des masses ajoutees ellipsoıdales.

DRDC Atlantic CR 2007–020 i

Executive Summary

Introduction

DRDC Atlantic is collaborating with ANSYS Canada to develop an unsteady, 6 degree-of-freedom (DOF), computational fluid dynamics (CFD) submarine maneuvering simulation ca-pability. This is needed to validate fast, coefficient based simulations used to investigate ma-neuvering limitations and establish safe operating envelopes for underwater vehicles. Severalcountries (eg, the US, UK, France) validate their simulations using a free swimming scale modelwhich “is currently the best predictor of full scale submarine maneuvering performance.” Sucha facility is unaffordable by Canada. Validating with CFD is affordable and can provide betterdetail. The disadvantage to using CFD is that its predictions are not as reliable as experimentalmeasurements. But CFD technology is evolving quickly and it is worth evaluating the capabil-ity now. By collaborating with a successful commercial CFD vendor, there is the potential forcommercialization which would minimize ongoing maintenance and development costs.

This report describes the current ANSYS Canada implementation of the simulation capa-bility 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, bodyfixed mesh extending from the surface of the vehicle out to the far field. The mesh and boatmove together controlled by the same 6 DOF solid body equations of motion used by theDRDC 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 next timestep. Theoretically, any maneuver can be modelled in which the boat is deeply submerged andisolated from any other vehicle or boundary.

Significance of Results

Evaluation of the CFX simulation capability has begun. It is being used to investigate asubmarine rising maneuver that generates a roll instability that can result in excessive rollas the submarine surfaces, a maneuver operators have asked DRDC about in the recent past.Conventional quasi-steady coefficient based hydrodynamic models have difficulty modelling thismaneuver. The submarine can be modelled as an isolated deeply submerged body throughoutthe maneuver because the free surface is unimportant in the development of the underwaterroll instability. The evaluation is taking place at the University of New Brunswick using theDRDC generic submarine shape for which extensive experimental data are available.

Future Plans

Preliminary work has shown that an alternative approach to the problem using a moving meshformulation is feasible. This would allow the mesh to deform with time as the boat movestoward or away from a boundary or other vehicle, as would be the case for littoral or two-bodyproblems. Minimal development work is required to implement moving mesh because the basiccapability already exists in CFX.

Finally, it is desirable to incorporate a generalized version of this capability in subsequentcommercial versions of CFX. A commercial capability supported by all CFX users is an eco-nomical way to handle future maintenance and development costs.

ANSYS Canada Ltd., 2007, Software Design Document for a Six DOF Unsteady Simulation

Capability in ANSYS-CFX, DRDC Atlantic CR 2007-020.

ii DRDC Atlantic CR 2007–020

Sommaire

Introduction

RDDC Atlantique s’est associe a ANSYS Canada pour mettre au point une capacite instablede simulation de manœuvres de sous-marin de dynamique des fluides computationnelle (CFD)selon 6 degres de liberte (DDL). Cette mise au point est necessaire en vue de la validationde simulations rapides, fondees sur des coefficients, qui servent a l’etude des restrictions desmanœuvres et a l’etablissement d’enveloppes de fonctionnement sur pour vehicules sous-marins.Plusieurs pays (dont les Etats-Unis, le Royaume-Uni et la France) valident leurs simulations aumoyen d’un modele-echelle autonome qui “est actuellement le meilleur outil de prediction durendement de manœuvre d’un sous-marin pleine grandeur.” Le Canada n’a pas les moyens dese doter d’une telle installation. La validation a l’aide de la CFD est cependant abordable etpermet d’obtenir de meilleurs details. L’inconvenient de la CFD, c’est que les predictions qu’ellepermet d’obtenir ne sont pas aussi sures que les mesures obtenues lors d’essais. La technologieCFD evolue cependant rapidement, et il vaut la peine d’evaluer la capacite maintenant. Unecollaboration avec un fournisseur commercial prospere de CFD ayant reussi offre une possibilitede commercialisation, ce qui reduirait au minimum les frais permanents de developpement etde maintenance.

Le present rapport decrit la mise en œuvre qu’effectue actuellement ANSYS Canada dela capacite de simulation au moyen de son logiciel en code CFD commercial. Le travail devalidation preliminaire est aussi presente.

Resultats

Dans le cadre des travaux en cours de mise en œuvre du logiciel CFX, le champ d’ecoulementdoit etre discretise au moyen d’une maille fixe de corps rigide qui s’etend de la surface du vehiculejusqu’au champ lointain. La maille et le navire se deplacent ensemble et sont commandes parles memes equations de mouvement de corps solide selon 6 DDL dont RDDC se sert dansle cadre de son programme de simulation de sous-marins (DSSP). Le logiciel CFX permet desolutionner l’ecoulement autour du sous-marin, fait passer les forces hydrodynamiques instablesaux equations de corps solide (pour tenir compte de l’inertie, de la flottabilite, de la propulsion,des forces exercees sur les commandes et d’autres facteurs) et reoit les vitesses de retour pourl’intervalle de temps qui suit. En theorie, n’importe quelle manœuvre peut etre modelisee demaniere a ce que le navire soit immerge a une grande profondeur et isole de tout autre vehiculeou limite.

Portee

L’evaluation de la capacite de simulation du logiciel CFX a commence. Elle sert a l’etude d’unemanœuvre de remontee d’un sous-marin qui genere une instabilite susceptible d’entraıner unroulis excessif a mesure que le sous-marin remonte a la surface, manœuvre que les operateursont demande a RDDC d’etudier recemment. Des modeles hydrodynamiques fondes sur des co-efficients quasi stables classiques posent des difficultes pour la modelisation de cette manœuvre.Le sous-marin peut etre modelise comme corps isole immerge a une grande profondeur danstoute la manœuvre parce que la surface libre n’a pas d’importance dans le developpement del’instabilite du roulis sous l’eau. L’evaluation a lieu a l’Universite du Nouveau-Brunswick aumoyen de la forme generique de sous-marin de RDDC, a l’egard de laquelle on dispose d’unegrande quantite de donnees experimentales.

DRDC Atlantic CR 2007–020 iii

Recherches futures

Les travaux preliminaires ont montre qu’une autre approche au probleme, qui fait appel a unepreparation de maille mobile, est possible. Cela permettrait une deformation de la maille amesure que le navire se deplace en direction d’une limite ou d’un autre vehicule ou qu’il s’eneloigne, ce qui serait le cas des problemes a proximite du littoral ou en presence de deux corps.Il faut mener des travaux minimes de developpement en vue de la mise en œuvre de la maillemobile, la capacite de base etant deja prevue dans le logiciel CFX.

Enfin, il est souhaitable d’integrer une version generalisee de la capacite a des versionscommerciales subsequentes du logiciel CFX. Une capacite commerciale dont se serviraient tousles utilisateurs du logiciel CFX constitue un moyen economique de s’occuper des frais ulterieursde developpement et de maintenance.

ANSYS Canada Ltd., 2007, document de conception de logiciel pour une capacite de simulation

instable de sixDDL en ANSYS-CFX, document CR2007-020 de RDDC Atlantique.

iv DRDC Atlantic CR 2007–020

Software Design Document for a Six

DOF Unsteady Simulation Capability

in ANSYS CFX

Prepared for DRDC Atlantic

CFX Report # 01

ANSYS Canada Ltd.

554 Parkside Drive

Waterloo, Ontario N2L 5Z4

Phone: (519) 886 8435

Fax: (519) 886 7580

www.ansys.com/cfx [email protected]

December 22, 2006

Page 2 of 55

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: 2073

ANSYS Contract Number: W7707 032219/001/HAL

Report Number: 01

Date: December 12, 2006

This report written by:

____________________________

Stephen Dajka

Manager, Directed Development

Dr. Philippe Godin

CFD Integration Developer

Dr. Andrew Gerber

CFD Consultant, University of New Brunswick

Page 3 of 55

Contents

1 Introduction............................................................................................................... 4

2 Theory ........................................................................................................................ 5

Nomenclature...................................................................................................................... 5

Kinematic Relationships ..................................................................................................... 7

Rigid Body Equations of Motion...................................................................................... 11

Fluid Equations of Motion ................................................................................................ 18

Tracking Body in Inertial Frame....................................................................................... 20

Boundary Conditions ........................................................................................................ 22

3 ANSYS CFX Implementation................................................................................ 26

Solution Procedure............................................................................................................ 26

Variable Time Steps.......................................................................................................... 28

Internal Data Arrays.......................................................................................................... 29

Sequencing of CFD and Solid Body EOM Solution ........................................................ 30

Solid Body Solver Organization ....................................................................................... 31

4 Validation................................................................................................................. 32

Validation Using Ellipsoid Shapes ................................................................................... 33

Validation with Fully Appended Submarine Shapes ........................................................ 48

Full Navier Stokes Six DOF Submarine Simulations ...................................................... 53

5 Recommendations for Future Work ................................................................. 55

Page 4 of 55

1 Introduction

This report presents the detailed software design of a six degree of freedom (DOF)

submarine simulation capability in ANSYS CFX. 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 ANSYS CFX 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 explore and document

alternative approaches, amenable to ANSYS CFX, for the solving six DOF submarine

motions. It is possible that all approaches will ultimately be available for use. This work

is outlined in Chapter 2. This is followed by a description of the software

implementation into ANSYS CFX in Chapter 3. The software implementation is on

going and therefore this section is subject to change, for example as the ANSYS CFX

version level changes (currently at V10) aspects of the implementation details will need

to be updated.

The next chapter in the report covers validation (Chapter 4) which up to the present time

has focused on clearly defined analytical or semi analytical benchmarks. This

methodology, using derived added mass coefficients with viscous corrections and highly

accurate explicit 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 six

DOF simulations fully coupled to the governing fluid equations.

Page 5 of 55

2 Theory

This section describes the rigid body equations of motion (EOM) as generally used by

DRDC in their underwater vehicle simulations. Since the ANSYS CFX implementation

is exploring two alternatives to six 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 translating rotating 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 auxiliary

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

fixed coordinate system for the submarine is placed along the hull axis (chosen during the

CFD mesh generation process) with the x axis aligned forward, the y axis to starboard,

and the z axis through the keel. Since the submarine coordinate system is not located at

either the center of mass (G) or buoyancy (CB) the derivation of the EOM for the

submarine must take this into account. In Figure 1 is shown the body fixed coordinate

system in relation to the G as well as the linear (u,v,w) and angular (p,q,r) velocities.

Figure 1 Body fixed coordinate system

z,w,r y,v,q

rG

x,u,p O1

G Hull axis

Page 6 of 55

B=ρVg Buoyancy

g Gravitational constant

K,M,N Body axis moments

Iij 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 axis submarine angular velocities

r Position vector

t Time

u,v,w Body axis submarine velocities

U Submarine speed

V Volume of external hydrodynamic envelope

W Submarine weight including ballast

x,y,z Body axis coordinates

xG,yG,zG Submarine center of gravity in body coordinates

xB,yB,zB Submarine center of buoyancy in body coordinates

xo,yo,zo Inertial coordinates

X,Y,Z Body axis forces

A,α krjqipv&

v&

v& ++=

φ Roll, rotation about body x axis

θ Pitch, rotation about body y axis

ψ Yaw, rotation about body z axis

ρ Fluid density

,ω krjqipvvv

++=

Subscripts/Superscripts

G 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 7 of 55

Kinematic Relationships

2.1.1 Particle Motion in a Translating Rotating Frame of Reference

In the development of the six DOF motion equations, a number of kinematic relationships

are needed for describing general particle motion in a translating rotating frame of

reference, and rigid body motion in translation and rotation. We will start with the most

general motion situation first.

Figure 2 Particle displacement vectors for A with O1 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 O1. We want the fluid particle position, velocity and acceleration

relative to the axes at O1. 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:

1rRro += (2.1)

and the velocity by differentiation with respect to the origin O:

dt

rd

dt

Rd

dt

rd oooo 1+= (2.2)

For the velocity of point A it is better to replace the last term with one differentiated with

respect to origin O1 (shown by dtad1 ), which results in:

111 r

dt

rd

dt

Rd

dt

rd ooo ×++= (2.3)

xo O

zo yo

x O1

z y

A

ro

r1

R

Page 8 of 55

where:

1111 r

dt

rd

dt

rd o ×+= (2.4)

The angular rate of rotation of the moving axes, , now appears in Eq. 2.3 with the axis

of rotation acting through O1, and a direction defined relative to the inertial coordinates.

Now by further differentiation the acceleration appears:

dt

rdr

dt

d

dt

rd

dt

d

dt

Rd

dt

rd oooooo 11

11

2

2

2

2

×+×+

+= (2.5)

applying the rule and result of Eq. 2.4 to the third and fifth terms respectively and

combining like terms results in:

( )dt

rdrr

dt

d

dt

rd

dt

Rd

dt

rd oooo 11112

1

2

1

2

2

2

2

2 ×+××+×++= (2.6)

Equation 2.6 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 O1 of the moving axes relative to the inertial axes

• The acceleration of point A relative to the moving axes

• 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 axes.

To simulate flow in the moving frame in ANSYS CFX the user would be expected to

define the following quantities at any instant in time:

• The angular velocity and acceleration and dtdo respectively. Note that this

includes defining the rotation vector relative to the inertial frame.

• The velocity and acceleration of O1, namely dtRdo and 22 dtRdo respectively.

• The position of A relative to the origin O1, i.e. r1, which is available from the

coordinates of each node in the CFD mesh.

Page 9 of 55

The quantities appearing in Eq. 2.6 that are not predefined are the acceleration,

dtuddtrd 11

2

1

2

1 = , and velocity, 111 udtrd = (that appears in the Coriolis term), of

position A relative to O1 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. 2.6 in a manner emphasizing the unknown velocity components u1:

( ) 11111

2

2

2 urrdt

d

dt

ud

dt

Rd

dt

ud oooo ×+××+×++= (2.7)

2.1.2 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 translating rotating

coordinate system, , fixed to the submarine at O1.

For rigid body motion around a fixed point O1, the velocity and acceleration of point A

located on the body at a distance r1 from O1 is obtained by:

111 r

dt

rd×= ω (2.8)

( )112

1

2

1 rrdt

rd××+×= ωωω& (2.9)

However, for general rigid body motion relative to a fixed (inertial) frame of reference at

O we have, using Fig. 2 as a reference:

10 rRr += (2.10)

dt

rd

dt

Rd

dt

rd 10000 += (2.11)

where the third term from the left is replaced using Eq. 2.4 while noting that for a rigid

body d1r1/dt=0. This gives for velocity and acceleration:

Page 10 of 55

1000 rdt

Rd

dt

rd×+= ω (2.12)

( )112

2

0

2

0

2

0 rrdt

Rd

dt

rd××+×+= ωωω& (2.13)

or writing in terms of the velocity u0:

( )112

2

000 rrdt

Rd

dt

ud××+×+= ωωω& (2.14)

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. 2.7),

where differences relate to the fluid particles relative velocity to O1.

What remains is to relate Eq. 2.14 to the situation where a local coordinate system is

fixed at point O1 on the submarine body. Consider the case where the coordinate system

at O1 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 fixed to the submarine

body, although the axes origin is fixed at O1. The x axis over time does not remain

aligned with the hull axis and occurs because ≠ω. The other case relevant to

submarine motion is the situation where the submarine rotational state defines the O1

coordinate system orientation such that =ω. This case maintains the O1 x axis aligned

with the hull axis, the y axis oriented starboard and the z axis 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. 2.4) that the angular acceleration of the rigid body in the

O1 reference frame is the same in the inertial frame O, and therefore ω&& = .

Page 11 of 55

Rigid Body Equations of Motion

2.1.3 EOM with Translating Rotating Coordinate System

The equations of motion for the submarine are formulated around a body fixed coordinate

system, with the coordinate system on the axis 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:

∑ = GumF & (2.15)

where Gu& 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. 2.14:

( )GGOG rruu ××+×+= ωωω&&& (2.16)

The acceleration, Ou& , is more convenient, for analysis purposes, to have in the frame of

the rotating body which leads to:

( ) ( )GGOxyzOG rruuu ××+×+×+= ωωωω &&& (2.17)

where ω in this case is the angular velocity of the body and fixed coordinate system so

that ω= ( is the angular velocity of the coordinate system as outlined in section 2.2).

The result is that ( Ou& )xyz and uO are relative to the body fixed coordinate system xyz.

Substituting this equation into Eq. (2.15) results in three equations for each component of

acceleration as presented in full in Eqs. (2.24).

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:

∑ = OO HM & (2.18)

However we are interested in using moments of inertia at the center of gravity, which is

obtained by the relation:

GGGO umrHH &&& ×+= (2.19)

Page 12 of 55

Substituting Eq. (2.19) into Eq. (2.18) and noting that this only applies to a stationary

frame. For a rotating frame the time derivative of HG must be expanded to give:

( ) GGGxyzGGGGO umrHHumrHM &&&& ×+×+=×+=∑ ω (2.20)

where

[ ]αIH G =& (2.21)

[ ]ωIH G = (2.22)

and

[ ]

−−

−−

−−

=

zzzyzx

yzyyyx

xzxyxx

III

III

III

I (2.23)

The moments of inertia, [I], are evaluated at the center of gravity of the submarine. On

this basis substituting Eqs. (2.17) and (2.21) thru (2.23) into Eq. (2.20) , and neglecting

small terms involving the square of coordinates (e.g. 2

Gx ), results in the three equations

for angular acceleration as presented in Eq. (2.25). Alternatively the parallel axis

theorem can be applied to transform the moments of inertia to a parallel axis acting

through body fixed location O. In this case it can be shown that terms involving the

square of the coordinates cancel out to exactly zero. In either treatment of [I], the form of

Eq. (2.25) is the same, however it would be preferable to have [I] supplied relative to

parallel axis O to avoid any approximation.

2.1.4 System of Equations for Solution

Axial (x axis), lateral (y axis) and normal (z axis) forces:

( ) ( ) ( )[ ] PSCFDGGG XXXqprzrpqyrqxwqvrum ++=++−++−+− &&&22 (2.24a)

( ) ( ) ( )[ ] PSCFDGGG YYYrqpxpqrzpryurwpvm ++=++−++−+− &&&22 (2.24b)

( ) ( ) ( )[ ] PSCFDGGG ZZZprqyqrpxqpzvpuqwm ++=++−++−+− &&&22 (2.24c)

Rolling (x axis), pitching (y axis) and yawing (z axis) moments:

Page 13 of 55

( ) ( ) ( ) ( )( ) ( )[ ] PSCFDGG

xyyzzxyzx

KKKurwpvzvpuqwym

IqprIqrIpqrqrIIpI

++=+−−+−

+−+−++−−+

&&

&&&22

(2.25a)

( ) ( ) ( ) ( )( ) ( )[ ] PSCFDGG

yzzxxyzxy

MMMvpuqwxwqvruzm

IrqpIrpIqrprpIIqI

++=+−−+−

+−+−++−−+

&&

&&&22

(2.25b)

( ) ( ) ( ) ( )( ) ( )[ ] PSCFDGG

zxxyyzxyz

NNNwqvruyurwpvxm

IprqIpqIrpqpqIIrI

++=+−−+−

+−+−++−−+

&&

&&&22

(2.25c)

Applied forces:

XCFD, YCFD, ZCFD Force vector obtained from integrated solid body surface

forces (pressure and shear)

( ) θsinBWX S −−= Weight and buoyancy component based on direction of

gravity vector

( ) φθ sincosBWYS −= Weight and buoyancy component based on direction of

gravity vector

( ) φθ coscosBWZ S −= Weight and buoyancy component based on direction of

gravity vector

XP, YP, ZP Thrust vector obtained from a model of the propulsion

system for the submarine

Applied moments:

KCFD, MCFD, NCFD

Moment vector obtained from

integrated solid body surface

forces (pressure and shear)

( ) ( ) φθφθ sincoscoscos BzWzByWyK BGBGS −−−= Weight and buoyancy

component based on direction

of gravity vector

( ) ( ) θφθ sincoscos BzWzBxWxM BGBGS −−−−= Weight and buoyancy


of gravity vector

( ) ( ) θφθ sinsincos ByWyBxWxN BGBGS −+−= Weight and buoyancy


of gravity vector

KP, MP, NP

Torque/moment component

obtained from a model of the

propulsion system for the

submarine.

Page 14 of 55

Auxiliary relations:

Submarine velocity in inertial coordinates

( ) ( )ψθφψφψφψθφψθ cossincossinsinsincoscossinsincoscos ++−+= wvuxo&

( ) ( )ψφψθφψθφψφψθ cossinsinsincossinsinsincoscossincos −+++= wvuyo&

φθφθθ coscossincossin wvuzo ++−=&

Submarine acceleration in inertial coordinates

( ) ( )ψθφψφψφψθφψθ cossincossinsinsincoscossinsincoscos ++−+= wvuuo&&&&

( ) ( )ψφψθφψθφψφψθ cossinsinsincossinsinsincoscossincos −+++= wvuvo&&&&

φθφθθ coscossincossin wvuwo&&&& ++−=

Body fixed angular velocities

( ) θφφφ tansincos qrp ++=&

φφθ sincos rq −=&

θφφ

ψcos

sincos qr +=&

System of equations:

−

−

−

−

−

−

=

−−−

−−−

−−−

−

−

−

LHSRHS

LHSRHS

LHSRHS

LHSRHS

LHSRHS

LHSRHS

zyzzxGG

yzyxyGG

zxxyxGG

GG

GG

GG

NN

MM

KK

ZZ

YY

XX

r

q

p

w

v

u

IIImxmy

IIImxmz

IIImymz

mxmym

mxmzm

mymzm

&

&

&

&

&

&

0

0

0

000

000

000

(2.26)

Page 15 of 55

The right hand side matrix is comprised of the applied forces and moments (designated

by the subscript RHS), and the non acceleration terms on the left hand side, of Eqs. 2.24

and 2.25. The non acceleration terms are identified with the subscript LHS.

The solution of Eq. 2.26 results in estimates at the new time level for accelerations:

rqpwvu &&&&&& ,,,,,

which in turn, with the time step know, allow for calculation of velocities:

rqpwvu ,,,,,

Finally the calculation of auxiliary derivative quantities (in the inertial frame) is then

possible:

ψθφ &&&&&&&&& ,,,,,,,, OOOOOO wvuzyx

and from these the integrated quantities for position and angular movement:

ψθφ ,,,,, OOO zyx

2.1.5 EOM with a Translating Coordinate System

∑ = GumF & (2.27)

( ) ( )GGxyzOG rruu ××+×+= ωωω&&& (2.28)

Noting that ( )xyzOO uu && = when the coordinate system at O is not rotating (i.e. =0).

GGGOO umrHHM &&& ×+==∑ (2.29)

( ) GGxyzGGGGO umrHumrHM &&&& ×+=×+=∑ (2.30)

Noting again that ( )xyzGG HH && = when the coordinate system at O is not rotating.

Since the axis 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:

[ ]( ) [ ] [ ] [ ] [ ]αωωωω

IIIIdt

IdH +=+== &&&& (2.31)

Page 16 of 55

Therefore the equation for rotational motion should be stated as:

[ ] [ ]( ) GGxyzGGGO umrIIumrHM &&&& ×++=×+=∑ αω (2.32)

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 (CG). If [I] is supplied relative to a body fixed axis acting through

O, then Eq. (2.34) should be modified by substituting [I] based on the parallel axis

theorem. The parallel axis theorem relates moments of inertia about a parallel axis at O to

that acting through the CG.

2.1.6 System of Equations for Solution

Axial (x axis), lateral (y axis) and normal (z axis) forces:

( ) ( ) ( )[ ] PSCFDGGG XXXqprzrpqyrqxum ++=++−++− &&&22 (2.33a)

( ) ( ) ( )[ ] PSCFDGGG YYYrqpxpqrzpryvm ++=++−++− &&&22 (2.33b)

( ) ( ) ( )[ ] PSCFDGGG ZZZprqyqrpxqpzwm ++=++−++− &&&22 (2.33c)

Rolling (x axis), pitching (y axis) and yawing (z axis) moments:

[ ] PSCFDGGxyzxxxyzxx KKKvzwymqIrIpIqIrIpI ++=−+−−+−− &&&&&&&& (2.34a)

[ ] PSCFDGGyzxyyyzxyy MMMwxuzmrIpIqIrIpIqI ++=−+−−+−− &&&&&&&& (2.34b)

[ ] PSCFDGGzxyzzzxyzz NNNuyvxmpIqIrIpIqIrI ++=−+−−+−− &&&&&&&& (2.34c)

This results in a solution matrix ordered as follows:

Page 17 of 55

−

−

−

−

−

−

=

−−−

−−−

−−−

−

−

−

LHSRHS

LHSRHS

LHSRHS

LHSRHS

LHSRHS

LHSRHS

zyzzxGG

yzyxyGG

zxxyxGG

GG

GG

GG

NN

MM

KK

ZZ

YY

XX

r

q

p

w

v

u

IIImxmy

IIImxmz

IIImymz

mxmym

mxmzm

mymzm

&

&

&

&

&

&

0

0

0

000

000

000

(2.35)

This matrix is the same as the matrix 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. 2.35 results in estimates at the new time level for accelerations:

rqpwvu &&&&&& ,,,,,

which in turn, with the time step know, allow for calculation of velocities:

rqpwvu ,,,,,

Finally the calculation of auxiliary derivative quantities (in the inertial frame) is then

possible:

ψθφ &&&&&&&&& ,,,,,,,, OOOOOO wvuzyx

and from these the integrated quantities for position and angular movement:

ψθφ ,,,,, OOO zyx

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.

Page 18 of 55

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. 2.7 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.

2.1.7 Fluid EOM for Translating Rotating Coordinate System

In this section the fluid EOM for the case of a translating rotating coordinate system is

presented. Since the coordinate system is free to move in all six degrees of freedom, the

apparent body force contribution (Fb) contains four terms. This is different then with a

translating coordinate system where the result is only one term in Fb, however the EOM

must then be cast in an Arbitrary Lagrangian Eulerian (ALE) form. These equations are

shown in the next section.

The conservation equations for mass and momentum are as follows where ui represents

the velocity field relative to a local coordinate system:

Mass Conservation:

0=∂

∂+

∂∂

j

j

x

u

t

ρρ (2.36)

Momentum:

BS

j

jii FFx

uu

t

u+=

∂

∂+

∂

∂ ρρ (2.37)

where

∂

∂

∂∂

+∂∂

−=j

i

ji

Sx

u

xx

PF (2.38)

and the apparent body force

( )

×+××+×+−= i

oo

b urrdt

d

dt

RdF 2112

2

ρ (2.39)

Page 19 of 55

For turbulent simulations the k ω based Shear Stress Transport (SST) is used and remain

unmodified from their standard form.

k equation:

∂∂

+

∂∂

+−=∂

∂+

∂∂

jk

t

j

k

j

j

x

k

xkP

x

ku

t

k

σωρβ

ρρ* (2.40)

ω equation:

∂∂

+

∂∂

+−=∂

∂+

∂∂

j

t

j

k

j

j

xxP

kx

u

t

ωσ

βρωω

αρωρω

ω

2 (2.41)

For details on the turbulence model source terms the ANSYS CFX documentation can be

consulted.

2.1.8 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 fixed 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 umj, 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

mjj

x

uu

t

ρρ (2.42)

Momentum:

BS

j

mjjii FFx

uuu

t

u+=

∂

−∂+

∂

∂ )(ρρ (2.43)

where

∂

∂

∂∂

+∂∂

−=j

i

ji

Sx

u

xx

PF (2.44)

Page 20 of 55

and

2

2

dt

RdF o

b ρ−= (2.45)

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:

k equation:

∂∂

+

∂∂

+−=∂

−∂+

∂∂

jk

t

j

k

j

mjj

x

k

xkP

x

uuk

t

k

σωρβ

ρρ*

)( (2.46)

ω equation:

∂∂

+

∂∂

+−=∂

−∂+

∂∂

j

t

j

k

j

mjj

xxP

kx

uu

t

ωσ

βρωω

αρωρω

ω

2)(

(2.47)

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

extraction of the mesh velocity umj. The mesh displacement equations have the form:

Mesh displacement equations:

0'

=

∂

∂Γ

∂∂

j

i

j x

x

x (2.48)

where

o

iii xxx −=' (2.49)

Note that the displacement diffusion coefficient, Γ, in Eq. (2.48) 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.

Page 21 of 55

When the body axis is oriented relative to the inertial axis 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 x). This order is accounted for through an appropriate

transformation matrix, 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

auxiliary equations previously described for the angular motion (body fixed angular

velocities):

( ) θφφφ tansincos qrp ++=&

φφθ sincos rq −=&

θφφ

ψcos

sincos qr +=&

and translational motion (submarine velocity in inertial coordinates):

( ) ( )ψθφψφψφψθφψθ cossincossinsinsincoscossinsincoscos ++−+= wvuxo&

( ) ( )ψφψθφψθφψφψθ cossinsinsincossinsinsincoscossincos −+++= wvuyo&

φθφθθ coscossincossin wvuzo ++−=&

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 Fb in the fluid EOM. In this case the auxiliary

equations are (submarine acceleration in inertial coordinates):

( ) ( )ψθφψφψφψθφψθ cossincossinsinsincoscossinsincoscos ++−+= wvuuo&&&&

( ) ( )ψφψθφψθφψφψθ cossinsinsincossinsinsincoscossincos −+++= wvuvo&&&&

φθφθθ coscossincossin wvuwo&&&& ++−=

Page 22 of 55

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 next 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 +x, x, +y, y, +z and z when flow is entering the

domain. When flow is exiting, 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:

2

2

rels

o

rel

VPP += (2.50)

with the static pressure at the boundary, Ps, taken to be zero (i.e. the reference pressure

for the solution) on the premise of quiescent flow, and is independent of the fame of

reference of the solution. The velocity Vrel is calculated on the basis of Eq. 2.3 and state

information which is described more fully for Option 2 in the next section. For exiting

flow velocity and direction is computed as part of the flow solution.

In this boundary condition treatment, adjacent control volumes 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. 2.50 based on the solution of the solid body EOM rather

than control volume values.

The remaining boundary condition is the submarine surface which can be treated as a no

slip “wall”.

Page 23 of 55

Figure 4a Opening boundary conditions employed at all faces relative to the coordinate

system.

Option 2

Another boundary condition possibility is one where the +x 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 side to side motion. The submarine

surface is treated as a no slip ‘wall’ while the remaining boundaries ( x, +y, y, +z) are

Page 24 of 55

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 control volumes 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 +x and –z inlets, possible

problems with this are described later.

Equation 2.3 is used to compute the applied velocity components (magnitude and

direction) at the inlet boundaries for Option 2, and is repeated here for clarity:

111 r

dt

rd

dt

Rd

dt

rd ooo ++= (2.3)

or rewriting in a more familiar nomenclature as:

11 ruuuO ++= (2.51)

where uO 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 u1 is the relative

velocity of the fluid particle at the boundary. The angular velocity can be either zero or

fixed at the angular velocity of the submarine. In either case the applied boundary

condition velocity (and direction) must be relative to the body fixed coordinate system

and so Eq.2.51 is rearranged to solve for u1:

11 ruuu O −−= (2.52)

In solving this equation r1 is the distance between the boundary mesh point and the

coordinate system origin located on the axis 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 uO 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:

uuu O −=1 (2.53)

and for =ω:

11 ruuu O ω−−= (2.53)

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 25 of 55

This second boundary condition is less desirable since inconsistencies can arise at the

inlets between the calculated control volume 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 26 of 55

3 ANSYS CFX Implementation

This section includes a description of how the theory has been translated into a ANSYS

CFX six DOF simulation. A primary aim of the section is to outline in detail the

approach in resolving discretely the highly non linear 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

( 1+→kky& ) and the time step ( tk), current velocity state (y

k) of the submarine, and the

applied forces (FCFD) and moments (MCFD) as summarized in:

( )111 ,,, +→+→+→ = kk

CFD

kk

CFD

kkkk MFytfy& (3.1)

where

2

11

++→ +

=k

CFD

k

CFDkk

CFD

FFF (3.2a)

2

11

++→ +

=k

CFD

k

CFDkk

CFD

MMM (3.2b)

[ ] 11 +→+→ = kkkk rqpwvuy &&&&&&& (3.2c)

and

[ ]kk rqpwvuy = (3.2d)

The solution procedure involves iterating within a time step to obtain the average force

/moment ( CFDF and CFDM ) conditions over the time step that results in a new predicted

velocity state (yk+1

= uk+1

,vk+1

,wk+1

,pk+1

,qk+1

and rk+1

) at the next time level (k+1). The

computed submarine accelerations ( 1+→kky& ) are always the average for the time step and

are used as follows:

kkkkk ytyy += +→+ 11

& . (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 ( 1+k

CFDF and 1+k

CFDM ) to use in the

Page 27 of 55

solid body solution. The repetition of coefficient loops, and re evaluation of the

submarine state, is continued until the RMS residuals on the fluid equations are reduced

to below a specified limit.

The fluid forces FCFD evolve with time and can be viewed, when integrated over time

step tk

as an impulse applied to the submarine. For a functional relationship F(t), and

assuming a linear profile for F between tk and t

k+1 noting that t

k =t

k+1tk, the trapezoidal

rule can be applied to compute the integral yielding:

kkk

kkk tFF

tF+

=+

+→

2

11 (3.4)

The average force over the time step is then:

2

11

++→ +

=kk

kk FFF (3.5)

and gives the form used in Eq. 3.2a. Therefore the integration of fluid forces FCFD is

treated with a first order approximation consistent with the trapezoidal rule. A higher

order scheme could be employed by using information on F available at time t<tk.

During the validation exercise solutions were obtained using Eqs. 3.2a and b to represent

the FCFD and MCFD influence, and yielded excellent results as long as abrupt changes due

to blowing ballast were not present. With blowing ballast, as the rate of change in Vti/Vi

(volume of blown air to initial volume of ballast) increased, divergent behavior was

found. The only remedy for this was to modify Eqs. 3.2a and b by applying a zero order

approximation of the form:

2

11

k

CFD

k

CFDkk

CFD

FFF

+=

−→− (3.6a)

2

11

k

CFD

k

CFDkk

CFD

MMM

+=

−→− (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 exercise 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 28 of 55

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 ANSYS CFX 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 ANSYS CFX 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 executed

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 experience it has been shown that the

most efficient time step is one that uses approximately 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 expended within a time step.

Further to the variable time step nominally second order time integration is employed

with ANSYS CFX as well. This second order method uses information from the

previous two time levels to reduce errors in the calculation at the next time level. A

Runge Kutta high order time integration is not practical in implicit solvers such as

employed in ANSYS CFX.

The details of the time step 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 Nmin<N <Nmax. The time step T is to be bounded by Tmin and

Tmax.

The new time step is incremented or decremented based on scaling factors calculated

using functions:

Finc = 101/inc

(3.7a)

Fdec= 0.11/dec

(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 Nmin N or N Nmax being non zero. In such

cases:

Page 29 of 55

Nmin – N > 0 → t = MIN(Finc(Nmin N)

* t, tmax) (3.8a)

N Nmax > 0 → t = MAX(Fdec(N Nmax)

* t, tmin) (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:

Finc = inc (3.9a)

Fdec= dec (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(2) V yp(2) 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) xo yp(7)

ox&

y(8) yo yp(8) oy&

y(9) zo yp(9) oz&

y(10) φ yp(10) φ&

y(11) θ yp(11) θ&

y(12) ψ yp(12) ψ&

Page 30 of 55

Sequencing of CFD and Solid Body EOM Solution

Page 31 of 55

Solid Body Solver Organization

Page 32 of 55

4 Validation

The validation of the six DOF 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 Multi Component, Deeply

Submerged Vehicle Part I: Theory and Program Description” by George Watt.

From this report the six added mass forces are calculated as:

( ) 22 qZwqZrYrpYvrYqXuqwXuXX qwrpvqwu &&&&&&&&&&& ++−−−+++= 4.1a

( ) pqZwpZqrXupwrXurXrYpYvYY qwqwurpv &&&&&&&&&&& −−+−++++= 4.1b

( ) rpYpYvpYqXuqXqZwZwquXZ rpvquqww &&&&&&&&&&& +++−−++−= 22 4.1c

Page 33 of 55

( ) ( )( )( ) ( )qrNMwrvqZY

vwYZurXuvXpqrKpKwpvYK

rqqr

vwqwrpp

&&&&

&&&&&&&&&&

−−−++

−+−++++−= )( 4.1d

( ) ( ) ( ) ( )( ) ( )22

22

rpKprNKvpY

vrYuwXuwZXqMuqwZwquXM

rrpr

pwwuqqq

−+−+−

+−+−++−++=

&&&&

&&&&&&&&&&

4.1e

( ) ( ) ( ) ( )( )

( )pqMKwpZ

vqupYXvwXuvYXrNqrpKurvYN

qpq

pqwvurrr

&&&

&&&&&&&&&&&

−−+

−++−−−+−++= 4.1f

The coefficients rqpwvu XXXXXX&&&&&&

,,,,, etc. are calculated from potential flow theory for

a fixed 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 non dimensionalized with the following:

Coefficient for non

dimensionalizing

wvu XXX&&&

,, wvu YYY&&&

,, wvu ZZZ&&&

,, 3

2

1Lρ

rqp XXX&&&

,, rqp YYY&&&

,, rqp ZZZ&&&

,, 4

2

1Lρ

rqp KKK&&&

,, rqp MMM&&&

,, rqp NMN&&&

,, 5

2

1Lρ

wvu KKK&&&

,, wvu MMM&&&

,, wvu NNN&&&

,, 4

2

1Lρ

The density ρ is that of the fluid medium and L is the length of the ellipsoid along its

primary axis.

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 6x6 matrix of coefficients of which all are

zero except the diagonal ones in the case of an ellipsoid geometry. These coefficients are

read into ANSYS CFX 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 34 of 55

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.

2. 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 approximated. 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 Geometry

To start a 6:1 ellipsoid shape can be chosen. The added mass coefficients for this shape

have been provided by DRDC as follows:

uX& 1.3143171E 3

vY& 2.6678039E 2

wZ& 2.6678039E 2

pK& 0.0

qM& 1.1395404E 3

rN& 1.1395404E 3

Page 35 of 55

Note that the first three coefficient have been nondimensionalized by C=1/2ρL3, and the

last three by B=1/2ρL5. Here L is the ellipsoid length equal to 2a, and the density ρ is

that of the fluid.

The geometric shape of the ellipsoid is governed by:

12

2

2

2

2

2

=++c

z

b

y

a

x (4.2)

with the x axis chosen as the principle axis with xmax = 0.5 m and xmin= 0.5 m. For a 6:1

ellipsoid then a = 0.5 m, b = 0.08333 m and c = 0.08333 m.

The moments of inertia are:

( )22

15

4cbabcI xx += ρπ (4.3a)

( )22

15

4caabcI yy += ρπ (4.3b)

( )22

15

4baabcI zz += ρπ (4.3c)

where ρ is the density of the material here taken as 997 kg/m3.

The volume of the ellipsoid is given as:

abcV π3

4= (4.4)

so that the mass of the ellipsoid is calculated by:

Vm ρ= (4.5)

The center of gravity G 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:

Ixx 0.040279937 kgm2

Iyy 0.74517883 kgm2

Izz 0.74517883 kgm2

Page 36 of 55

and the mass M=14.500777 kg. These quantities are requires as input to the ANSYS

CFX solid body equation of motion model, including the locations of CB and G.

Page 37 of 55

Validation Cases: Method 1

Case 1a

Linear acceleration of the 6:1 ellipsoid forward in the x direction by applying a

propulsion force with neutral buoyancy. The acceleration of the ellipsoid can be obtained

analytically and compared to the ANSYS CFX solid body prediction. From Newton’s

2nd

law, if an applied propulsive force of 100 N in the positive x direction is applied

(XCFD=100 N), and using the added mass equation 4.1a for the added mass force, the

acceleration can be calculated as:

u

CFD

CXm

Xu

&

&−

=

The ANSYS CFX prediction for this case is an acceleration of 6.5980625 m/s2, which

compares to the analytical result of 6.5980625 m/s2. This is an error of 0.0%.

Case 1b

Linear acceleration of the 6:1 ellipsoid forward in the y direction by applying a

propulsive force with neutral buoyancy. The acceleration of the ellipsoid can be obtained

analytically and compared to the ANSYS CFX solid body prediction. From Newton’s 2nd

law, if an applied propulsive force of 100N in the positive y direction is applied

(YCFD=100 N), and using the added mass equation 4.1b for the added mass force, the


v

CFD

CYm

Yv

&

&−

=

The ANSYS CFX prediction for this case is an acceleration of 3.5971508∗ m/s

2, which


Case 1c

Linear acceleration of the 6:1 ellipsoid forward in the z direction by applying a

propulsive force with neutral buoyancy. The acceleration of the ellipsoid can be obtained


law, if an applied propulsive force of 100N in the positive z direction is applied

(ZCFD=100 N), and using the added mass equation 4.1c for the added mass force, the


w

CFD

CZm

Zw

&

&−

=


2, which


∗ The last two digits represent the average of a small oscillation (+/ 0.0000010) that was observed on the

output.

Page 38 of 55

Case 1d

Angular acceleration of the 6:1 ellipsoid in the ϕ direction by applying a propulsive

moment with neutral buoyancy. The acceleration of the ellipsoid can be obtained


law, if an applied propulsive moment of 100Nm around the x axis is applied (KCFD=100

N), and using the added mass equation 4.1d for the added mass force, the acceleration can

be calculated as:

p

CFD

BKm

Mp

&

&−

=



Case 1e

Angular acceleration of the 6:1 ellipsoid in the θ direction by applying a propulsive



law, if an applied propulsive moment of 100Nm around the y axis is applied (MCFD=100

Nm), and using the added mass equation 4.1e for the added mass force, the acceleration

can be calculated as:

q

CFD

BMm

Mq

&

&−

=


2, which


Case 1f

Angular acceleration of the 6:1 ellipsoid in the ψ direction by applying a propulsive



law, if an applied propulsive moment of 100Nm around the z axis is applied (NCFD=100

Nm), and using the added mass equation 4.1f for the added mass force, the acceleration

can be calculated as:

r

CFD

BNm

Nr

&

&−

=


2, which


∗ The last two digits represent the average of a small oscillation (+/ 0.000008) that was observed on the

output.

Page 39 of 55

Validation against Octave results

The solid body equations of motion using apparent mass terms simplified for an ellipsoid

are presented in Eq. 4.2. These are solved using Octave and compared to ANSYS CFX

results. The algorithm used for the integration in Octave was DASSL1 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 complex cases when analytical results are not

available.

rzz

gg

rzz

xxyybgbgqvuCFD

qyy

gg

qyy

zzxxbgbgrwuCFD

pxx

ggbgbgCFD

w

gggvuCFD

v

gggwuCFD

u

gggwvCFD

NI

wqvruyurwpvxm

NI

pqIIByWyBxWxpqMuvYXNr

MI

vpuqwxwqvruzm

MI

prIIBzWzBxWxprNuwZXMq

KI

urwpvzvpuqwymBzWzByWyKp

Zm

prqyqrpxqpzvpuqmBWvpYuqXZw

Ym

rqpxpqrzpryurwpmBWwpZurXYv

Xm

qprzrpqyrqxwqvrmBWwqZvrYXu

&

&

&&&

&

&

&&&

&

&

&&

&

&&

&

&&

&&

&

&&

&

&&&

&&&

&&&

&&&

−

+−−+−−

−

−−−−−+−−−=

−

+−−+−−

−

−−−−−−−−+=

−

+−−+−−−−−−=

−

++−++−+−−−++−=

−

++−++−+−−−+−+=

−

++−++−+−−−−+−=

))()((

)(sin)(sincos)()(

))()((

)(sin)(coscos)()(

))()((sincos)(coscos)(

))()()((coscos)(

))()()((sincos)(

))()()((sin)(

22

22

22

θϑθ

θϑθ

ϑθϑθ

ϑθ

ϑθ

θ

4.6

1 K. E. Brenan, et al., Numerical Solution of Initial Value Problems in Differential Algebraic Equations,

North Holland (1989) for more information about the implementation of DASSL.

Page 40 of 55

Case 1g

This test case uses a sphere with dimensions a=b=c=0.5m.

Linear acceleration of the sphere upward in the z direction 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 ANSYS CFX solid body prediction. From Newton’s 2nd

law, if the

buoyant force is twice its weight and the added mass coefficient is calculated2 to be

6/π−=wZ&

the acceleration can be calculated as:

wCZm

Ww

&

&−

−=


2, which

compares to the analytical result of –6.5377667 m/s2. This is an error of 0.000003%.

Integrating this twice to obtain the z displacement gives:

2

2

1)( t

CZm

Wtz

w&−−=

Using ANSYS CFX to predict the value of the z displacement at t=0.1s using a time step

of 0.001s gives z(0.1)= 3.3015724E 2m while the analytical result gives z(0.01) =

3.2688833E 2m. This is an error of 1% over the integration of 100 timesteps. Octave

predicted z(0.1)= 3.2688888E 2m for this test case. This is an error of 0.00016%.

Validation Cases: Method 2

These test cases replicate those undertaken for Method 1 except 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. 2.39) applied as a source term to each control

volume in the solution:

( )

×+××+×+−= i

oo

b urrdt

d

dt

RdF 2112

2

ρ

each term is tested in the following six 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 axes. The final case (Case

g) considers the first term applied in two directions simultaneously.

2 using equations presented in the DRDC document: “Estimates for the Added Mass of a Multi Component,

Deeply Submerged Vehicle Part I: Theory and Program Description” by George Watt

∗ The last digit represents the average of a small oscillation (+/ 0.0000002) that was observed on the

output.

Page 41 of 55

Case 2a

Linear acceleration of the 6:1 ellipsoid forward in the x 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 ANSYS CFX solid body prediction. From

Newton’s 2nd

law, if an applied propulsive force of 100 N in the positive x direction is

applied (XCFD=100 N), and using the added mass equation 4.1a for the added mass force,


u

CFD

CXm

Xu

&

&−

=



In the Figure below is shown the velocity field predicted in this case, which is relative to

the body fixed coordinate system. The velocity field is opposite to the acceleration of the

body in the positive x direction.

Page 42 of 55

Case 2b

Linear acceleration of the 6:1 ellipsoid in the y direction (see Fig. 4) by applying a

propulsion force with neutral buoyancy. The acceleration of the ellipsoid can also be


Newton’s 2nd

law, if an applied propulsive force of 100 N in the positive y direction is

applied (YCFD=100 N), and using the added mass equation 4.1b for the added mass force,


v

CFD

CYm

Yv

&

&−

=


compares to the analytical result of 3.591 m/s2. This is an error of 0.306% which is much

larger than for Case a, however motion in the y axis 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 body fixed coordinate system. Again, the velocity field is in the opposite sense to the

motion of the solid body.

Page 43 of 55

Case 2c

Linear acceleration of the 6:1 ellipsoid in the z direction (see Fig. 4) by applying a

propulsion force with neutral buoyancy. The acceleration of the ellipsoid can also be


Newton’s 2nd

law, if an applied propulsive force of 100 N in the positive z direction is

applied (ZCFD=100 N), and using the added mass equation 4.1c for the added mass force,


w

CFD

CZm

Zw

&

&−

=


compares to the analytical result of 3.591 m/s2. 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 z direction.

Page 44 of 55

Case 2d

This test case applies a fixed propulsive moment to the vehicle of 100 Nm around the x

axis of Fig. 4. The solution is now coupled to the CFD solver, which calculates the

forces experienced 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:

p

CFD

BKm

Kp

&

&−

=

the analytical solution is 2475.178 rad/s2. The CFD prediction for this case is 2475.153

rad/s2 with a percentage error of 0.001%.

The velocity field associated with this prediction is given in the Figure below, where the

counter rotating flow about the y axis is as expected with the frame of reference fixed to

the ellipsoid body. Note that the opening boundary conditions treat both the inflow and

outflows correctly around the entire flow domain.

Page 45 of 55

Case 2e

This test case applies a fixed propulsive moment to the vehicle of 100 Nm around the y

axis of Fig. 4. The solution follows as for Case 2d.


angular momentum with the apparent body force based on the application of Eq. 4.1(e).


q

CFD

BMm

Mq

&

&−

=

the analytical solution is 76.005 rad/s2. The CFD prediction for this case is 76.280 rad/s

2

with a percentage error of 0.362%. This is considered good keeping in mind that the

ellipsoid surface is not perfectly smooth and that this motion, as opposed to Case 2d,

involves a much larger displacement of the surrounding fluid.

The velocity field associated with this prediction is given in the Figure below, where the

counter rotating flow about the y axis is as expected with the frame of reference fixed to

the ellipsoid body. It is clear that much more fluid is displaced than when the body

rotates about its x axis. Note that the opening boundary conditions treat both the inflow

and outflows correctly around the entire flow domain.

Page 46 of 55

Case 2f

This test case applies a fixed propulsive moment to the vehicle of 100 Nm around the z

axis of Fig. 4. The solution follows as for Case 2d.


angular momentum with the apparent body force based on the application of Eq. 4.1(f).


r

CFD

BNm

Nr

&

&−

=

the analytical solution is 76.005 rad/s2. The CFD prediction for this case is 76.292 rad/s

2

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 z axis

involves the same displacement of fluid as that for motion about the y axis the result

should be nearly identical to that for Case 2e.

The velocity field associated with this prediction is given in the Figure below where the

counter rotating flow about the z axis is as expected with the frame of reference fixed to

the ellipsoid body. Note that the opening boundary conditions again treat both the inflow

and outflows correctly around the entire flow domain.

Page 47 of 55

Case 2g

This test case applies a fixed propulsive force (100 N in the x axis) to the same ellipsoid

geometry of the previous tests. Furthermore the submerged ellipsoid is treated as heavier

than water via a reduced buoyancy force applied in the z direction. The heavier than

water motion is obtained by setting the mass to 14.50078 kg and the buoyancy force to

13.83597891 N. The center of gravity and buoyancy remain located at the ellipsoid

origin. No rotation or out of plan motion is allowed in the CFD simulation so as to

reflect the idealized solution as close as possible.

In this case there is no analytical solution available for the resultant motion, and therefore

comparison of the CFD predictions is against the numerical integrator Octave described

earlier.

For the Octave solution a simulation time of 1 millisecond is used, with a time step of

0.001 milliseconds. This numerical result is used as a baseline comparison for the CFD

simulation.

The predicted steady acceleration in the x and z directions from Octave is respectively,

6.5981 m/s2 and 4.6176 m/s

2. Meanwhile the CFD predictions give 6.6015 m/s

2 and

4.6316 m/s2. The percentage difference from the Octave result for the two accelerations

is 0.05% and 0.30%. Differences can in part be attributed to the mesh surface not being

perfectly smooth. In addition the motion in the Z direction involves a larger

displacement of fluid and the error scales accordingly in that direction.

In the Figure below is shown the trajectory of the submersed ellipsoid using results from

Octave and the CFD solution. In both cases a fixed time step is used of 0.001

milliseconds for a simulation time of 1 millisecond.

Page 48 of 55

Validation with Fully Appended Submarine Shapes

In this method of validation the potential flow coefficients for added mass are again used.

However in this instance the 6x6 matrix has additional terms since a fully appended

submarine shape is approximated by several ellipsoids. These same coefficients are

used in a DSSP (DRDC Submarine Simulation Program) style submarine model

developed by ANSYS and written in the Octave programming language/environment.

The coefficients and any specialized models for viscous effects, propulsion,

ballast/blowing and rudder/stern deflections are provided by DRDC. These simulations

provide submarine trajectories under applied forces (buoyancy, gravity, propulsion etc.)

and added mass forces based on coefficients. The time integration scheme is a high order

explicit scheme that is used in the Octave program. Note that the DSSP program from

DRDC is considered too complex for use in this process of verifying the ANSYS CFX

six DOF model.

DRDC provided 8 test cases2 to validate the Octave based simulator with the DRDC’s

maple based DSSP program. The comparison between these was excellent as all state

variables and model specific variable were reproduced to plotting accuracy, for the cases

where the data files were provided, and for the others visual inspection sufficed to

acknowledge the agreement. For brevity the plots are not shown here as these same test

cases are used in the next section.

The results from the Octave based simulator are then used for comparison against a CFX

solution where the same added mass coefficients (6x6 matrix) are used to provide forces

to the solid body solver (other force information for viscous effects, propulsion etc. were

also be included). A full submarine trajectory is obtained using the CFX fluid/solid body

coupling scheme (based on an implicit time integration method) and compared to the

explicit high order solution obtained by Octave. Note that both the Octave based

simulator and the CFX model use the same moment of inertia data to represent the fully

appended sub, centers of gravity and buoyancy, and applied forces.

These simulations are very fast compared to a fully coupled CFD/solid body solution

where the forces are generated by the Navier Stokes equations. Furthermore, and

importantly, a CFD mesh of the full submarine with appendages is not required for this

validation phase.

As mentioned previously, DRDC provided 8 test cases that are, along with the Octave

based simulator mentioned above, used to help in the implementation and validation of

the Standard Model 6 DOF Equations of Motion in CFX. The results shown below

represent the three translational and three rotational coordinates plotted as a function of

2 More details about the test cases below are available in the DRDC report entitled

“Standard Model 6 DOF Equations of Motion: HST: Axisymmetric Hull, Sail,

Symmetrical “+” Tailplanes” written by G. Watt.

Page 49 of 55

time. The Octave results (points) are overlaid on top of the CFX results (line) in each plot

to facilitate comparison. As can be seen excellent agreement was obtained.

Simulation 1 – Base Case

The first simulation reproduces the emergency rise procedure that some of the larger

boats use to avoid the roll instability. Start with straight and level flight at a depth of

100m and a speed of 3 m/s. At t=0s, pitch the nose up, blow emergency ballast, and

increase speed to 6 m/s to get to the surface fast with incidence (Theta) minimized.

0501001502002500 10 20 30 40 50Xo[m℄ time [s℄O tavebbbbbbbbbbbbbbbbbbbb

b bCFX �0:3�0:25�0:2�0:15�0:1�0:0500:050 10 20 30 40 50Yo[m℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �40�20020406080

1000 10 20 30 40 50Zo[m℄ time [s℄bbbbbbbbbbbbbbbbbbbbb

�2�1:6�1:2�0:8�0:400 10 20 30 40 50�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �505101520

250 10 20 30 40 50�[deg℄ time [s℄bb bb bb bb bb bb bb bb bb bbb �0:4�0:3�0:2�0:100:1

0 10 20 30 40 50 [deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb

Page 50 of 55

Simulation 2 – No Speed Increase

The simulation is the same as S1 (Simulation 1) except that the commanded speed is kept

at 3 m/s.

040801201602000 10 20 30 40 50Xo[m℄ time [s℄O tavebbbbbbbbbbbbbbbbbbb

bb bCFX �0:3�0:25�0:2�0:15�0:1�0:0500:050 10 20 30 40 50Yo[m℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �40�20020406080


�0:15�0:1�0:0500:050:10:150:20:250 10 20 30 40 50�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbb

b�50510152025

0 10 20 30 40 50�[deg℄ time [s℄bb bb bb bb bb bb bb bb bb bbb �0:08�0:06�0:04�0:0200:020 10 20 30 40 50 [deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb

Simulation 3 – Double Dynamic Pressure

The simulation is the same as S1 (Simulation 1) except that forward speed is increased by

50%. That is, the initial speed is 4.5 m/s and it is increased to 9 m/s.


bb bCFX �1:6�1:2�0:8�0:400:40 10 20 30 40 50Yo[m℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �120�80�4004080120


�3:5�3�2:5�2�1:5�1�0:500 10 20 30 40 50�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �100102030

400 10 20 30 40 50�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �1�0:8�0:6�0:4�0:200:2


Page 51 of 55

Simulation 4 – No Sternplane Control

This simulation is the same as S1 (Simulation 1) except that sternplane control is not

used. This means the pitch angle remains well under 10 degrees.


bb bCFX �0:04�0:0200:020:040:060:080 10 20 30 40 50Yo[m℄ time [s℄bbbbbbbbbbbbbbbbbbbbb 20406080


�2�1:6�1:2�0:8�0:400 10 20 30 40 50�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �10123

40 10 20 30 40 50�[deg℄ time [s℄bb bb bb bb bb bb bb bb bb bbb �0:0200:020:040:060:080:1


Simulation 5 – Using Normal Air for Blowing

This simulation is the same as S1 (Simulation 1) except that the MBT’s are blown using

the “normal” air supply which is half that of the emergency supply.


bb bCFX �0:25�0:2�0:15�0:1�0:0500:050 10 20 30 40 50Yo[m℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �20020406080


�1:4�1:2�1�0:8�0:6�0:4�0:200 10 20 30 40 50�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �505101520

250 10 20 30 40 50�[deg℄ time [s℄bb bb bb bb bb bb bb bb bb bbb �0:25�0:2�0:15�0:1�0:0500:05


Page 52 of 55

Simulation 6 – Curtailed Pitch Angle

This simulation is the same as S1 (Simulation 1) except that the pitch angle is curtailed at

the end of the manoeuver so the boat surfaces with a low angle of 5 degrees or so. Some

operators feel the boat will be more stable if it surfaces horizontally.


bb bCFX �0:0200:020:040:060:080 10 20 30 40 50Yo[m℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �20020406080


�2:5�2�1:5�1�0:500 10 20 30 40 50�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �505101520

250 10 20 30 40 50�[deg℄ time [s℄bb bb bb bb bb bb bb bb bb bbb �0:2�0:100:10:20:30:4


Simulation 7 – High Pitch Angle

This simulation is the same as S1 (Simulation 1) except the pitch angle is allowed to

increase indefinitely without sternplane curtailment.

040801201600 10 20 30 40 50Xo[m℄ time [s℄O tavebbbbbbbbbbbbbbbbb

bbbb bCFX �1:2�1�0:8�0:6�0:4�0:200:20 10 20 30 40 50Yo[m℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �80�4004080120


�14�12�10�8�6�4�20 0 10 20 30 40 50�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �20020406080

1000 10 20 30 40 50�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �16�12�8�404 0 10 20 30 40 50 [deg℄ time [s℄

bbbbbbbbbbbbbbbbbbbbb

Page 53 of 55

Simulation 8 – Turning Spiral, No Sternplanes

The simulation applies 25 degrees of rudder without blowing ballast or applying

sternplane control. The boat turns and begins a downwards spiral which is unrealistic it

should be an upwards spiral. This happens because the out of plane force resulting from

rotation r is currently not well modeled. This doesn’t matter for the RS problem.

040801201602000 20 40 60 80 100Xo[m℄ time [s℄O tavebbbbbbbbbbbb

bbbbbbbbb bCFX �300�250�200�150�100�500500 20 40 60 80100Yo[m℄ time [s℄bbbbbbbbbbbbbbbbbbbbb 98100102104106108110112


�3�2:5�2�1:5�1�0:500:50 20 40 60 80100�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �0:2�0:100:10:20:30:4

0 20 40 60 80100�[deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb �160�120�80�4000 20 40 60 80100 [deg℄ time [s℄bbbbbbbbbbbbbbbbbbbbb

Full Navier Stokes Six DOF Submarine Simulations

The activities of the preceding section are to be repeated with full coupling to the Navier

Stokes equations active. In this case the forces supplied to the solid body EOM solver

are obtained by integrating the pressure and shear stress distribution over the submarine

surface. In these simulations a complete submarine mesh with appendages will need to

be supplied. A fully appended mesh has been built at ANSYS CFX for this purpose and

will be used in this next phase of testing to be conducted at the University of New

Brunswick. It is expected however that this mesh will not be adequate for capturing all

of the important flow interactions influencing the Rising Stability, and that future mesh

refinement will be required.

Due to the small time steps required for accuracy in a transient simulation, as well as the

size of the mesh needed to represent the submarine affected flow field, these simulations

will be lengthy. The predicted motions can again be verified against the Octave

simulator results; however, differences may be significant because of limitations in the

coefficient based hydrodynamic model used by Octave. The fully coupled CFD model

considers flow interaction between appendages as well as viscous effects. If these

Page 54 of 55

influences are modeled in the Octave simulator they will be highly simplified.

Differences should therefore be expected and at times significant.

Page 55 of 55

5 Recommendations for Future Work

Some recommendations are presented here to guide future activities.

1. The next release of ANSYS CFX V11 uses exact volume calculations associated

with the ALE moving mesh option. This eliminates one serious obstacle to using

this option for 6 DOF simulations in CFX. With relatively minor work the

current 6 DOF capability can be extended to also use an ALE moving mesh

treatment. It should be noted that the moving mesh option introduces difficult to

assess discretization errors (if any) in the fluid flow equations (all equations are

affected). It therefore should be compared to the present full apparent mass

approach to see if both methods given similar answers.

2. The present development work has been done using ANSYS CFX V10. Some

minor work can be done to incorporate the apparent mass source terms into the

main CFX source repository. At present the source code outside of the user

source code envelope consists of only two files.

3. A second order treatment of the CFD forces applied in the solid body EOM

calculations should be investigated, although the first order/zero order approach

already gives very good results compared to the high order Octave solutions.

4. It would be advisable that development of a hybrid scripted mesh for the

appended submarine be started. With the 6 DOF model implemented correctly

the mesh quality remains an important aspect of solution accuracy. The current

fully appended mesh will allow simulations to be undertaken but it may not be

accurate enough for final calculations.


DRDC Atlantic mod. May 02

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ANSYS Canada Ltd.

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2. SECURITY CLASSIFICATION (overall security classification of the document

including special warning terms if applicable).

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3. TITLE (the complete document title as indicated on the title page. Its classification should be indicated by the appropriate

abbreviation (S,C,R or U) in parentheses after the title).

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

4. AUTHORS (Last name, first name, middle initial. If military, show rank, e.g. Doe, Maj. John E.)

A.G. Gerber, P.G. Godine, S. Dajka

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January 2007

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Project 11GP03

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DRDC Atlantic CR 2007-020

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DRDC Atlantic mod. May 02

13. ABSTRACT (a brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is

highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication

of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R),

or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual).

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 fixed 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 next 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.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterize a

document and could be helpful in cataloguing the document. They should be selected so that no security classification is required.

Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included.

If possible keywords should be selected from a published thesaurus. e.g. Thesaurus of Engineering and Scientific Terms (TEST) and

that thesaurus-identified. If it not possible to select indexing terms which are Unclassified, the classification of each should be

indicated as with the title).

Six Degrees-of-Freedom

Reynolds Averaged Navier-Stokes equations

Computation Fluid Dynamics

Unsteady

Maneuvering

Submarine Hydrodynamics


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