Date 2 0 1 0
Author S . L . T o x o p e u s
Address D e l f t U n i v e r s i t y o f T e c l i n o l o g y S h i p H y d r o m e c h a n i c s L a b o r a t o r y M e l < e l w e g 2 , 2 6 2 8 C D D e l f t
Delft University of Teclinology
TUDelft
Calculation of bottom clearance effects on
lyd//-//^ submarine hydrodynamics
by
M. S e t t l e , S . L . Toxopeus and A. Gerber
Report No. 1794-P 2010
Published in: I n t e r n a t i o n a l Shipbuilding P r o g r e s s , Volume 5 7 , Number 3-4, 2 0 1 0 , I S S N 0 0 2 0 - 8 6 8 X , I C S Press
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Calculation of bottom clearance effects on Walrus submarine hydrodynamics Journal Publisher ISSN Subject issue Pages DOI Pages Subject Group Online Date
International Shipbuilding Progress IDS Press
0020-868X (Print) 1566-2829 (Online)
Engineering and Technolcov and Civil and Structural Engineering
Volume 57. Number 3-4 / 2010 101-125
10.3233/ISP-2010-0065 101-125
Engineering and Technologv Monday, January 24, 2011
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Mark Beltle', Serge L. Toxopeus^, Andrew Gerber' 'Llniversity of New Brunswick, Fredericton, NB, Canada
^Maritime Research Institute Netheriands/Delft Universily of Technology, Wageningen, The Netheriands
Abstract
Due to changes in operations, the Royal Netheriands Navy (RNLN) is operating its submarines increasingly in brown water regions (i.e., waters with restriction in width and/or depth, such as littoral areas). To improve predictions for manoeuvring in restricted watera/ays, the RNLN has ordered the Maritime Research Institute Netherlands (MARIN) to conduct studies regarding the influence of the seabed on the behaviour of the submarine. In this wori<, viscous-flow calculations are used to predict the influence of bottom clearance on the hydrodynamio forces on the Walrus class submarines.
The simulations representing unrestricted-water conditions have been validated wilh available model test results and good agreement was found. The details of the methodology and results of the viscous-flow calculations are presented in this article. The bottom clearance effects on hydrodynamio forces and moments are summarised.
A strong non-linear influence of the bottom clearance on the vertical force and pitch moment was found. Wilh a trim by the bow, the vertical down force can increase up to the same order of magnitude as the vertical up force that can be generated by the sail planes.
Keywords
Waims class submarine, CFD, RANS, manoeuvring, bottom effects, shallow water
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Contents
Y. Tian and Z.-S. Ji
Continuum damage model of low-cycle fatigue and
computation of ultimate strength of locally fatigue damaged
plates under uniaxial compression
M. Bettle, S.L. Toxopeus and A. Gerber
Calculation of bottom clearance effects on Walrus
submarine hydrodynamics
J.FJ. Pruyn, H.G. Hekkenberg and CM. van Hooren
Determination of the Compensated Gross Tonnage factors
for superyachts
S. Masiuk and V. Gorban
Calculation of ship interaction forces in restricted waterway
using three-dimensional boundary element method
Inlemational Shipbuilding Progress 57 (2010) 101-125 DOI 10.3233/ISP-2010-0065
IDS Press
101
Calculation of bottom clearance effects on
Walrus
submarine hydrodynamics
M a r k B e t t l e ^, Serge L . T o x o p e u s and A n d r e w G e r b e r ^
" University of New Briins^viclt, Fredericton, NB, Canada
^ Maritime Researcli Institute Netherlands/Delft University ofTechnology, Wageningen, The Netherlands
Due to changes in operations, the Royal Netherlands Navy (RNLN) is operating its submarines increas-ingly in brown water regions (i.e., waters with restriction in width and/or depth, such as littoral areas). To improve predictions for manoeuvring in restricted waterways, the RNLN has ordered the Maritime Research Institute Netheriands (MARIN) to conduct studies regarding the influence of the seabed on the behaviour of the submarine. In this work, viscous-flow calculations are used to predict the influence of bottom clearance on the hydrodynamic forces on the Walrus class submarines.
The simulations representing unrestricted-water conditions have been validated with available model test results and good agreement was found. The details of the methodology and results of the viscous-flow calculations are presented in this article. The bottom clearance effects on hydrodynamic forces and moments are summarised.
A strong non-linear influence ofthe bottom clearance on the vertical force and pitch moment was found. With a U'ira by the bow, the vertical down force can increase up to the same order of magnitude as the vertical up force that can be generated by the sail planes.
Keywords: Walrus class submarine, CFD, RANS, manoeuvring, bottom effects, shallow water
1. Introduction
Due to changes i n operations, the Royal Netherlands Navy ( R N L N ) is operating its submarines increasingly i n brown water regions (i.e., waters w i t h restriction i n width and/or depth, such as littoral areas). The hydrodynamic forces and moments on the submarine change significantly as it approaches solid boundaries such as the seabed. This i n turn affects the manoeuvrability and control o f the boat and must be accounted for to ensure its safe operation. To improve predictions f o r manoeu-vring i n restricted waterways, the R N L N has ordered the M a r i t i m e Research Institute Netherlands ( M A R I N ) to conduct studies regarding the influence o f the seabed on the behaviour of the submarine. This article describes a computational study aimed Corresponding author: S.L. Toxopeus, Maritime Research Institute Netherlands/Delft University of Technology, RO. Box 28, 6700 A A Wageningen, The Netheriands. Tel.: -t-Sl 317 493 443; Fax: +31 317 493 245; E-mail: S.L.Toxopeus@marin.nl.
102 M. Bettle et al. / Calculation of bottom clearance effects
at improving tlie accuracy of submarine manoeuvring predictions i n restricted wa-terways.
I n this work, the viscous-flow solver R E F R E S C O is used to predict the influence o f bottom clearance on the hydrodynamic forces on the Walrus class submarines used by the R N L N . For this first study, the unappended hull-deck-sail configuration was used. Straight-sailing simulations were conducted f o r a wide range o f bottom clearances c (see Fig. 5), i.e., ranging f r o m 1/8 to 12 times the maximum submarine breadth B. The effects o f d r i f t angle and trim angle were also investigated. For un-restricted water and f o r one condition with a small bottom clearance, grid sensitivity studies were conducted i n order to investigate the uncertainty i n the results due to discretisation. The simulations representing unrestricted-water conditions have been validated with available model test results and good agreement was found.
From the results of these calculations, a model describing the bottom influence has been derived. This model is to be implemented i n M A R I N ' s submarine manoeu-vring simulation program S A M S O N (Safety A n d Manoeumanoeu-vring System O f Navy submarines), which did not account f o r bottom clearance effects before.
The details of the methodology and results of the viscous-flow calculations are presented in this article. The bottom clearance effects on hydrodynamic forces and moments are summarised. The article concludes w i t h recommendations f o r future study that could further improve manoeuvring predictions o f near-bottom operation.
2. Numerical procedures 2.1. Coordinate system
A l l the force and moment data presented i n this report have been transformed to a standard coordinate system used i n submarine manoeuvring, see Fig. 1. The origin of this right-handed coordinate system is located at the intersection o f the longitudinal axis of symmetry of the hull, midship and centre-plane, w i t h x directed forward, y to starboard and z vertically downward. A l l integral forces X , Y , Z and moments K, M, N are directed as shown i n Fig. 1 and made non-dimensional with the length between perpendiculars Lpp o f the submarine using:
where p is density and Voo is the velocity of the submarine {Voo = (u, v, w)). The d r i f t angle is defined by /? = a r c t a n = ^ , with u and v directed according to the
X and y axes of Fig. 1, respectively, which means P is positive f o r flow coming f r o m port side. The angle o f attack is defined by a = ai'ctan ^ , which means a is positive f o r flow coming f r o m below. I n this work, the angle o f attack develops as a result of the submarine pitch or trim angle, 9, which is the angle the hull axis makes
M. Bettle et al. / Calculation of bottom clearance effects 103
z, w, Z
Fig. 1. Coordinate system.
with the horizontal plane. The angle o f attack a is equal to the pitch angle 9 i n these simulations because the submarine is always translating parallel to a horizontal seabed (cf. Fig. 5).
2.2. Botiiidary conditions
A t the submarine and seabed surfaces, no-slip and impermeable boundary condi-tions are used. The velocity is set to zero at the submarine surface while the seabed is given a tangential velocity equal i n magnitude to the free-stream velocity far f r o m the submarine. This is consistent with using a body-fixed coordinate system.
A cylindrical domain surrounds the submarine as shown i n F i g . 2. A symmetry boundary condition is used f o r the top surface. I n such cases, the velocity component normal to the boundary and the gradients of Uf,Vf and Wf normal to the boundary are set to zero. For simulations with zero-drift angle, flow is only simulated on the port side and the symmetry boundary condition is also used on the longitudinal plane of symmetry. When a drift angle is present, the symmetry condition no longer appUes at the longitudinal plane o f geometric symmetry and the f u l l cylindrical grid is used. The boundary condition on the curved exterior surface is set by calculating the angle between the inflow and the normal vector on a cell face on the boundary. For angles larger than 9 0 ° , the boundary condition o f the cell face is set to inflow, other-wise to outflow. A Neumann boundary condition is applied on outflow faces:
^ = 0, ^ = 0,
dn dn
where M j represents each component o f velocity, p is pressure and n is a vector normal to the boundary. The pressure level is set to zero at a point far above and i n front o f the submarine. The velocity components i n the inflow faces are set to the undisturbed velocity components, while the pressure is extrapolated f r o m the interior. The turbulent intensity I = ^ is set to 0.05 and the eddy viscosity p,t to 10 times the molecular viscosity p. The values appUed at the inflow plane are also used as initial condition i n the complete flow domain.
104 M. Bettle et al. / Calculation of bottom clearance effects
Max. Breadth = B = 8.4 m
Fig. 2. Grid layout.
The advantage of using a cylindrical domain i n combination w i t h an "auto-detect" boundary condition f o r i n f l o w and outflow is that the same grid can be used f o r various d r i f t angles. This, i n combination with an automatically changing drift angle, as discussed i n Section 2.4, results i n considerable time savings.
2.3. Solver set-tip
In this study, the flows were calculated with R E F R E S C O , see, e.g., [11]. A p p h -cation and validation cases can be found i n , e.g., [10] or [12]. The code is a spin-o f f spin-o f the F R E S C O cspin-ode which started as a j spin-o i n t develspin-opment by M A R I N , Ham-burgische Schiffbau-Versuchanstalt ( H S V A ) and Technische Universitat
Hamburg-Harburg ( T U H H ) w i t h i n the E U project V I R T U E i n 2005. R E F R E S C O solves the
multi-phase unsteady incompressible Reynolds-averaged Navier-Stokes equations complemented with turbulence models. The equations are discretised using a finite-volume approach w i t h cell-centred collocated variables. The implementation is face-based, which permits grids w i t h elements with an arbitrary number o f faces (hexa-hedrals, tetra(hexa-hedrals, prisms, pyramids, etc.). The code is parallehsed using Message Passing Interface ( M P I ) and subdomain decomposition. First order and second order spatial and temporal discretisation schemes are available i n the code. The equations are solved i n a segregated approach and the pressure-velocity coupling is solved using the S I M P L E algorithm. R E F R E S C O is intended to solve ship-building and
M. Settle et al. / Calculation of bottom cleanince effect.^ 105
offshore related flow problems, and is therefore optimised f o r and continuously ap-plied to these types of flows.
For all calculations, use was made of Menter's SST version of the two-equation k — u) turbulence model [ 8 ] . The boundary layer on the submarine and seabed is resolved down to t/"*" = 1 or less rather than using wall fimctions to estimate the boundary layer. The Reynolds number Re based on submarine length was 5.2 x 10^ f o r all computations. This is chosen to be consistent with the free-sailing experi-ments with variable bottom clearances, conducted at M A R I N . The Q U I C K scheme (Quadratic U p w i n d Interpolation f o r Convective Kinetics) was used to discretise the convection terms f o r the momentum equation. A l l simulations were run i n steady mode.
2.4. Sweeping ofthe drift angle
I n order to efficiently generate results for many drift angles, a routine was used to automatically increment the d r i f t angle during a single simulation. Simulations begin with the d r i f t angle set at —20° until the solution stagnates at 7000 iterations. Next the drift angle is incremented by 2 . 5 ° , by changing the inflow and seabed boundary conditions, and the solution is continued f r o m the converged solution f r o m the previ-ous drift angle. The d r i f t angle is held fixed f o r 2000 iterations before incrementing f3 again by 2 . 5 ° . For some cases, P was changed f r o m - 2 0 ° to 2 0 ° i n 2.5° incre-ments. From symmetry considerations, the results should theoreticaUy be symmetric about zero drift. This fact was used to provide an assessment o f the convergence error and hysteresis i n the simulations. It was found that the asymmetry i n the calculated results is small when 2000 iterations were used f o r each d r i f t angle and thus only negative drift angles were simulated in subsequent calculations w i t h the same proce-dure. Starting the calculations f r o m a converged solution at a sUghtly different drift angle saves times compared to performing each calculation separately. This routine was implemented and tested i n R E F R E S C O in previous work, see [12].
3. Grid generation 3.1. Hull form
Calculations were conducted using the unappended hull-deck-sail configuration of the Walrus submarine. The main particulars of the Walrus are specified i n Table 1. 3.2. Grid
A structured grid was generated around the submarine using the commercial mesh-ing software, I C E M CFD. A n overall view of the mesh is shown i n Fig. 2. A large domain was created i n order to ensure that the influence o f applying undisturbed
106 M. Bettle et al. / Calculation of bottom clearance effects
Table 1
Main particulars of the bare hull Walrus submarine
Description Symbol Magnitude Unit Length overall 67.725 m Length between perpendiculars Lpp 67.725 m Maximum hull radius .Rmax 4.200 m Centre of buoyancy (aft of nose) FB 29.884 m Volume of displacement V 3099.096 n?
Wetted surface 1688.947 m2
Fig. 3. Grid near the submarine (side and top view).
velocities at ttie opening and top surfaces on the solution is small. The influence o f the domain size has been studied previously by Ega et al. [5]. Based on that study, a domain radius of 4 • Lpp is found to be appropriate f o r the present work, taldng also the obUque flow conditions into account. A multi-block 0 - 0 type grid was created, as shown i n Fig. 3. Block locations were chosen such that the grid lines followed the submarine contours as closely as possible. Grid refinement was applied to regions o f high curvature on the deck and sail as shown i n Fig. 4. The grid is stretched towards the submarine and sea bottom surfaces i n order to achieve values o f less than 1 to resolve the boundary layers without using wall functions.
3.3. Adjusting tlie bottom cleaiwice
I n order to adjust the bottom clearance, the submarine boundary was held fixed while the location o f the bottom boundary was shifted. Also, when changing the pitch angle o f the submarine, the submarine boundary was held fixed while the outer cyUndrical boundary was rotated by the submarine pitch angle as illustrated i n Fig. 5. This procedure kept the solution i n the body-fixed coordinate system. When chang-ing the clearance at zero pitch, the mesh was unchanged except for the bottom block
M. Bettle etal. / Calculation of bottom clearance effects 107
Fig. 4. Surface grid around ttie sail.
z X v „ ( e = o ) ~ —-——^ J Ik V„(9) O II Sea Floor (6) u O Sea Floor (9 = 0)
Fig. 5. Definition of bottom clearance and angle of attack a.
that extends to the seabed. The distribution of cells i n the bottom blocks was adjusted with each clearance to give a maximum y + of around 1 on the seabed and a smooth transition to the grid blocks above i t . When changing the pitch angle, the mesh re-mained unchanged i n a box surrounding the submarine (Fig. 3) so that the near-body mesh was unaltered. The mesh outside this box was rotated w i t h the outer boundary so that it continued to conform to the exterior domain.
3.4. Siiininary of grids generated
A grid-refinement study was conducted at the largest clearance (c = 11.9S) with zero-pitch and drift angles. Five grids were generated f o r this study w i t h the designa-tions coarse (c), medium-coarse (mc), medium (m), medium-fine ( m f ) and fine ( f ) . Beginning with a coarse mesh having 470 thousand cells (half mesh), each sub-sequent grid was generated by increasing the number of cells i n each block edge by approximately \/2, resulting i n the total mesh cells increasing by approximately 2 each time. The spacing of cells was adjusted at every grid block intersection to maintain smooth transitions. From the grid dependence study, the medium (2 x 10^ cells) mesh was selected f o r studying the effect of bottom clearance. One o f the
108 M. Beltle et al. / Calculation of bottom clearance effects
meshes with c/B = 1 / 4 was also generated f r o m the coarse mesh to compare coarse and medium meshes w i t h a small bottom clearance. Some f u l l meshes were created f r o m the half meshes i n order to perform calculations w i t h non-zero d r i f t angles.
4. Review of the calculations
The computations conducted i n this study are summarised i n Table 2. The mesh used f o r each simulation is indicated i n the matrix. A l l twelve clearances ranging f r o m c/B = 1/8 to c/B = 12 were simulated for the zero-drift, zero-pitch case using the medium mesh whereas selected clearances were simulated f o r pitch t r i m angles at - 2 ° and 2 ° . Five calculations w i t h different grids were performed f o r the deepest clearance and zero-tlow incidence f o r the grid-refinement study. Computa-rions that swept f r o m - 2 0 ° to 0 ° d r i f t angle were performed f o r seven different clearances using the medium mesh. One d r i f t sweep was also performed w i t h the coarse mesh and a clearance o f c = Ö / 4 .
5. Numerical results
5.7. Verification and validation
Verification and vahdation studies are very important i n estimating the accuracy o f the computations and thus they are presented before discussing bottom clearance
Table 2 Calculation matrix No. c(m) c/B - 2 Zero drift {() = 0), a ( ° ) 0 2 Drift sweep ( a = 0) - 2 0 ° < / 3 s S 0 ° , A / 3 = 2.5° 1 1.05 0.125 ni 2 1.40 0.167 m 3 1.68 0.200 m ni ni m 4 2.10 0.250 m m ni c*, m .5 2.80 0.333 m m m m 6 4.20 0.500 m m m m 7 5.60 0.667 ni m 8 8.40 1.000 m m m m 9 11.20 1.333 m 10 16.80 2.000 m m m m 11 25.20 3.000 ni m m 12 100.00 11.905 m c, mc, m, mf, f m c * , m *
Notes: Mesh designations: c - coarse, mc - medium-coarse, m - medium, mf - medium-fine, f - fine.
M. Bettle et al. / Calculation of bottom clearance effects 109
effects. The uncertainty of a C F D prediction can be divided into two categories: numerical uncertainty and modelhng error. The numerical uncertainty relates to how well the governing equations are solved while the modelling error is a measure o f the differences between the mathematical representation of the physical problem and reahty.
The numerical uncertainty can be divided into three components: round-off error, iterative error and discretisation error [ 4 ] . The roundoff error is a result of the f i -nite precision o f the computer and is generally negligible compared to the other two components. The iterative error is a consequence of the non-linearity of the equa-tions solved i n the calculaequa-tions. I t is theoretically possible to reduce the iterative error to the same order of magnitude as the round-off error. However this is often excessively time consuming f o r complex turbulent flows, such as the ones simulated in this work. The discretisation error arises out of the approximations used i n the finite-volume scheme that transforms the partial differential R A N S equations to a set o f algebraic equations. I t is generally the largest source of numerical uncertainty. However, the discretisation error decreases with grid refinement and its magnitude can be estimated with a grid-refinement study. The iterative error and discretisation error are assessed in verification studies i n this work.
A primary source of modelling error i n CFD is the turbulence modelling. The accuracy of the CFD calculations performed i n this work is assessed by validating unrestricted-water calculations with experiments. I n the subsections that follow, esti-mates o f the numerical uncertainty o f the CFD solutions are presented first, followed by validation against experimental data.
5.2. Iterative error
A quaUtative measure o f the iterative error can be seen i n the convergence histories of the simuladons. The residuals presented i n this section are non-dimensionalised using the undisturbed inflow velocity ^ 0 0 - Both the LQO norm and L2 norm resid-uals are given f o r selected calculations. The
Z/Qo norm residuals are the maximum changes for a given variable f r o m iteration to iteration i n the whole domain. The L2 norm residual is an RMS value f o r the domain and is calculated as follows:
where A(f> is the normaUsed change i n a given variable and rip is the number of grid cells. As a general guideline, the iterative errors are generally assumed to become negUgible compared to the discretisation errors when the Loo residuals drop below around 10"'* or 1 0 " ^
Figure 6 shows the convergence histories o f a selected calculation w i t h a d r i f t angle of zero.
110 M. Bertie et al. / Calculation of bottom clearance effects
The maximum of the L2 and Loo norms f o r u, v, w and p at the end of each calculation are tabulated i n Table 3. Iterative error estimates f o r X', Z' and M' are also listed. These error estimates are calculated by evaluating the percent difference between the integral quantities at a given iteration, <pi, with the value at the end o f the calculation, (^end- The values listed are the maximum of the differences observed over the last 200 iterations of a calculation.
Excellent convergence was achieved f o r the grid-refinement study, w i t h Loo and L2 residuals dropping below 10~^ and 10~^, respectively, i n all cases. For all grids, the integrated forces and moments were unchanging to 7 significant digits (the pre-cision w i t h which this data was written to the results file) f o r the last several hundred iterations. As a result, the estimation o f discretisation error presented i n the next sec-tion was performed i n the absence of any significant iterative error. Excellent con-vergence was also obtained f o r the zero-drift calculations with ,0 = 0 ° and a = 2 ° . For all clearances, the Loo residuals converged to 10~^ or lower and the integrated forces and moments converged to 7 significant digits.
The only calculations f o r which poor convergence was observed were w i t h a = —2° (trim by the bow) f o r intermediate bottom clearances. A n analysis of the resid-uals showed that the regions of highest residresid-uals i n the poorly converged simula-tions were near the bottom boundary. I t is unknown why the residuals were high i n some simulations with a = —2° and not others and this should be further studied. However, the well-converged results f o r bow t r i m at c = 11.95 and at the small clearances make it possible to draw some conclusions about the effect of a.
Figure 7 shows a convergence plot f o r one o f the calculations with variable d r i f t angle. The first 7000 iterations apply to a d r i f t angle of - 2 0 ° , after which the drift angle was incremented every 2000 iterations by 2 . 5 ° . Jumps i n the residuals are observed every time the d r i f t angle is changed. For large values of P, the Loo and L2 residuals stagnate at around 0.1 and 10"'*, respectively. The large difference between Loo and L2 residuals indicate that a few cells have high residuals. Examination of the
M. Beltle et al. / Calculation of bottom clearance effects 111
Table 3 Iterative convergence
c c/B ricells "iter ^2 residuals Loo residuals Fluctuations, last 200 iters (m) (m) xlO"-^ m a x ( u , t l , p ) max(M, t),iu,p) 100 x max(|</>i — (/ijmjl/i/ijnd)
X' Z' M' Grid study, a = 0 ° , / 3 = 0° 100 11.90 470 17,030 6.2 X 1 0 - " 1.4 X 10-11 0 0 0 100 11.90 954 15,000 6.3 X 10-^ 3.1 X 10-^ 0 0 0 100 11.90 1909 7375 1.0 X 1 0 ^ " 1.1 X 10-5 0 0 0 100 11.90 3761 8160 1.3 X 10-^ 7.6 X l O - ' ' 0 0 0 100 11.90 7509 15,000 2.8 X 10-5 4.7 X 10"^ 0 0 0 Clearance study. a = 0 °,/3 = 0 ° 1.05 0.13 1994 6000 2.1 X 10-* 1.6 X 10-5 0 0 0 1.4 0.17 1994 6000 6.6 X 10-12 4.6 X 1 0 - 1 ° 0 0 0 1.68 0.20 1994 5738 4.0 X 10-11 1.4 X 10-5 0 0 0 2.1 0.25 1994 13,721 2.5 X 10-* 1.4 X 10-5 0 0 0 2.8 0.33 2031 10,579 7.7 X l O - l ' l 4.2 X 10-12 0 0 0 4.2 0.50 2007 10,665 1.1 X 10-13 1.1 X 10-12 0 0 0 5.6 0.67 1994 6000 6.2 X 10~11 2.5 X 10-5 0 0 0 8.4 1.00 2105 8139 6.3 X 10-13 2.8 X 10-11 0 0 0 11.2 1.33 2093 6000 5.0 X 10-11 2.2 X 10-5 0 0 0 16.8 2.00 2399 6002 4.7 X 10-11 1.7 X 10-5 0 0 0 25.2 3.00 2240 6000 4.9 X 10-11 2.0 X 10-5 0 0 0 Clearance study. a = 2 ",(3 = 0° 1.68 0.20 1994 8000 1.4 X 10-* 8.2 X IO-*" 0 0 0 2.1 0.25 1994 8000 1.4 X 10-* 8.4 X 10-^ 0 0 0 2.8 0.33 1994 5398 1.5 X 10-* 9.6 X IO-** 0 0 0 4.2 0.50 1994 7449 1.0 X 10-* 6.1 X 10-Ö 0 0 0 5.6 0.67 2056 8000 1.4 X 10-* 8.6 X 10-"= 0 0 0 8.4 1.00 2056 7463 1.5 X 10-5 9.9 X 10-7 0 0 0 16.8 2.00 2093 8000 1.5 X 10-11 6.3 X 10-10 0 0 0 25.2 3.00 2240 6008 5.7 X 10-11 2.3 X 10-5 0 0 0 100 11.90 2301 8000 7.8 X 10-12 5.1 X 1 0 - 1 ° 0 0 0 Clearance study. a = — 2 ° , ,3 = 0° 1.68 0.20 1994 8000 6.2 X 10-0 5.1 X 10-"! 0 O.OI 0.01 2.1 0.25 1994 7088 2.4 X 10-* 1.2 X 10-5 0 0 0 2.8 0.33 1994 5077 4.0 X IO-"! 4.3 X 10-3 0.11 0.43 0.16 4.2 0.50 1994 7576 7.5 X IO-"! 4.2 X 10-2 1.03 10.52 2.78 8.4 1.00 2056 6477 7.2 X IO-"* 1.0 X 10-2 0.84 14.46 2.99 16.8 2.00 2093 4865 2.1 X l O - ' l 2.9 X 10-3 0.28 7.88 0.83 25.2 3.00 2240 8000 5.7 X IO-"! 1.5 X 10-2 0.67 41.51 7.16 100 11.90 2301 4525 2.9 X 1 0 - * 1.1 X 10-5 0 0 0
112 M. Bettle et al. / Calculation of bottom clearance effects
residuals showed that the high residuals are indeed localised to small regions on the sail and deck. The effect on the integral quantities is small: the forces and moments are fluctuating by less than 0.01% at /? = - 2 0 ° . This convergence is similar for all the clearances.
A second estimate of the combined round-off and iterative error can be obtained f r o m symmetry considerations about (3 = Q°. For example, the maximum difference between the integral results f o r /? = - 2 0 ° and /3 = 2 0 ° was found to be 1.4% f o r X' which is very acceptable f o r the purpose of this study. The difference for the other forces or moments was 0.5% or less.
5.3. Discretisation error
I t was possible to analyse the discretisation errors i n the absence of any signifi-cant iterative errors because o f the excellent convergence obtained for the grid study calculations. I n the present work, the uncertainty analysis proposed by E j a , Vaz and Hoekstra [6] was used. This method is based on the least squares version of the Grid Convergence Index (GCI) method (see [9]). This method uses Richai^dson extrapo-lation to estimate the value o f a specific quantity f o r an infinitely dense grid. This extrapolated value and the apparent order of convergence, p, are found by applying a least-squares method using data f r o m grids w i t h different levels o f refinement. Er-ror estimates are evaluated using values f r o m the least-squares method with safety margins applied according to experience. The f u l l details of the method can be found i n [6].
The results of the discretisation error analysis are presented i n Fig. 8. The relative step size h shown on the horizontal axis indicates the coai'seness of the grid w i t h respect to the finest grid. I t is defined as / i = ^/N^JNI, where A'^i is the number o f points i n the finest grid and A^, is the number of points i n grid i. Thus a relative step size o f one is f o r the fine grid and the larger the value, the coarser the grid. A relative
M. Bettle et al. / Calculation of bottom clearance effects 113 0.5 p=2.0 : I M . 0 % -> I 1.5 2 relative step size
2.5 U=1.7% • 0 . : -0 -0 . ^ O O O 0.5 1 1.5 2 2.5 3 relative step size
-0.00019 -0.0002 •0.00021 -0.00022 -0.00023 -0.00024 -0.00025 -0.00026 -0.00027 p=7.6 0 05 1 1.5 2 relative step siai
25
Fig. 8. Discretisation error analysis, a = 0 ° , /3 = 0 ° , c = 11.9B.
Step size o f 0 is tlie l i m i t o f infinite refinement, at which point the discretisation error is theoretically zero. For this analysis, the results f o r the coarsest grid were taken as an outlier and the least-squares method was applied to the four finest grids only.
The results f o r the viscous component o f axial force, X^, f o l l o w the best fit curve closely. The apparent order o f convergence p was found to be 2, which agrees with the expected order o f accuracy f o r R E F R E S C O when using the second order Q U I C K discretisation scheme f o r convective fluxes. The uncertainty U for Xf was estimated to be 1.0%.
The pressure component o f axial force, X'^, was found to decrease w i t h grid re-finement as expected. However, the scatter i n X!p is more noticeable than f o r X'^. The estimated order o f convergence f r o m the least squares regression was 7.6. This is much higher than the expected value o f 2 and may be a result o f scatter or the use o f too coarse grids. Since the obtained order o f convergence was not realistic, an adopted order o f convergence p* o f 2 and a safety factor Fg o f 3 were used i n the estimation o f the uncertainty, according to the verification procedure. The estimated uncertainty f o r X'^ was found to be 11.2%. Note that the magnitude of Xp is approx-imately l / 6 t h that o f X'^ and thus the contribution o f this uncertainty to the overall axial force X' is small.
I n the analysis o f total axial force X', a curve f o r monotonic convergence was not found by the least-squares analysis. This could be a result o f scatter i n the data, in combination with only four data points being used i n the analysis. However, the difference i n predicted axial force changes by only 0.5% f o r grids ranging f r o m
114 M. Bettle etal. /Calculation of bottom clearance effects
0.95 X 10^ cells to 7.5 x 10^ cells. The conservative error bounds indicate an uncer-tainty o f only 1.7% i n X'.
A comparison of results f o r different drift angles w i t h the coarsest grid and the medium grid with a bottom clearance of c = 0.25B is shown i n Fig. 9. The differ-ences between the forces and moments i n other directions are smaller than f o r Z' or M' and judged to be negligible. For Z' and M' i t can be seen that the difference between the coarsest grid and the medium grid results is small f o r the f u l l range o f d r i f t angles simulated i n this work. The main discrepancies were seen i n Z' at /3 > 17.5° and i n M' at /9 > 7 . 5 ° . Note that M' is numerically small i n compar-ison w i t h the other quantities. As seen i n Fig. 8, the integral quantities change less between the medium and fine grids than between the coarse and medium grids. This suggests that using the finest grid would change the forces and moments by less than the discrepancies between the coarse and medium mesh seen i n Fig. 9.
From this analysis, i t was judged that the discretisation errors are reasonably small for the grid with medium refinement (2 x 10^ cells half mesh, relative stepsize h
approximately 1.55) and thus it was used f o r the study of bottom clearance effects discussed further i n this article.
5.4. Comparison of unrestricted-water calculations with experiments
It was established i n the previous two subsections that the iterative and discretisa-tion errors are reasonably small for the purpose o f this work (with the excepdiscretisa-tion of a few simulations at a = - 2 ° ) . I t is still important to validate the results to ensure that the numerical modelling of the flow is a good approximation of reality. This is done by comparing unrestricted-water calculations at various d r i f t angles with experimen-tal data.
M. Settle et al. / Calculation of bottom clearance effects 115
Table 4
Designations and descriptions of experimental configurations [3] No. Hull with deck Bridge fairwater (sail),
sailplanes, sonar dome
X-tail, propeller
1
+
+
+
2
+
+
3
+
The experiments f o r comparison were conducted by the David W. Taylor Naval Ship Research and Development Center (DTNSRDC) [3]. Experimental results were obtained f o r the three configurations o f the early Walrus design listed i n Table 4.
It should be noted that the design o f the Walrus as used f o r the D T N S R D C model tests differs slightly f r o m the real Walrus class submarine design, which was used f o r the present study. The largest differences compared to the real design are a sUghtly smaller length (0.9%Loa) and the absence of the Toekan (exhaust diffuser). I t is expected that this discrepancy w i l l have only a small effect on the overall forces and moments.
The calculations were performed at Re = 5.2 x 10^ (to resemble the condition of the free-sailing experiments) whereas the D T N S R D C experiments were conducted at two higher Reynolds numbers: 9 x 10^ and 14 x 10^. The main effect of Reynolds number i n this range is to reduce the viscous drag. I n order to better compare the results, the axial force evaluated i n the calculations at zero-drift angle was scaled to 7?e = 14 X 10^ using the I T T C 1957 friction line:
^ / ^ , _^ ,w g / , / 7 T c ( ^ g = 1 4 x l 0 6 ) ^Re=l4xW^ - ^ P + ^ / , R . = 5 . 2 X 1 0 6 ' C f j j j c ( R e = 5.2 X 10^)
w i t h
^ f J T r c ( R e ) = j j ~ ^ ^ ^ .
This was done f o r the overall force on the hull and sail as well as on the hull sur-face alone. Tables 5 and 6 compare the scaled CFD results with experimental values (Re = 14 X 10^) for the hull-sail (configuration 2) and h u l l only (configuration 3) configurations, respectively. The computations predict the resistance on the hull to within 3% of the experimental data. This is expected to be within experimental er-ror bounds based on the scatter i n the experimental data. The total resistance was predicted to be 9% lower than configuration 2. I t was expected that the calculated resistance would be lower than the experimental value because the computations do not account f o r the drag on the sailplanes, which is present i n experimental results f o r configuration 2.
The unrestricted-water calculations are also compared with configuration 2 ex-periments for a range of drift angles i n Fig. 10. The agreement is considered to be
116 M. Bettle et al. / Calculation of bottom clearance effects Table 5
Grid-refinement study: Total forces and moments ( x lO^)
Cells X' X' Form factor X\Re = UM) X' exp. % diff. 1 +k config. 2 i n X ' 470 -0.266 -1.275 -1.546 1.209 -1.295 8.0 954 -0.236 -1.282 -1.523 1.184 -1.275 9.4 1909 -0.220 -1.297 -1.521 1.170 -1.274 -1.407 9.4 3761 -0.221 -1.302 -1.527 1.170 -1.279 9.1 7509 -0.216 -1.309 -1.530 1.165 -1.281 8.9
Notes: Cf jjj-ciRe = 14 X 10^) = 2.832 x 10"^ (non-dimensional with wetted surface).
Cƒ ijTcil^e = 5.2 x 10^) = 3.372 X 10"^ (non-dimensional with wetted surface).
Table 6
Grid-refinement study: Hull forces and moments ( x 10^) Cells X!p X' Form factor
1 + fc
X'iRe = UM) X' exp.
config. 3 % diff. inX' 470 -0.214 -1.101 -1.314 1.194 -1.104 - 2 . 4 954 -0.199 -1.107 -1.306 1.179 -1.097 -1.8 1909 -0.190 -1.120 -1.311 1.170 -1.101 -1.078 - 2 . 2 3761 -0.187 -1.125 -1.312 1.166 -1.102 - 2 . 2 7509 -0.189 -1.131 -1.320 1.167 -1.109 -2.9
Notes: Cfjxrc^Re = 14 x 10^) = 2.832 X 10"^ (non-dimensional with wetted surface). Cf jjjciRe = 5.2 X 10^) = 3.372 x lO^^ (non-dimensional with wetted surface).
very good f o r all components of forces and moments, w i t h the exception of the pitch-ing moment M'. The pitchpitch-ing moment i n the calculations follows the same trend as the experiments but is shifted down to smaller values. I t is possible that this dis-crepancy is a result of the sailplanes being present i n the experiments but not i n the calculations. The drag and l i f t on the sailplanes w o u l d tend to increase M ' , consistent w i t h the shift observed i n F i g . 10. I t also appears f r o m the scatter i n the data and the differences between positive and negative d r i f t angles that there is more uncertainty in M' (relative to the scale used f o r the plot) than for the other integral quantities.
The comparison w i t h experiments showed that CFD gives accurate predictions for the forces and moments on the submarine for the case o f um-estricted water over a range o f d r i f t angles. No experimental data w i t h the Walrus at different bottom clear-ances are available to compare against at this time. However, the good agreement w i t h a large bottom clearance w i t h the experiments and the results of the mesh sen-sitivity studies at both a large and a small bottom clearance (see Figs 8 and 9) give confidence that the calculations w i l l also provide reahstic results w i t h the various bottom clearances analysed i n this work. Furthermore, i t w i l l be shown below that the results agree w i t h trends found i n other publications.
118 M. Bettle etal. / Calculation of bottom clearance effects
5.5. Effect of bottom cleamiice and pitch angle for zero drift
Figure 11 gives an impression of tiie results obtained with a small bottom clear-ance (c = 0.25S) at zero-drift angle. I t can be seen that a low pressure region devel-ops below the submarine. This is a result of the flow accelerating i n the small channel between the submarine and seabed.
The low pressure region below the submarine generates a down force and affects the pitching moment and axial force on the submarine. These effects are shown i n Fig. 12. This figure shows that the changes i n forces and pitching moment occur gradually as the bottom clearance is decreased f r o m large clearances (c = 11.9B) to around c = B. However, a further reduction i n clearance results i n rapid changes. In other words, the influence increases f o r increasing B/c and therefore the data has been plotted against B / c i n the right-hand side o f t h e figure.
The largest effect is an increase i n down force w i t h decreasing bottom clearance. For both positive and negative pitch angles the down force increases non-linearly with increasing B/c. I t begins with a nearly quadratic relationship at small B/c values (large clearances). This is consistent w i t h a previous study [13] that looked at the effect of bottom clearance on down force f o r S / c ranging up to 1. However, it was found i n this w o r k that the quadratic relationship transforms into more of an exponential decay relationship at smaller clearances.
To provide a reference f o r the magnitude of the down force, the upward force that can be achieved with a 2 0 ° sailplane deflection is also shown i n Fig. 12. I t can be seen that at small clearances and negative pitch angles, the down force approaches the force of a 2 0 ° sternplane deflection. A t a pitch angle of - 2 ° and a clearance of B / 5 , the downforce is at 75% the magnitude that can be offset with sailplane deflection.
M. Bettle et al. / Calculation of bottom clearance effects 119
A further reduction i n clearance and/or shift to more negative pitch angles could result i n a condition where sailplane control alone is not sufficient for preventing the submarine f r o m sinking to, and colhding with, the seabed.
120 M. Bettle et al. / Calculation of bottom clearance effects
The effect o f bottom clearance on pitching moment also has important implica-tions on manoeuvring. I t was found that a small negative (bow-down) pitching mo-ment develops as the bottom clearance is reduced with zero angle of attack. The magnitude of the pitching moment is small for a = 0 ° even at the smallest clear-ance, indicating the point o f apphcation of the down force is close to mid-ship. I n unrestricted waters (approximated by c = 1 1 . 9 B ) , it was found that a negative pitch-ing moment develops f o r negative pitch angles while a positive pitchpitch-ing moment results for a positive pitch angle, as expected. The rate of change i n pitching moment w i t h a, dM/da, is such that it destabilises the submarine i n the vertical plane of motion. This is because the pitching moment tends to increase the magnitude of the pitch angle whether i t is positive or negative. The results show that as the clearance is reduced, the pitching moment curves diverge f o r the different angles of attack. This increases dM/da and results i n lower stability i n the vertical plane. The explanation for this effect is as follows: as the submarine is rotated towards negative pitch angles, the flow channel between the nose and seabed becomes smaller, resulting i n lower pressures below the nose. Also, the channel between the tail and the seabed becomes larger, resulting i n higher pressure on the bottom of the tail. This results i n a larger rate of change towai-ds more negative pitching moments than f o r unrestricted waters. When the submarine is rotated i n the positive direction the opposite occurs - the larger gap below the nose increases the pressure there while the pressure decreases under the tail due to the narrowing gap below it. This shifts M' towards more posi-tive values at a faster rate than i n unrestricted water. I t is anticipated that the reduced stability i n pitch would require more frequent adjustments of the control surfaces to maintain straight and level flight when underway w i t h a small bottom clearance. For the purposes of manoeuvring simulations, the change i n M' w i t h B/c is modelled using the hnear functions shown i n Fig. 12.
The change i n axial force showed a complex dependence on bottom clearance. As the bottom clearance was reduced, X' increased i n magnitude f o r a = 0 ° and a = 2° but decreases i n magnitude slightly for a = - 2 ° . This finding is i n agreement w i t h previous studies with a different submarine model, see [ 2 ] .
5.6. Effect of bottom clearance and drift angle for zero-pitch angle
The effect o f bottom clearance and d r i f t angle on the hydrodynamic forces and moments f o r a = 0 ° is presented in F i g . 13. The sohd curves were generated using a hydrodynamic model derived f r o m the data. The f o l l o w i n g trends are observed:
1. The axial force increases i n magnitude slightly as the clearance is reduced f o r
\I3\ < 5 ° . A t high d r i f t angles, the thrust i n the axial direction is magnified as the clearance is reduced.
2. The lateral force Y', rolling moment K' and yawing moment N' that develop with a d r i f t angle are all magnified as the bottom clearance is reduced. These changes w i l l have an effect on the stability and minimum turning radius of the submarine. Manoeuvring simulations are needed to quantify this effect.
M. Bettle et al./ Calculation of bottom clearance effects 121
o c/B = 11.9
P(deg) p(deg)
Fig. 13. Forces and moments as a function of j3 and c(a = 0 ° ) . Solid curves were generated from the hydrodynamic model.
3. A significant down force on tiie submarine is generated when a d r i f t angle is present, even w i t h large bottom clearances. This out-of-plane force is discussed
122 M. Bettle et al. / Calculation of bottom clearance effects
in detail by Bridges [1]. I t is a result of the vorticity generated at the sail when at a d r i f t angle. Ckculation is generated around the hull downstream of the sail. This circulation interacts with the crossflow to generate a hydrodynamic pres-sure difference between the deck and keel - the Magnus effect. The prespres-sure at the deck increases relative to the pressure at the keel, resulting in a down force. Since the pressure difference due to the Magnus effect only occurs aft of the sail, the location of the resultant down force is aft of the midship, giving a positive pitching moment. Both the down force and pitching moment vary with angle of d r i f t squared f o r moderate d r i f t angles [ 1 ] . When the bottom clearance is reduced, the down force due to the Magnus effect is shifted to even larger values, at least f o r d r i f t angles ranging up to 15°. The combination of small bottom clearances and moderate drift angles that occur when the submarine is turning can result i n down forces that exceed the upward force due to a 20° sailplane deflection. It is noted that f o r large drift angles near 2 0 ° , decreasing the clearance below B/3 resulted i n a decrease of the magnitude of the down force. Although this is qualitatively consistent w i t h results found during model tests by Mackay [7], i t is uncertain why this change i n trend occurs and further analysis of the flow field is needed.
4. The pitching moment was consistently shifted towards more negative values as the bottom clearance was reduced. This shift becomes larger as the drift angle increases. I t is expected that the rapid change i n M w i t h c at high drift angles would add to the difficulties i n controlhng pitch when performing a tum w i t h a small bottom clearance. Manoeuvring simulations are needed to investigate the significance of this effect. Note that this effect (and other observed trends with 13) applies to a = 0 ° . W i t h a = 2 ° , it was found that the pitching moments shift slightly towards more positive values at zero drift. M o r e data is needed to see the combined effect o f a and /? when f3 is not small.
5.7. Hydrodynamic model incorporating bottom clearance effects
A coefficient-based hydrodynamic model incorporating bottom clearance effects was created f r o m the data i n this work. The results of the model were evaluated and plotted along w i t h the data i n Figs 12 and 13. The coefficients f o r the model were determined using least-squares fits to the data. I t can be seen that there is good agree-ment between the model and the data over the f u l l range o f d r i f t angles investigated.
This model can be used f o r manoeuvring simulations, however the range of ap-plicabihty should be emphasised. It is based on clearances f r o m c = 11.9B (ap-proximating unrestricted water) to c = 0.20B and f o r d r i f t angles i n the range - 2 0 ° ^ /? < 2 0 ° . Smaller clearances were investigated, but only at zero drift. The effect of angle of attack is also included, based on - 2 ° < a < 2 ° , but i t only apphes at zero drift.
M. Bettle et al. / Calculation of bottom clearance effecLt 123
6. Conclusions
A n extensive number of viscous-flow simulations f o r the Walrus-class subma-rine has been conducted. The calculations comprised a broad range o f inflow angles and clearances to the sea bed. Numerical uncertainties were quantified by studying the iterative and discretisation errors. The unrestricted-water calculations showed very good agreement with experimentally-determined hydrodynamic forces and mo-ments. This demonstrated that the viscous-flow solver R E F R E S C O is capable of generating accurate predictions of hydrodynamic coefficients f o r the purpose of ma-noeuvring simulations.
A t a drift angle of zero, the primary effect of reducing the bottom clearance is the development of a down force on the submarine. This down force changes rapidly as the bottom clearance is reduced below one submarine breadth. A t small clearances, the magnitude of the down force becomes a high percentage o f the upward force that can be achieved with a 2 0 ° sailplane deflection, particularly w i t h a negative pitch angle (bow down). A t a bottom clearance of 1/5 of the breadth o f t h e submarine and an angle of attack of - 2 ° , the down force was calculated to be around 75% o f the upward force resulting f r o m a 2 0 ° sailplane deflection. A second important effect observed at zero drift was an increase i n the destabihsing rate o f change dM/da as the bottom clearance was reduced. As a result, i t is expected that it becomes more difficult to control the pitch angle at small bottom clearances.
Calculations performed f o r d r i f t angles ranging up to 2 0 ° showed that the down force on the submarine increases with drift angle f o r unrestricted water, i n agree-ment with experiagree-ment. This out-of-plane force is magnified as the bottom clearance is reduced, for d r i f t angles up to 15°. A t higher drift angles, the relationship between down force and bottom clearance becomes complex, with the down force decreasing in magnitude at the smallest clearances. The combination o f small clearances and moderate-to-high d r i f t angles experienced i n a tight turn can result i n down forces significantiy larger than a 2 0 ° sailplane deflection. The pitching moment was found to shift i n the negative direction as the bottom clearance is reduced w i t h an angle o f attack of 0 ° . This effect became much more pronounced as the d r i f t angle was i n -creased. The lateral force, rolling moment and yawing moment increase i n magnitude over the f u l l range o f d r i f t angles investigated as the bottom clearance is reduced.
A coefficient-based model incorporating bottom clearance effects was derived and comparison w i t h the C F D data shows that i t gives a very good approximation to all force and moment data over the f u l l range of drift angles analysed i n this work ( - 2 0 ° < /5 ^ 2 0 ° ) . I t is intended to implement the model i n the simulation program S A M S O N to quantify the effects o f bottom clearance on submarine manoeuvring and to investigate the implications of the findings on the manoeuvrabiUty and safe operating limits f o r the submarine when operating near the seabed.
This is a first computational investigation on the effect o f bottom clearance on the Walrus submarine hydrodynamics. Several recommendations are suggested for further study:
124 M. Bettle etal. /Calculation of bottom clearance effects
• This study dealt with steady translation with different angles of d r i f t and angles of attack. The effect o f bottom clearance on rotational hydrodynamic coeffi-cients is needed to complete the hydrodynamic model f o r more accurate simu-lations.
• Similarly, the influence of drift angle combined with various trim angles should be investigated to expand the range of applicability of the model.
• The effect o f bottom clearance on the tailplane and sailplane forces is not i n -cluded i n this study. I t is suggested that a future study investigates the influence of the bottom clearance on the tailplanes i n particular due to their close prox-imity to the bottom.
• For more extreme manoeuvring conditions, unsteady effects might play a role. Therefore, investigations of unsteady effects on the forces and moments on the ship are recommended.
• To further demonstrate the applicability of CFD, calculations f o r near-bottom conditions should be validated w i t h experimental results.
Acknowledgements
The work presented here was funded through T N O . Defence, Security and Safety within the framework o f Programma V705 carried out f o r D M 0 o f R N L N . Their support is greatly acknowledged.
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Calculation of bottom clearance effects on Walrus submarine hydrodynamics Journal Publisher ISSN Subject Issue Pages DOI Pages Subject Group Online Date
IntemaUonal Shipbuilding Progress lOS Press
0020-868X (Print) 1566-2829 (Online)
Enaineering and Technoloov and Civil and Stmctural Engineering
Volume 57. Number 3-4 / 2010 101-125
10.3233/ISP-2010-0065 101-125
Engineering and Technologv Monday, January 24, 2011
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Mark Bettle', Serge L. Toxopeus^, Andrew Gerber' 'University of New Brunswick, Fredericton, NB, Canada
^Maritime Research Institute Netheriands/Delft University of Technology, Wageningen, The Netheriands
Abstract
Due to changes in operalions, the Royal Netherlands Navy (RNLN) is operating its submarines increasingly in brown water regions (I.e., waters with restriction in width and/or depth, such as littoral areas). To improve predictions for manoeuvring in restricted waterways, the RNLN has ordered the Maritime Research Institute Netheriands (MARIN) to conduct studies regarding the influence of the seabed on the behaviour of the submarine. In this work, viscous-flow calculations are used to predict the influence of bottom clearance on the hydrodynamic forces on the Walrus class submarines.
The simulations representing unrestricted-water conditions have been validated with available model test results and good agreement was found. The details of the methodology and results of the viscous-flow calculations are presented in this article. The bottom clearance effects on hydrodynamic forces and moments are summarised.
A strong non-linear influence of the bottom clearance on the vertical force and pitch moment was found. With a trim by the bow, the vertical down force can increase up to the same order of magnitude as the vertical up force that can be generated by the sail planes.
Keywords
Walnjs class submarine, CFD, RANS, manoeuvring, bottom effects, shallow water
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