N Kimber and P Crossland, QinetiQ Ltd, UK SUMMARY
As the complexity of modern submarine design increases, it is sometimes necessary to develop or modify the mathematical models which describe the manoeuvring characteristics. Submarine manoeuvring simulations have been developed from quite simple models that can predict basic manoeuvres accurately into more complex models that aim to improve the quality of the prediction when the submarine is undergoing more extreme manoeuvres. These complex, coefficient based, models demand a more extensive experimental test programme in order to derive the terms in the mathematical model; a change to the mathematical model that increases the number of independent variables can dramatically increase the number, and hence the overall duration and cost, of model tests. This paper describes some of the shortcomings in the mathematical models and how new experimental techniques can be developed to help understand these deficiencies and address them.
NOMENCLATURE
CFD Computational Fluid Dynamics DNS Direct Navier-Stokes DVL Doppler Velocity Log LES Large-scale Eddy Simulations MLD Manoeuvring Limitation Diagram RANS Reynolds-Averaged Navier-Stokes RLG Ring Laser Gyro
RN Royal Navy
RPM Revolutions Per Minute SRM Submarine Research Model UK United Kingdom B buoyancy force (N)
m mass (kg)
ǻm excess mass (kg)
g acceleration due to gravity (m/s2)
xG longitudinal location of centre of gravity (m) yG lateral location of centre of gravity (m) zG vertical location of centre of gravity (m) xB longitudinal location of centre of buoyancy (m) yB lateral location of centre of buoyancy (m) zB vertical location of centre of buoyancy (m) Zturn force acting on hull during turn (N) Mturn moment acting on hull during turn (Nm) u forward velocity (m/s)
v sideslip velocity (m/s) w vertical velocity (m/s)
X cross-flow velocity (m/s) = ¥(v2+w2) p roll rate (rad/s)
q pitch rate (rad/s) r yaw rate (rad/s) I roll angle (rad) T pitch angle (rad)
Xuu etc. hydrodynamic axial force coefficient Yuv etc. hydrodynamic side force coefficient Zvv etc. hydrodynamic lift force coefficient Kur etc. hydrodynamic roll moment coefficient Mvv etc. hydrodynamic pitch moment coefficient Nuv etc. hydrodynamic yaw moment coefficient Gb,Gs bow/stern hydroplane angle (rad) Gr rudder angle (rad)
1. INTRODUCTION
Numerical submarine manoeuvring simulation models have been developed by QinetiQ Haslar over many years for use by the UK Ministry of Defence to set safety and operational constraints for the RN fleet. The developments in technology and the requirements of modern submarines have meant that the simulation tools are being used in areas beyond their validated limits.
Experience with full-scale trials and experiments has shown that the quality of the prediction of the manoeuvring envelope of a submarine degrades as the manoeuvre becomes more extreme. Such manoeuvres are typical of situations where the submarine has experienced an emergency due to flooding or a plane jam incident.
Submarine manoeuvring simulation codes have developed from quite simple models that can predict moderate manoeuvres with some accuracy into more complex models that aim to improve the fidelity of simulations of more extreme manoeuvres.
It appears that there are some features of the depth and pitch response of a turning submarine that are not modelled sufficiently accurately. The authors attribute this to the cross-coupling terms in the coefficient based model that are derived from captive model tests.
Furthermore, the algorithms used in the simulation tools to estimate the forces on the submarine are essentially not for extreme angles of attack or extremes of propulsor state.
As a result of these shortfalls in the prediction capability, potentially excessive safety factors are applied to Manoeuvring Limitation Diagrams which in turn apply conservative operational limitations on the submarine [1].
In extreme emergency manoeuvres there are complex hydrodynamic flows that may not be amenable to numerical simulation using the existing coefficient based approach. Clearly, it would be advantageous to be able to improve the fidelity of the numerical simulation
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©2008: The Royal Institution of Naval Architects capability, to reduce the safety factors and potentially
relax the operational constraints.
The fidelity of the numerical methods needs to be improved to better understand the forces and moments acting on a manoeuvring submarine. Providing data for these more complex models will potentially require a more extensive experimental test programme; a change to the mathematical model that increases the number of independent variables can dramatically increase the number, and hence the overall duration and cost, of model tests. Thus, new techniques that can reduce the duration of experiments without compromising the quality of the data would be beneficial.
2. THE FORCES AND MOMENTS ON A SUBMARINE
2.1 COEFFICIENT BASED MODELS
The essence of submarine manoeuvring codes is that, for each time step, the state variables associated with the rigid body are equated to the external hydrodynamic forces and moments X, Y, Z, K, M and N. Based on the submarine's mass properties, the forces and moments are converted to accelerations, which are integrated to provide velocities and displacements. The mathematical approach to determining the quasi steady state forces and moments on a manoeuvring submarine are described by Gertler and Hagen [2]. For completeness the equations describing the hydrodynamic forces and moments are presented here.
Axial force
c c
c T c c
¹
·
©
§ cG
G G cG G G cG G G
¸¹
¨ ·
©
§ c c c c cc c
¸¹
¨ ·
©
§ c c c cc c c c
¸¹
¨ ·
©
§ c c c c c
c
Xwave Xn
sin g m B
s2 s s u Xu b2 b b u Xu r2 r r u Xu
p rpr X rrr 2 X qqq X
q wqw X r vrv X uu X
w2 Xww v2 Xvv Xuu X
2
Side force
c
T I c c c
¸
¹
¨ ·
©
§ c c c c c c c c c c c cc
¸
¹
¨ ·
©
§ c cG
G c Xc c cc X
¸¹
¨ ·
©
§ cc c c cc cc c c c c c cc
¹
·
©
§ cG
Xc G c c c X cc c c c c
Ywave +
cos sin B g m
r qrq Y + q pqp Y p p p Yp rr Y pp Y
r r r r Yu v r v v r Yv
r wrw Y + p wpw Y + q vqv Y + urr Y upp Y vv Y
r r u Yu + v v Y + w vwv Y uvv uu Y Y Y
Heave force
c c c I T c
¸¹
¨ ·
©
§ c c c c cc cc c
¸
¹
¨ ·
©
§ c cG
G c Xc c c c X
¸¹
¨ ·
©
§ c c c c c c c ccc
¸
¹
¨ ·
©
§ c cQc
Q c c cXc Xc
¸¹
¨ ·
©
§ cG
G G G c cc
c c c c
Zwave + cos cos B g m
p rpr Z 2+ rrr 2 Z ppp Z qq Z
s s q q Zu w q ww q Zw
r vrv Z + p vpv Z uqq Z ww Z
w w Z w w Zu + w w Z
s s Zuu + b b Zuu 2+ vvv Z uww uu Z Z Z
Roll moment
c c
T
¸ I
¹
¨ ·
©
§ c c c c c T
¸ I
¹
¨ ·
©
§ c c c c c
¸¹
¨ ·
©
§ c c c c c c c c cc c cc
¸¹
¨ ·
©
§ c c c c c c c c c c c c c cc
¸¹
¨ ·
©
§ cG
Xc G c c c X c c cc c c
Kn Kwave +
cos B sin z G B z g m cos B cos y G B y g m
r qrq K + q pqp K p p p Kp rr K pp K
r wrw K + p wpw K + q vqv K + urr K up p K vv K
r r u Ku + v v K + w vwv K uvv uu K K K
Pitch moment
c
¸ T
¹
¨ ·
©
§ c c c c c T
¸ I
¹
¨ ·
©
§ c c c c c
¸
¹
¨ ·
©
§ c c c c c c c c c c c c
¸
¹
¨ ·
©
§ c cG
G cXc Xc
¸¹
¨ ·
©
§ c c c c c c c c cc
¸
¹
¨ ·
©
§ c cXc
X c c Xc c c X
¸¹
¨ ·
©
§ cG
G G G c c c
c c c c
Mwave +
B sin z G B z g m cos B cos x G B x g m
q q q Mq + p rpr M 2+ rrr 2 M ppp M qq M
s s q q Mu q q
M
r vrv M + p vpv M uqq M ww M
w w M w w Mu + w w M
s s u Mu + b b u Mu 2+ vwv M uww uu M M M
Yaw moment
c
¸ T
¹
¨ ·
©
§ c c c c c
T
¸ I
¹
¨ ·
©
§ c c c c c
¸
¹
¨ ·
©
§ c c c c c c c c c c c cc
¸¹
¨ ·
©
§ c cG
G cXc Xc
¸¹
¨ ·
©
§ c c c c c c c c c c c c c c c
¹
·
©
§ cG
Xc G c c c X c c c c c c
Nwave +
B sin y G B y g m cos B sin x G B x g m
r qrq N + q pq p N p p p Np rr N pp N
r r r r Nu r r N
r wrw N + p wpw N + q vqv N + urr N up p N vv N
r r u Nu + v v N + w vwv N uvv uu N N N
Current methods of determining this coefficient set include physical captive model tests, numerical methods or a combination of both. In each case the model or geometry is constrained at a fixed angle of attack with or without the control surfaces at a fixed angle of attack; the resultant forces and moments are then measured or predicted.
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©2008: The Royal Institution of Naval Architects
Low fidelit y
High fidelit y LES CFD
Idealised met hods
Lift ing panel met hods Vort ex
met hods
Increase in empiricism
Increase in complexity
Direct NS Solvers RANS
CFD
Low fidelit y
High fidelit y LES CFD
Idealised met hods
Vort ex met hods
Increase in empiricism
Increase in complexity
Direct NS Solvers RANS
CFD
Low fidelit y
High fidelit y LES CFD
Idealised met hods
Lift ing panel met hods Vort ex
met hods
Increase in empiricism
Increase in complexity
Direct NS Solvers RANS
CFD
Low fidelit y
High fidelit y LES CFD
Idealised met hods
Vort ex met hods
Increase in empiricism
Increase in complexity
Direct NS Solvers RANS
CFD
2.2 REVIEW OF NUMERICAL METHODS There are various levels of sophistication associated with the numerical techniques that are available for deriving the quasi steady state forces and moments, ranging from methods involving solutions of the fundamental Navier- Stokes equations to those involving highly idealised methods. The range of techniques can be represented as the schematic in Figure 1. As would be expected, the more sophisticated methods provide the highest level of fidelity but are incredibly complex techniques requiring high levels of computing power.
Figure 1: Schematic of numerical techniques Direct Navier Stokes solvers (DNS) are presently only used by experts within academia, see [3] for example;
usually applied to a solution domain that is orders of magnitude smaller than those required for a submarine. A relatively less complex, albeit still computationally intensive, approach is Large-scale Eddy Simulations (LES), see [4] for example. This methodology revolves around the use of large-scale eddies that are associated with the geometry of the submarine. The benefit of this approach is the ability to describe the evolution of vortices and how they impact downstream on the submarine hull or appendages.
The Reynolds-Averaged Navier Stokes solvers (RANS) approach attempts to time-average the Navier Stokes equation by defining the instantaneous velocity components and pressure to be the sum of a mean component and a fluctuating component, see [5]. One issue associated with these CFD techniques is that a significant amount of effort is still required to create a satisfactory grid for the fully appended submarine.
A level of fidelity down from full Navier Stokes type techniques are inviscid methods, [6] for example, which assumes the viscosity is set to zero and so the Navier Stokes equations can be simplified significantly. These types of methods can be generally successful for hulls at modest angles of attack, but break down for any significant manoeuvre. In these instances, the increase in force with angle of attack becomes strongly non-linear due to the centre of pressure moving aft as a result of the flow separation. Thus, the non-linearities present in the moment have different characteristics to the underlying force.
This inability to identify accurately the position of flow separation can influence greatly the resulting prediction of forces and moments. Indeed, much of the data relating to separation points are derived from experiments and thus contain, to a certain extent, embedded empiricism.
Mathematical models with even greater simplification than lifting panel methods are vortex methods and purely idealised methods utilising classical theories. These methods are heavily dependent upon empirical data with the forces and moments on the hull determined using empirical expressions derived from experiments on a standard series such as bodies of revolution with varying fineness ratios and prismatic coefficients.
The obvious limitation of such methods is that any predictions of forces and moments on a submarine based on the empirical expressions must be evaluated in the context of how similar the hull form is to those used in that empiricism.
So, at the present time, physical model experiments remain a necessity for generating the majority of the coefficients which populate the mathematical model.
2.3 CONSTRAINED MODEL EXPERIMENTS The concept of a constrained model experiment is simple and well-established. Internal instrumentation consists of a strain-gauge balance capable of measuring the forces and moments acting on the body in all six degrees of freedom.
The model is towed in a Ship Tank at a range of body incidences in both the horizontal and vertical plane. This provides the relationship between the forces, moments and velocities. Then, with the model straight and level, the various control surfaces are exercised over their working range to measure forces and moments generated.
Typically, the model is then transferred to a Rotating Arm where the forces and moments due to rotational velocities are established.
The data from the two sets of experiments are combined and a least-squares regression is performed to provide the coefficients for the mathematical model.
The choice of range and combinations of rates and angles is generally determined by the form of the mathematical model. Some combinations cannot be physically achieved by a towed model. For example, despite there being an Nvwƍ coefficient in the numerical model, it is not easily possible to generate simultaneous vertical and horizontal velocities. This is where the numerical techniques have been used to provide an insight into the significance of such terms.
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©2008: The Royal Institution of Naval Architects 2.4 FREE MANOEUVRING MODEL
EXPERIMENTS
The conduct of free-manoeuvring model experiments can fulfil several purposes, including:
x confirmation of basic manoeuvring parameters such as turning circles and zig-zag overshoots x a measure of performance in waves
x confirmation of autopilot performance in a real- time, non-linear environment [7]
x providing manoeuvring data for simulation validation
The QinetiQ Submarine Research Model (SRM) is capable of all the above, but is chiefly used for exploring the extremes of the manoeuvring envelope [1]. Since that reference, the SRM has undergone a significant upgrade to its systems, improving functionality and operability.
Accurate motion measurements (angles and angular rates) continue to be provided by a Ring Laser Gyro (RLG). For the most recent experiment programme, the RLG was augmented with a Doppler Velocity Log (DVL) which provides accurate 3-dimensional velocity measurements. Integration of these velocities provides a reconstruction of the model's track, as demonstrated in Figure 2, which shows the model returning to within 1 m of the launch position, following a run in excess of 1400 m.
-300 -200 -100 0 100 200 300 400 500
-100
0
100
200
300
(m)
(m)
launch raft and position
circle manoeuvres recovery of model
Figure 2: Reconstructed track using DVL output Free-manoeuvring experiments are not routinely used for direct measurement of the forces and moments acting on the body. However, with accurate trajectory information, techniques are being developed which allow the reconstruction of the forces and moments acting on the body at each instant in time.
3. MODELLING CROSS-COUPLING BEHAVIOUR
The term cross-coupling refers to the vertical force and moment acting on the body during a horizontal plane turn,
sometimes referred to as the out-of-plane force and moment.
3.1 CURRENT MATHEMATICAL MODEL In the model of Gertler and Hagen [2], the force and moment are assumed to be the following functions of side-slip velocity, v, and yaw rate, r:
2 2
2 2
r M vr M v M M
r Z vr Z v Z Z
rr vr
vv turn
rr vr
vv turn
(1)
The form of these equations dictates the particular constrained experiments to be conducted in order to determine the coefficients. This case requires a range of pure side-slip velocities (or drift angles) on a Ship Tank carriage, a range of pure yaw rates on a Rotating Arm, and combinations of drift angles and yaw rates, also on the Rotating Arm. A regression on the full set of measurements then yields the coefficients. An example is plotted in Figure 3, showing the measurements and curve fits from a typical experiment.
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
v'
M'
r' = 0.2 r' = -0.2 r' = 0.35 r' = -0.35 r' = -0.5 r' = 0.5 r' = 0
Figure 3: Example measurement of M' as a function of v' and r'
3.2 INITIAL OBSERVATION
A routine test programme consists of constrained model experiments to determine the coefficients which populate the mathematical model. This is followed by free- running model experiments to explore the manoeuvring performance, such as turning circles, autopilot response and depth-keeping under waves. This data is also used for simulation validation.
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During one such programme the free-running model experiment included some free-turns. These are open- loop manoeuvres which apply a fixed rudder angle without any depth control during the turn. Following the natural excursions, the hydroplanes are re-activated and the turn continues under depth control.
Simulations of these manoeuvres using the coefficients derived from the constrained model experiments predicted far greater pitch and depth excursions than those demonstrated by the free-running model. There was historical evidence that previous data sets had required modifications to the cross-coupling coefficients, but for this programme, a rigorous method and justification were required.
The first assessment of the cross-coupling coefficient model was to explore a range of variations around the nominal values and observe their effect on the prediction of the pitch and depth response in the free turns.
It was observed for several runs that the coefficients which gave the best prediction of pitch and depth in the open-loop part of the turn did not predict the correct hydroplane angles required to subsequently maintain depth. Conversely, coefficients could be found which predicted the hydroplane angles in the controlled part of the turn, but did not exhibit the open-loop excursions.
Figure 4 plots an example of the experiment data and simulations with the two different modifications to the cross-coupling coefficients.
350 400 450 500 550 600 650 700 750 800 850
-30 -10 +10
depth (m)
350 400 450 500 550 600 650 700 750 800 850
-10 0 10 20
pitch (deg)
350 400 450 500 550 600 650 700 750 800 850
-20 0 20
bow (deg)
350 400 450 500 550 600 650 700 750 800 850
-20 0 20
stern (deg)
350 400 450 500 550 600 650
-20 0 20
rudder (deg)
Time (s)
experiment data open-loop best-fit closed-loop best-fit
Figure 4: Effect of revised coefficients on open-loop and closed-loop predictions On the assumption that the open-loop modification is correct (there are no other terms in the mathematical model which affect depth and pitch in a turn), the
conclusion was that the sternplane effectiveness on the free-running model must have been greater than that predicted by the simulation. However, for straight-line running and depth changing, simulations agreed very well with the experiment data, including the hydroplane angles required to achieve those trajectories. Any enhanced sternplane effectiveness must therefore only occur in a turn.
There is no coefficient in the mathematical model to account for this, nor were any constrained model tests conducted on a Rotating Arm to explore sternplane effectiveness during a turn. However, it is considered that due to the magnitude of the drift angle during a turn, the aft end of the submarine has a higher local velocity through the water than the centre, and therefore there is a greater flow over the stern hydroplanes.
3.3 REVISED FORMULATION
The current mathematical model assumes that the sternplane lift is proportional to forward velocity (u) squared, and is calculated using
s u Z
Z uuGs 2G (2)
Taking into account the local velocity, an alternative calculation is
s x r v u Z
Z uus s¸ G
¹
¨ ·
©
§ G
G
2 2
2 (3)
where v is the sideslip velocity, r is the yaw-rate and xGs is the effective longitudinal location of the stern hydroplanes.
400 500 600 700 800
-30 -10 depth (m) +10
400 500 600 700 800
-10 0 10 20
pitch (deg)
400 500 600 700 800
-20 0 20
bow (deg)
400 500 600 700 800
-20 0 20
stern (deg)
400 500 600
-20 0 20
rudder (deg)
Time (s)
experiment data revised model
Figure 5 : Effect of revised stern hydroplane model
Warship 2008: Naval Submarines 9, Glasgow, UK
©2008: The Royal Institution of Naval Architects By making these changes to both the lift and moment
calculations for the stern hydroplanes, and re-running the simulation with the original cross-coupling coefficients, the results for the example are plotted in Figure 5.
There is still a little over-prediction of the depth and pitch excursions, but there is now far better agreement for both the open-loop and closed-loop phases of the run.
4. IDENTIFICATION OF CROSS- COUPLING FORCE AND MOMENT FROM FREE-RUNNING MODEL DATA The closed-loop depth-controlled turns, if run for a sufficiently long period to obtain steady state conditions, provide enough information to calculate the cross- coupling force and moment acting on the hull. In part, this is due to the control algorithm in use, which guarantees that the ordered depth is maintained at zero pitch angle. In this condition, the following equations hold (incorporating equation (3)):
m mg
x uq M
s x r v u M
b u M uw M u M M
cos cos mg uq m m Z
s x r v u Z
b u Z uw Z u Z Z
g uq
s s
uu
b uu uw
uu turn
uq
s s
uu
b uu uw
uu turn
'
¸ G
¹
¨ ·
©
§
G
T I ' '
¸ G
¹
¨ ·
©
§
G
G G
G G G
G
2 2
2
2 2
2 2
2
2 2
(4)
where Zturn and Mturn are the unknown force and moment acting on the hull as a result of turning.
Note that in a steady turn, although the depth and pitch are constant, there are non-zero values of w and q in body axes due to the steady roll angle (I).
I I
tan r q
tan v
w (5)
In order to provide data points for equation (4), the SRM was programmed to conduct turning circles at three different speeds, using three different rudder angles to both port and starboard.
The steady state values of all the contributing parameters were extracted for each turn, and the right-hand side of equation (4) tabulated for each combination of speed and rudder angle.
4.1 FITTING A MODEL TO THE DATA
The standard cross-coupling model of equation (1) calculates the force and moment as functions of sideslip velocity, v, and yaw rate, r. Manoeuvring measurements
show that there is such a strong correlation between v andr that it is not possible to separate out the effects of these two variables independently (as can be done with a constrained model). Therefore, this analysis considers the force and moment as a function of a single parameter, v.
The measured force data points are plotted in Figure 6, with the moment data points in Figure 7.
The equations of the best fit lines are given by:
' v ' u ' M ' v ' v ' M ' M
' v ' u ' Z ' v ' v ' Z ' Z
v vv u
turn
v vv u
turn
(6)
This formulation provides a reasonable fit to each of the data sets analysed.
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 v'
Z'
8 knots 12 knots 18 knots original fit
Figure 6 : Z' as a function of v'
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 v'
M'
8 knots 12 knots 18 knots original fit
Figure 7 : M' as a function of v' The results confirm that the current mathematical model, with the coefficients derived from the constrained model experiments, does over-predict the force and moment when turning. The low speed (8 knot) runs show different characteristic curve-fits compared with the 12