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An inverse design approach for minimising wake at propeller plane using CFD
Ocean Engineering 33 (2006) 119–136 www.elsevier.com/locate/oceaneng
An inverse design approach for minimising wake at propeller plane using CFD V. Anantha Subramanian*, R. Vijayakumar Department of Ocean Engineering, Indian Institute of Technology, Chennai 600 036, India Received 15 November 2004; accepted 27 April 2005 Available online 12 July 2005
Abstract Knowledge of wake characteristics in the stern region is important for ensuring good propeller design and performance. This work examines the utility of CFD in the analysis of flow in the case of full aft beam vessels having characteristic cut stern shape to facilitate propeller aperture. The underwater stern shape may be more complex due to the occurrence of stern appendages such as bossings, strut supports and local shape variations. To this extent, CFD offers an effective tool for both qualitative as well as quantitative assessment of the local geometry. Wake estimate is required for choice of the most favorable propeller geometry. In the present method, the analysis quantifies the effects of small changes in stern rake angles and offers an inverse design approach towards finalising the stern shape. The method consists of solving the standard k–3 turbulent model of RANS equations in cell centered finite volume multi zone grid in the flow domain. This approach has been used in estimating the velocity at the propeller plane. The results have been compared with experimentally obtained values of nominal wake. The approach suggests that CFD can provide a cost effective and quick assessment of flow. It is also an attractive means of pre-empting heterogeneous flow related problems such as vibration and noise due to unfavorable wake in the stern region. q 2005 Elsevier Ltd. All rights reserved. Keywords: Wake fraction; Computational model; Inverse design; Surface modelling
1. Introduction The loss of inflow to the propeller, commonly known as wake, is an important propeller-hull interaction factor for consideration in obtaining effective propeller thrust
* Corresponding author. E-mail address: [email protected] (V.A. Subramanian).
0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.04.014
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power. For optimum propeller performance and minimised flow induced vibration, it is essential to have smooth homogeneous inflow to the propeller. In the case of normal ship shapes, the wake can be estimated fairly accurately at an early design stage on the basis of data from past experience. In the case of broad beamed vessels having their form more in the nature of full beam self propelled barges, the propeller aperture is created by providing a cut stern with an appropriate rake angle. The recent case study of newly built prototypes, Subramanian (2002), have shown that inappropriate stern design can result in potentially serious problem of propeller performance, with attendant noise and vibration mainly due to inadequate as well as highly heterogeneous inflow at the propeller plane. Heterogeneous inflow can also be made worse due to the unfavorable location of appendages such as bossing and struts. Since the consequences of the detailed design layout of the above components at the aft end cannot be exactly quantified otherwise at an early design stage, CFD offers a potential tool for quick estimation of the inflow characteristics. Wake distribution is either measured by laser-Doppler velocimetry or computed by CFD. Current wake predictions using CFD are not yet at a mature stage, whereas the integral of the wake over the propeller plane is well predicted, Bertram (2000). The inverse design approach has been adopted for obtaining more optimal hull shaping Daniele et al. (2001), and Chen and Cheng (2002). The motivation for this work is therefore to investigate a numerical cum experimental approach towards better estimation of the wake phenomena, especially in the presence of the complex aft geometry. As mentioned above, it is known from real field experiences that inappropriate design and location of stern appendages can result in manifestation of propeller flow induced vibration while running at modest engine speeds. This can become severe vibration at higher engine speeds near the rated rotational propeller speed. This study has been undertaken to understand the flow dynamics by combining numerical flow modelling with experimental wake determination. The modelling results have been reaffirmed by observing the fluid appendage interaction. For this purpose, dye injection method has been used and underwater video obtained to characterize the flow in the region of the propeller. The results of the investigation indicate that the numerical method can be successfully used in an inverse design approach.
2. Methodology The flow field in the propeller region, around the hull is simulated employing commercial CFD code (FLUENT 6.0.20) and the obtained flow velocity compared with experimental studies. The CFD simulation is based on Reynolds Averaged Navier Stokes (RANS) equations solver using a finite-volume method. The flow field domain (solution domain) is subdivided into finite number of control volumes (CV), using the modeler Gambit 2.0. The conservation equations, the general form of which is given in Eq. (1), are then applied to each control volume. In actual implementation this integral equation is converted into linearized algebraic difference form [Eq. (2)] for the discrete dependent variable and applied to the computational node positioned at the centroid of each control volume.
V.A. Subramanian, R. Vijayakumar / Ocean Engineering 33 (2006) 119–136
ððð
v vt
ðð Rate of increase of f
C
ðð convection term
A A rfVd
QðrfÞdQ
Z
121
ððð diffusion term
C
A Gf VfdA
source term Q Sf dQ
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} (1) Where f is any variable and can take a value of 1 to give the continuity equation u,v,w to give the momentum equation in three direction h to give the energy equation The general conservation equation in difference form can be written as N N faces faces X X ðrfp ÞtCDt K ðrfp Þt DQ C rf nf ff A f Z Gf ðVfÞn A f C SfQ Dt f f
(2)
Interpolation is used to express variable values at the control volume (CV) surface in terms of CV-centre node values. The standard k–3 model which is a semi-empirical model based on equations for turbulent kinetic energy (k) and its dissipation rate (3), is then used. The transport equations are vk v nt vk 0 0 vU Ui Z Kui uj K3 C (3) vxi vxj vxi sk vxi Z KC31 k3 ui0 uj0 vU h i 2 v3 vxj K C32 3k C vxv i snt3 vx i Ui
v3 vxi
(4)
where CmZ0.09, skZ1.0, Cx1Z1.44, Cx2Z1.92 sk Zturbulent Prandtl number for k(1.0) s3 Zturbulent Prandtl number for 3(1.3) These are the constants for the standard k–3 model. The eddy or turbulent viscosity, nt is computed by combining k and 3 as follows nt Z rCm
k2 3
(5)
where Cm Zconstant (0.09). The zone of interest around the hull, i.e. the flow field domain has initially been taken as one ship length at the front of the ship, 5 ship lengths behind the ship and 5 times the breadth and depth along the beam and depth directions, respectively. On successive analysis with reduced domain size, an optimal size was arrived at. The final domain size
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Fig. 1. Different zones of fluid domain.
was taken with two ship lengths behind and three times the breadth and depth along the respective directions. This has effectively reduced the computation domain by 80%. The reduced domain size allowed better flexibility and finer meshing size as required. The flow field domain is discretized into finite control volumes as shown in Fig. 1. Multi zone unstructured grid has been used for domain discretization, for this purpose, the complete flow domain is discretized into zones and these are further discretized in turn with tetrahedral cells. Using graded mesh sizes, a total number of finite volume cells (approximately 2.5 million) have been generated with minimum edge size of 0.37 cm. The hull surface is also discretized with relatively finer meshing at the forward and aft ends (Figs. 2 and 3). The boundary condition for the fluid domain is described in Figs. 4 and 5. 2.1. Solution The analysis is performed using a commercial CFD code (FLUENT 6.0.20) using a segregated implicit RANS solver with standard k–3 turbulent model as described above. The simulation is performed on a 64 way, 32 GB SGI Origin 3800 machine. The procedure uses unstructured zonal meshing for discretization with tetrahedral type cells. The solution process requires iterations with check on the residuals on the right hand side of the governing equations. This iterative procedure is continued till convergence. For typical
Fig. 2. View showing Aft hull meshing.
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Fig. 3. View showing A bracket meshing.
Fig. 4. Boundary condition.
values, for specific run, convergence was achieved in 156 iterations, with seven hours running time. 2.1.1. Flow effect due to stern appendages The bossing is a significant appendage underwater forward of the propeller and it serves as an envelope at the region where the propeller shafting pierces the hull. Although bossings are generally beneficial in smoothening and streamlining the shapes, there are exceptions where bossings are best eliminated. Fig. 6 shows the modelling with bossing
Fig. 5. Top surface boundary condition.
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Fig. 6. Contours of velocity at propeller plane.
and shaft brackets. The bossing structures (Fig. 6) and the shaft support brackets were integrated into the CFD model for analysis. 2.1.2. Evaluation of wake at the propeller location (with and without bossing) The presence of bossing ahead of the propeller location influences the inflow to the propeller. Therefore the CFD analysis has been done for ship speeds with and without bossing. Table 1 gives the intensity of turbulent kinetic energy. The non dimensional Taylor’s wake fraction is defined as: w Z ðV VA Þ=V;
(6)
V being the free stream velocity and VA the inflow velocity at a given point. The wake fractions are calculated and presented in Table 2. Based on experiments described later below, the experimental values are also given along side for comparison. The mean difference in wake fraction is G0.016 and the wake values bear close comparison. The turbulent intensity is defined as rffiffiffiffiffiffi 2 IZ kV (7) 3 where
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Table 1 Results of Turbulent kinetic energy at propeller location for the cases of model with and without bossing Sl no CFD results with bossing 1 2 3 4 5 6 7 8 9 10 11 12 13 14 CFD results no bossing 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Fn
Tub KE (m2/s2)
0.1181 0.1263 0.1432 0.1518 0.1589 0.1653 0.1720 0.1807 0.1871 0.1945 0.2035 0.2106 0.2166 0.2235
0.3462 0.3468 0.3574 0.3490 0.3440 0.3521 0.3521 0.3562 0.3680 0.3672 0.3693 0.3691 0.3700 0.3710
0.1206 0.1268 0.1354 0.1435 0.1494 0.1609 0.1657 0.1732 0.1769 0.1905 0.1973 0.2052 0.2155 0.2218
0.2361 0.2374 0.2370 0.2390 0.2405 0.2450 0.2480 0.2502 0.2520 0.2539 0.2541 0.2555 0.2560 0.2567
I turbulent Intensity k turbulent kinetic energy V free stream velocity This represents intensity of turbulence around the bossing region and is a measure of the interference to flow. Turbulent intensity is shown Fig. 7, where the interference due to bossing is evident. The numerical results indicate there is a 33% drop in turbulent kinetic energy after removal of bossing. Table 1 shows turbulent kinetic energy as a function of Froude numbers.
3. Experimental analysis Experimental benchmarking is vital for the progress of CFD modelling and investigations. In recent years, (Valkhof et al., 1998) combination of the two, have
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Table 2 Experimental and CFD wake at propeller location for conditions with and without bossing Fn (V/sqrt(g*L)) With bossing 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Without bossing 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Wake fraction Exp
CFD
0.118 0.127 0.135 0.144 0.152 0.160 0.166 0.173 0.181 0.188 0.195 0.204 0.211 0.217 0.224 0.233 0.242
0.281 0.272 0.280 0.281 0.286 0.278 0.294 0.293 0.291 0.288 0.303 0.279 0.298 0.301 0.304 0.292 0.282
0.278 0.274 0.271 0.269 0.266 0.267 0.266 0.268 0.270 0.269 0.271 0.273 0.275 0.276 0.271 0.279 0.279
0.121 0.127 0.136 0.144 0.150 0.161 0.166 0.174 0.177 0.191 0.198 0.206 0.216 0.222 0.237 0.241 0.250
0.138 0.134 0.142 0.149 0.135 0.143 0.147 0.143 0.137 0.137 0.137 0.154 0.161 0.143 0.156 0.148 0.149
0.145 0.155 0.152 0.149 0.146 0.148 0.153 0.160 0.160 0.164 0.160 0.167 0.168 0.170 0.176 0.178 0.175
provided advances in hull form optimization. The prime motivation for the experimental study was the necessity to validate the computational assessment of wake characteristics. In addition, to achieve complete characterization of flow influences, the experiment was devised with appropriate fluorescein dye injection and underwater video observation of the flow towards the active running propeller. For this purpose, a 1:12 scale model with removable bossings was set up, complete with twin shaft electric motor drive and propellers. See Table 3 for ship particulars. The model was run in the towing carriage (82m!3.2m!2.8m depth) at speeds corresponding to different self propulsion J values.
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CFD RESULTS OF TURB KE WITH AND WITHOUT BOSSING 0.4000 Turb KE with bossing Turb KE without bossing
0.3800
Turb KE (m^2/Sec^2)
0.3600 0.3400 0.3200 0.3000 0.2800 0.2600 0.2400 0.2200 0.2000 0.0800
0.1000
0.1200
0.1400
0.1600
0.1800
0.2000
0.2200
0.2400
0.2600
Fn(V/sqrt(gL))
Fig. 7. Turbulent kinetic energy for cases of ship with and without bossing.
Model details are shown in Figs. 8–10. A near frictionless vane type impeller of 50 mm diameter was used for nominal wake measurement at the averaged centre point location of the propeller. All geometric modelling incorporated in the CFD flow simulations in the stern region (stern tubes, A-brackets, with and without bossings) were included in the experimental model. Laminar flow conditions were avoided by using turbulence stimulators on the model. Since wake has spatial variation, the comparison is meaningful only when restricted to the same size zone in the propeller plane. The measured wake values are given in Table 1. It is interesting to note that measured values and numerically obtained values Table 3 Vessel particulars
Length overall LOA Length on the waterline LWL Length between perpendicular LBP Moulded breadth B Moulded depth to upper deck D Design draft T Block coefficient CB Midship coefficient CM Design speed Volume displacement P
Prototype
Model
44.84 m 44.04 m 42.28 m 12.0 m 3.5 m 2.4 m 0.81 0.99 10 knots 1060 m3
3.85 m 3.78 m 3.63 m 1.03 m 0.30 m 0.20 m 0.81 0.99 2.93 knots 0.67 m3
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Fig. 8. Profile view of the model.
Fig. 9. Model fitted with propeller.
compare very well through out the speed range. The numerical model has been validated for the specific flow conditions in the stern of the vessel. Wake values were measured using a vane wheel. In the case of measurements with bossing in place, an average wake fraction of 0.29 was obtained and in the case of measurements without bossing, an average value of 0.14 was obtained. The numerically obtained values in these cases were, respectively, 0.27 and 0.16 (average values). These numbers bear very close match, thus validating the computed values (Fig. 11). The succession of instantaneous frame shots from the video are shown in Figs. 12 and 13 for the case of the designed speed condition of the ship. The growth of the flow from
Fig. 10. Model fitted with vane.
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WAKE COMPARISON 0.35 0.3
Wake {(V-Va)/V}
0.25 0.2 0.15 0.1
exp wake with bossing CFD wake with bossing exp wake without bossing CFD wake without bossing
0.05 0 0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
Fn {V/sqrt(g*L)}
Fig. 11. Comparison of wake for the cases of ship with and without bossing.
Fig. 12. View of frames at Fn 0.22 for the model with bossing.
0.26
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Fig. 13. View of frames at Fn 0.22 for the case without bossing.
inception of dye injection with running propeller clearly shows the obstructive influence of the bossing in Fig. 12. In Fig. 13, the same successions of frames are shown in free running propeller condition at designed ship speed but with bossing removed. The dye injected flow path clearly hugs the stern shape in the absence of the bossing. The video frames in comparison provide conclusive evidence of the obstructive influence of the bossing in the design under consideration.
4. Numerical modelling of aft hull form The aft hull stern shape is represented by a sloping plane hull surface in the simplest case. The integration of the bossing shape requires the definition of the curvature of the bossing plate and definition of chine locations and discontinuities. The surface is obtained from the initial 3D mesh of polygon points and is transformed into consistent faired 3D surface data, obtained in any desirable closely spaced digitized format, Subramanian and Suchithran (1999). The original input points are weighting functions. The program uses 2D B-spline interpolation at every station to regenerate an equal number of control polygon points. After generation of the control polygon points, the program fairs
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a B-spline surface through them. A script file is generated as output containing the output points, which is formatted for directly interfacing with AutoCAD. The obtained design is exported for CFD analysis. A Cartesian product B-spline surface is defined, Rogers and Adams (1990),
Qðu; wÞ Z
nC1 m C1 X X
Bi;j Ni;k ðuÞMj;l ðwÞ
(8)
iZ1 jZ1
Bi,j’s are the vertices of a defining polygon net. The indices n and m are the number of defining polygon vertices in the u and w parametric directions, respectively. Ni,k(u) and Mj,l(w) are the B-spline basis functions in the bi-parametric u and w directions, respectively. k and l are the order of the curves in the u and w directions, respectively. The definition for the basis functions is given below. Ni;1 ðuÞ Z 1 if ai %u%aiC1 Z 0 otherwise Ni;k ðuÞ Z ðu Kai ÞNi;kK1 ðuÞ=ðaiCk Kai Þ C ðaiCk KuÞNiC1;kK1 ðuÞ=ðaiCk KaiC1 Þ Mj;1 ðwÞ Z 1 if bj w!bjC1 Z 0 otherwise Mj;1 ðwÞ Z ðw Kbj ÞMj;lK1 ðwÞ=ðbjClK1 Kbj Þ C ðbjCl KwÞMjC1;lK1 ðwÞ=ðbjCl KbjC1 Þ ai and bj are elements of knot vectors. The references give details for implementation of these schemes to rapidly evolve a faired form. This representation enables rapid redefinition of the surface for varying stern rake angles. The CFD analysis was carried out for a range of stern rake angle from 13 to 88. The wire frame models are shown in Fig. 14 The results are given in Table 4 and also in (Figs. 15 and 16).
Fig. 14. Wire Frame model of different stern rake (13.1 and 88) angles.
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Table 4 Wake and propeller aperture for different aft cut up angle S. No.
Aft cut up inclination (deg.)
Wake fraction
Aperture for the propeller (m)
1 2 3
8 10 13
0.112 0.137 0.155
0.983 1.234 1.62
4.1. Contours of velocity distribution along the hull The spatial velocity distribution is brought out by projecting the velocity contours, see Fig. 17. In Fig. 18, velocity contours at different yz plane along the hull are given. The contours plots show the variation of inflow velocity due to the aft cut.
0.16 0.15 0.14 Wake
0.13 0.12 0.11 0.1 0.09 0.08 7
8
9 10 11 12 Aft cut up inclination indeg
13
14
Fig. 15. Wake fraction vs aft cut up angle.
Aperture for propeller in m
1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 7
8
9 10 11 12 Aft cut up in clination in Deg
Fig. 16. Propeller aperture vs aft cut up angle.
13
14
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Fig. 17. Velocity contours along profile section (XZ plane).
Fig. 18. Velocity contour at different YZ plane.
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4.2. Wake results By modifying the stern rake through 8,9,11 degrees, respectively the analysis was carried out using implicit standard RANS k–3 turbulent model. Table 4 gives associated wake values. The results of velocity contours at the propeller location for different rake angles are given in Fig. 19–22.
5. Results and conclusion Stable results have been obtained for the wake fraction in relation to the complex aft under water body geometry. The adverse effect of the bossing in terms of heterogeneous flow and increased wake fraction are vindicated by the combined CFD modeling and experimental results together with video images of the flow in the propeller region. The study demonstrates new avenues for application of CFD in the early design stages. The numerical model shows that reduced rake angle at the stern, improves the wake fraction. In a realistic scenario, the rake angle cannot be reduced beyond a minimum value because of the consideration for minimum aperture for propeller, optimum propeller diameter and adequate propeller tip clearance from the hull. Using surface generation tools the inverse design process can be adapted to obtain favourable flow conditions in the aft and therefore improved propulsive efficiency.
Fig. 19. Velocity contours at propeller location for 88 aft cut up hull form.
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Fig. 20. Velocity contours at propeller location for 98 aft cut up hull form.
Fig. 21. Velocity contours at propeller location for 108 aft cut up hull form.
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Fig. 22. Velocity contours at propeller location for 138 aft cut up hull form.
References Bertram, V., 2000. Practical Ship Hydrodynamics. Butterworth–Heinemann, Oxford. Chen, P.F., Cheng, H.H., 2002. An inverse hull design problem in optimizing the desired wake of a ship. Journal of Ship Reserch 46 (2), 138–146. Daniele, P., Michele, R., Emilio, F.C., 2001. Design optimisation of ship hulls via CFD techniques. Journal of Ship Reserch 45 (2), 140–149. Rogers, D.F., Adams, J.A., 1990. Mathematical Elements for Computer Graphics. McGraw-Hill, New York, NY. Subramanian, V.A., 2002. Technical Report on propeller induced hull vibration of Lube Oil Tanker GAC Mumtaz, Report Nr.OEC047, IIT Madras, India. Subramanian, V.A., Suchithran, P.R., 1999. Interactive curve fairing and bi-quintic surface generation for ship design. International Shipbuilding Progress 46, 189–208. Valkhof, H.H., Hoekstra, M., Andersen, J.E., 1998. Model tests and CFD in Hull form optimization. SNAME Transaction 106, 391–412.
An inverse design approach for minimising wake at propeller plane using CFD V. Anantha Subramanian*, R. Vijayakumar Department of Ocean Engineering, Indian Institute of Technology, Chennai 600 036, India Received 15 November 2004; accepted 27 April 2005 Available online 12 July 2005
Abstract Knowledge of wake characteristics in the stern region is important for ensuring good propeller design and performance. This work examines the utility of CFD in the analysis of flow in the case of full aft beam vessels having characteristic cut stern shape to facilitate propeller aperture. The underwater stern shape may be more complex due to the occurrence of stern appendages such as bossings, strut supports and local shape variations. To this extent, CFD offers an effective tool for both qualitative as well as quantitative assessment of the local geometry. Wake estimate is required for choice of the most favorable propeller geometry. In the present method, the analysis quantifies the effects of small changes in stern rake angles and offers an inverse design approach towards finalising the stern shape. The method consists of solving the standard k–3 turbulent model of RANS equations in cell centered finite volume multi zone grid in the flow domain. This approach has been used in estimating the velocity at the propeller plane. The results have been compared with experimentally obtained values of nominal wake. The approach suggests that CFD can provide a cost effective and quick assessment of flow. It is also an attractive means of pre-empting heterogeneous flow related problems such as vibration and noise due to unfavorable wake in the stern region. q 2005 Elsevier Ltd. All rights reserved. Keywords: Wake fraction; Computational model; Inverse design; Surface modelling
1. Introduction The loss of inflow to the propeller, commonly known as wake, is an important propeller-hull interaction factor for consideration in obtaining effective propeller thrust
* Corresponding author. E-mail address: [email protected] (V.A. Subramanian).
0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.04.014
120
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power. For optimum propeller performance and minimised flow induced vibration, it is essential to have smooth homogeneous inflow to the propeller. In the case of normal ship shapes, the wake can be estimated fairly accurately at an early design stage on the basis of data from past experience. In the case of broad beamed vessels having their form more in the nature of full beam self propelled barges, the propeller aperture is created by providing a cut stern with an appropriate rake angle. The recent case study of newly built prototypes, Subramanian (2002), have shown that inappropriate stern design can result in potentially serious problem of propeller performance, with attendant noise and vibration mainly due to inadequate as well as highly heterogeneous inflow at the propeller plane. Heterogeneous inflow can also be made worse due to the unfavorable location of appendages such as bossing and struts. Since the consequences of the detailed design layout of the above components at the aft end cannot be exactly quantified otherwise at an early design stage, CFD offers a potential tool for quick estimation of the inflow characteristics. Wake distribution is either measured by laser-Doppler velocimetry or computed by CFD. Current wake predictions using CFD are not yet at a mature stage, whereas the integral of the wake over the propeller plane is well predicted, Bertram (2000). The inverse design approach has been adopted for obtaining more optimal hull shaping Daniele et al. (2001), and Chen and Cheng (2002). The motivation for this work is therefore to investigate a numerical cum experimental approach towards better estimation of the wake phenomena, especially in the presence of the complex aft geometry. As mentioned above, it is known from real field experiences that inappropriate design and location of stern appendages can result in manifestation of propeller flow induced vibration while running at modest engine speeds. This can become severe vibration at higher engine speeds near the rated rotational propeller speed. This study has been undertaken to understand the flow dynamics by combining numerical flow modelling with experimental wake determination. The modelling results have been reaffirmed by observing the fluid appendage interaction. For this purpose, dye injection method has been used and underwater video obtained to characterize the flow in the region of the propeller. The results of the investigation indicate that the numerical method can be successfully used in an inverse design approach.
2. Methodology The flow field in the propeller region, around the hull is simulated employing commercial CFD code (FLUENT 6.0.20) and the obtained flow velocity compared with experimental studies. The CFD simulation is based on Reynolds Averaged Navier Stokes (RANS) equations solver using a finite-volume method. The flow field domain (solution domain) is subdivided into finite number of control volumes (CV), using the modeler Gambit 2.0. The conservation equations, the general form of which is given in Eq. (1), are then applied to each control volume. In actual implementation this integral equation is converted into linearized algebraic difference form [Eq. (2)] for the discrete dependent variable and applied to the computational node positioned at the centroid of each control volume.
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ððð
v vt
ðð Rate of increase of f
C
ðð convection term
A A rfVd
QðrfÞdQ
Z
121
ððð diffusion term
C
A Gf VfdA
source term Q Sf dQ
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} (1) Where f is any variable and can take a value of 1 to give the continuity equation u,v,w to give the momentum equation in three direction h to give the energy equation The general conservation equation in difference form can be written as N N faces faces X X ðrfp ÞtCDt K ðrfp Þt DQ C rf nf ff A f Z Gf ðVfÞn A f C SfQ Dt f f
(2)
Interpolation is used to express variable values at the control volume (CV) surface in terms of CV-centre node values. The standard k–3 model which is a semi-empirical model based on equations for turbulent kinetic energy (k) and its dissipation rate (3), is then used. The transport equations are vk v nt vk 0 0 vU Ui Z Kui uj K3 C (3) vxi vxj vxi sk vxi Z KC31 k3 ui0 uj0 vU h i 2 v3 vxj K C32 3k C vxv i snt3 vx i Ui
v3 vxi
(4)
where CmZ0.09, skZ1.0, Cx1Z1.44, Cx2Z1.92 sk Zturbulent Prandtl number for k(1.0) s3 Zturbulent Prandtl number for 3(1.3) These are the constants for the standard k–3 model. The eddy or turbulent viscosity, nt is computed by combining k and 3 as follows nt Z rCm
k2 3
(5)
where Cm Zconstant (0.09). The zone of interest around the hull, i.e. the flow field domain has initially been taken as one ship length at the front of the ship, 5 ship lengths behind the ship and 5 times the breadth and depth along the beam and depth directions, respectively. On successive analysis with reduced domain size, an optimal size was arrived at. The final domain size
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Fig. 1. Different zones of fluid domain.
was taken with two ship lengths behind and three times the breadth and depth along the respective directions. This has effectively reduced the computation domain by 80%. The reduced domain size allowed better flexibility and finer meshing size as required. The flow field domain is discretized into finite control volumes as shown in Fig. 1. Multi zone unstructured grid has been used for domain discretization, for this purpose, the complete flow domain is discretized into zones and these are further discretized in turn with tetrahedral cells. Using graded mesh sizes, a total number of finite volume cells (approximately 2.5 million) have been generated with minimum edge size of 0.37 cm. The hull surface is also discretized with relatively finer meshing at the forward and aft ends (Figs. 2 and 3). The boundary condition for the fluid domain is described in Figs. 4 and 5. 2.1. Solution The analysis is performed using a commercial CFD code (FLUENT 6.0.20) using a segregated implicit RANS solver with standard k–3 turbulent model as described above. The simulation is performed on a 64 way, 32 GB SGI Origin 3800 machine. The procedure uses unstructured zonal meshing for discretization with tetrahedral type cells. The solution process requires iterations with check on the residuals on the right hand side of the governing equations. This iterative procedure is continued till convergence. For typical
Fig. 2. View showing Aft hull meshing.
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Fig. 3. View showing A bracket meshing.
Fig. 4. Boundary condition.
values, for specific run, convergence was achieved in 156 iterations, with seven hours running time. 2.1.1. Flow effect due to stern appendages The bossing is a significant appendage underwater forward of the propeller and it serves as an envelope at the region where the propeller shafting pierces the hull. Although bossings are generally beneficial in smoothening and streamlining the shapes, there are exceptions where bossings are best eliminated. Fig. 6 shows the modelling with bossing
Fig. 5. Top surface boundary condition.
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Fig. 6. Contours of velocity at propeller plane.
and shaft brackets. The bossing structures (Fig. 6) and the shaft support brackets were integrated into the CFD model for analysis. 2.1.2. Evaluation of wake at the propeller location (with and without bossing) The presence of bossing ahead of the propeller location influences the inflow to the propeller. Therefore the CFD analysis has been done for ship speeds with and without bossing. Table 1 gives the intensity of turbulent kinetic energy. The non dimensional Taylor’s wake fraction is defined as: w Z ðV VA Þ=V;
(6)
V being the free stream velocity and VA the inflow velocity at a given point. The wake fractions are calculated and presented in Table 2. Based on experiments described later below, the experimental values are also given along side for comparison. The mean difference in wake fraction is G0.016 and the wake values bear close comparison. The turbulent intensity is defined as rffiffiffiffiffiffi 2 IZ kV (7) 3 where
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Table 1 Results of Turbulent kinetic energy at propeller location for the cases of model with and without bossing Sl no CFD results with bossing 1 2 3 4 5 6 7 8 9 10 11 12 13 14 CFD results no bossing 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Fn
Tub KE (m2/s2)
0.1181 0.1263 0.1432 0.1518 0.1589 0.1653 0.1720 0.1807 0.1871 0.1945 0.2035 0.2106 0.2166 0.2235
0.3462 0.3468 0.3574 0.3490 0.3440 0.3521 0.3521 0.3562 0.3680 0.3672 0.3693 0.3691 0.3700 0.3710
0.1206 0.1268 0.1354 0.1435 0.1494 0.1609 0.1657 0.1732 0.1769 0.1905 0.1973 0.2052 0.2155 0.2218
0.2361 0.2374 0.2370 0.2390 0.2405 0.2450 0.2480 0.2502 0.2520 0.2539 0.2541 0.2555 0.2560 0.2567
I turbulent Intensity k turbulent kinetic energy V free stream velocity This represents intensity of turbulence around the bossing region and is a measure of the interference to flow. Turbulent intensity is shown Fig. 7, where the interference due to bossing is evident. The numerical results indicate there is a 33% drop in turbulent kinetic energy after removal of bossing. Table 1 shows turbulent kinetic energy as a function of Froude numbers.
3. Experimental analysis Experimental benchmarking is vital for the progress of CFD modelling and investigations. In recent years, (Valkhof et al., 1998) combination of the two, have
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Table 2 Experimental and CFD wake at propeller location for conditions with and without bossing Fn (V/sqrt(g*L)) With bossing 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Without bossing 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Wake fraction Exp
CFD
0.118 0.127 0.135 0.144 0.152 0.160 0.166 0.173 0.181 0.188 0.195 0.204 0.211 0.217 0.224 0.233 0.242
0.281 0.272 0.280 0.281 0.286 0.278 0.294 0.293 0.291 0.288 0.303 0.279 0.298 0.301 0.304 0.292 0.282
0.278 0.274 0.271 0.269 0.266 0.267 0.266 0.268 0.270 0.269 0.271 0.273 0.275 0.276 0.271 0.279 0.279
0.121 0.127 0.136 0.144 0.150 0.161 0.166 0.174 0.177 0.191 0.198 0.206 0.216 0.222 0.237 0.241 0.250
0.138 0.134 0.142 0.149 0.135 0.143 0.147 0.143 0.137 0.137 0.137 0.154 0.161 0.143 0.156 0.148 0.149
0.145 0.155 0.152 0.149 0.146 0.148 0.153 0.160 0.160 0.164 0.160 0.167 0.168 0.170 0.176 0.178 0.175
provided advances in hull form optimization. The prime motivation for the experimental study was the necessity to validate the computational assessment of wake characteristics. In addition, to achieve complete characterization of flow influences, the experiment was devised with appropriate fluorescein dye injection and underwater video observation of the flow towards the active running propeller. For this purpose, a 1:12 scale model with removable bossings was set up, complete with twin shaft electric motor drive and propellers. See Table 3 for ship particulars. The model was run in the towing carriage (82m!3.2m!2.8m depth) at speeds corresponding to different self propulsion J values.
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CFD RESULTS OF TURB KE WITH AND WITHOUT BOSSING 0.4000 Turb KE with bossing Turb KE without bossing
0.3800
Turb KE (m^2/Sec^2)
0.3600 0.3400 0.3200 0.3000 0.2800 0.2600 0.2400 0.2200 0.2000 0.0800
0.1000
0.1200
0.1400
0.1600
0.1800
0.2000
0.2200
0.2400
0.2600
Fn(V/sqrt(gL))
Fig. 7. Turbulent kinetic energy for cases of ship with and without bossing.
Model details are shown in Figs. 8–10. A near frictionless vane type impeller of 50 mm diameter was used for nominal wake measurement at the averaged centre point location of the propeller. All geometric modelling incorporated in the CFD flow simulations in the stern region (stern tubes, A-brackets, with and without bossings) were included in the experimental model. Laminar flow conditions were avoided by using turbulence stimulators on the model. Since wake has spatial variation, the comparison is meaningful only when restricted to the same size zone in the propeller plane. The measured wake values are given in Table 1. It is interesting to note that measured values and numerically obtained values Table 3 Vessel particulars
Length overall LOA Length on the waterline LWL Length between perpendicular LBP Moulded breadth B Moulded depth to upper deck D Design draft T Block coefficient CB Midship coefficient CM Design speed Volume displacement P
Prototype
Model
44.84 m 44.04 m 42.28 m 12.0 m 3.5 m 2.4 m 0.81 0.99 10 knots 1060 m3
3.85 m 3.78 m 3.63 m 1.03 m 0.30 m 0.20 m 0.81 0.99 2.93 knots 0.67 m3
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Fig. 8. Profile view of the model.
Fig. 9. Model fitted with propeller.
compare very well through out the speed range. The numerical model has been validated for the specific flow conditions in the stern of the vessel. Wake values were measured using a vane wheel. In the case of measurements with bossing in place, an average wake fraction of 0.29 was obtained and in the case of measurements without bossing, an average value of 0.14 was obtained. The numerically obtained values in these cases were, respectively, 0.27 and 0.16 (average values). These numbers bear very close match, thus validating the computed values (Fig. 11). The succession of instantaneous frame shots from the video are shown in Figs. 12 and 13 for the case of the designed speed condition of the ship. The growth of the flow from
Fig. 10. Model fitted with vane.
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WAKE COMPARISON 0.35 0.3
Wake {(V-Va)/V}
0.25 0.2 0.15 0.1
exp wake with bossing CFD wake with bossing exp wake without bossing CFD wake without bossing
0.05 0 0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
Fn {V/sqrt(g*L)}
Fig. 11. Comparison of wake for the cases of ship with and without bossing.
Fig. 12. View of frames at Fn 0.22 for the model with bossing.
0.26
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Fig. 13. View of frames at Fn 0.22 for the case without bossing.
inception of dye injection with running propeller clearly shows the obstructive influence of the bossing in Fig. 12. In Fig. 13, the same successions of frames are shown in free running propeller condition at designed ship speed but with bossing removed. The dye injected flow path clearly hugs the stern shape in the absence of the bossing. The video frames in comparison provide conclusive evidence of the obstructive influence of the bossing in the design under consideration.
4. Numerical modelling of aft hull form The aft hull stern shape is represented by a sloping plane hull surface in the simplest case. The integration of the bossing shape requires the definition of the curvature of the bossing plate and definition of chine locations and discontinuities. The surface is obtained from the initial 3D mesh of polygon points and is transformed into consistent faired 3D surface data, obtained in any desirable closely spaced digitized format, Subramanian and Suchithran (1999). The original input points are weighting functions. The program uses 2D B-spline interpolation at every station to regenerate an equal number of control polygon points. After generation of the control polygon points, the program fairs
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a B-spline surface through them. A script file is generated as output containing the output points, which is formatted for directly interfacing with AutoCAD. The obtained design is exported for CFD analysis. A Cartesian product B-spline surface is defined, Rogers and Adams (1990),
Qðu; wÞ Z
nC1 m C1 X X
Bi;j Ni;k ðuÞMj;l ðwÞ
(8)
iZ1 jZ1
Bi,j’s are the vertices of a defining polygon net. The indices n and m are the number of defining polygon vertices in the u and w parametric directions, respectively. Ni,k(u) and Mj,l(w) are the B-spline basis functions in the bi-parametric u and w directions, respectively. k and l are the order of the curves in the u and w directions, respectively. The definition for the basis functions is given below. Ni;1 ðuÞ Z 1 if ai %u%aiC1 Z 0 otherwise Ni;k ðuÞ Z ðu Kai ÞNi;kK1 ðuÞ=ðaiCk Kai Þ C ðaiCk KuÞNiC1;kK1 ðuÞ=ðaiCk KaiC1 Þ Mj;1 ðwÞ Z 1 if bj w!bjC1 Z 0 otherwise Mj;1 ðwÞ Z ðw Kbj ÞMj;lK1 ðwÞ=ðbjClK1 Kbj Þ C ðbjCl KwÞMjC1;lK1 ðwÞ=ðbjCl KbjC1 Þ ai and bj are elements of knot vectors. The references give details for implementation of these schemes to rapidly evolve a faired form. This representation enables rapid redefinition of the surface for varying stern rake angles. The CFD analysis was carried out for a range of stern rake angle from 13 to 88. The wire frame models are shown in Fig. 14 The results are given in Table 4 and also in (Figs. 15 and 16).
Fig. 14. Wire Frame model of different stern rake (13.1 and 88) angles.
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Table 4 Wake and propeller aperture for different aft cut up angle S. No.
Aft cut up inclination (deg.)
Wake fraction
Aperture for the propeller (m)
1 2 3
8 10 13
0.112 0.137 0.155
0.983 1.234 1.62
4.1. Contours of velocity distribution along the hull The spatial velocity distribution is brought out by projecting the velocity contours, see Fig. 17. In Fig. 18, velocity contours at different yz plane along the hull are given. The contours plots show the variation of inflow velocity due to the aft cut.
0.16 0.15 0.14 Wake
0.13 0.12 0.11 0.1 0.09 0.08 7
8
9 10 11 12 Aft cut up inclination indeg
13
14
Fig. 15. Wake fraction vs aft cut up angle.
Aperture for propeller in m
1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 7
8
9 10 11 12 Aft cut up in clination in Deg
Fig. 16. Propeller aperture vs aft cut up angle.
13
14
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Fig. 17. Velocity contours along profile section (XZ plane).
Fig. 18. Velocity contour at different YZ plane.
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4.2. Wake results By modifying the stern rake through 8,9,11 degrees, respectively the analysis was carried out using implicit standard RANS k–3 turbulent model. Table 4 gives associated wake values. The results of velocity contours at the propeller location for different rake angles are given in Fig. 19–22.
5. Results and conclusion Stable results have been obtained for the wake fraction in relation to the complex aft under water body geometry. The adverse effect of the bossing in terms of heterogeneous flow and increased wake fraction are vindicated by the combined CFD modeling and experimental results together with video images of the flow in the propeller region. The study demonstrates new avenues for application of CFD in the early design stages. The numerical model shows that reduced rake angle at the stern, improves the wake fraction. In a realistic scenario, the rake angle cannot be reduced beyond a minimum value because of the consideration for minimum aperture for propeller, optimum propeller diameter and adequate propeller tip clearance from the hull. Using surface generation tools the inverse design process can be adapted to obtain favourable flow conditions in the aft and therefore improved propulsive efficiency.
Fig. 19. Velocity contours at propeller location for 88 aft cut up hull form.
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Fig. 20. Velocity contours at propeller location for 98 aft cut up hull form.
Fig. 21. Velocity contours at propeller location for 108 aft cut up hull form.
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Fig. 22. Velocity contours at propeller location for 138 aft cut up hull form.
References Bertram, V., 2000. Practical Ship Hydrodynamics. Butterworth–Heinemann, Oxford. Chen, P.F., Cheng, H.H., 2002. An inverse hull design problem in optimizing the desired wake of a ship. Journal of Ship Reserch 46 (2), 138–146. Daniele, P., Michele, R., Emilio, F.C., 2001. Design optimisation of ship hulls via CFD techniques. Journal of Ship Reserch 45 (2), 140–149. Rogers, D.F., Adams, J.A., 1990. Mathematical Elements for Computer Graphics. McGraw-Hill, New York, NY. Subramanian, V.A., 2002. Technical Report on propeller induced hull vibration of Lube Oil Tanker GAC Mumtaz, Report Nr.OEC047, IIT Madras, India. Subramanian, V.A., Suchithran, P.R., 1999. Interactive curve fairing and bi-quintic surface generation for ship design. International Shipbuilding Progress 46, 189–208. Valkhof, H.H., Hoekstra, M., Andersen, J.E., 1998. Model tests and CFD in Hull form optimization. SNAME Transaction 106, 391–412.