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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 527
Simulation of Incompressible Viscous Flow Around a Ducted
Propeller Using a RANS Equation Solver
A.Sánchez-Caja, (VTT Manufacturing Technology, Finland)
P.Rautaheimo, T.Siikonen (Helsinki University of Technology, Finland)
ABSTRACT
The incompressible viscous flow around a marine ducted propeller is simulated by solving the RANS equations with
the k-ε turbulence model. The FINFLO solver developed at Helsinki University of Technology is used in the calculations.
FINFLO is a multiblock cell-centered finite-volume computer code with sliding mesh, moving-grid and free-surface
capabilities. In this paper, the flow over a Ka series propeller and NSMB nozzle 19A is analyzed. The calculated flow
patterns downstream of the propeller and duct are illustrated and compared with experiments for one advance number.
Calculated thrust and torque are also provided for several advance numbers. Good correlation with experiments is
obtained in terms of force coefficients and velocity distributions.
INTRODUCTION
Ducted propulsors are known to offer significant advantages for particular marine applications. Since 1931, they
have been first installed in tugs, push-boats, trawlers, and later in research vessels, drilling platforms, submersibles, etc.
There are some installations in commercial ships, like large tankers and bulk carriers, and warships like naval destroyers
and submarines. Among the benefits of ducted propulsors are remarkable increases in efficiency for high propeller
loadings with flow-accelerating ducts, or alternatively smaller propeller size; reduction of inflow velocity and,
consequently, of cavitation and noise with flow-decelerating ducts; better control over the inflow to the propeller;
improvement of maneuverability and position-keeping abilities of vessels; protection from damage to the propeller, etc.
From a theoretical standpoint, the hydrodynamic interaction between duct and propeller produces a twofold effect.
On the one hand, the presence of the duct permits to transfer the mean lifting force on the propeller blade closer to the
propeller tip, which in turn efficiently deflects the force to a direction near that of the ship's motion. On the other hand,
the radial contraction of the flow due to the propeller action results in an additional thrusting force on the duct, which
increases the total thrust of the propulsor unit provided that the loading is sufficiently high to overcome the duct viscous
drag. However, there is an upper limit also for the duct loading, which is determined by the risk of flow separation, as
well as for the propeller loading, which is determined by the risk of cavitation at the propeller tip. The design of a ducted
propeller is, therefore, a complicated process in which the designer often has to make a compromise between conflicting
requirements. In such cases, having access to information on the details of the flowfield in problematic areas is most
valuable for a successful design.
Most of the analysis methods for ducted propulsors have been based on potential theory, using an actuator disk
(Gibson and Lewis, 1973; Gibson, 1974; Falcão de Campos, 1983, etc.), lifting-line or lifting-surface approaches (Kerwin
et al. 1987; Hughes & Kinnas, 1991, etc.) for modeling the propeller. The more recent panel methods also belong
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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 528
to this class of potential-based theories (Hoshino 1989, Kawakita 1992, etc.). All these methods represent a great advance
in understanding the main features of the flow around ducted propulsors. However, they all have the shortcoming of
incorporating viscous effects artificially through empirical corrections external to the theory. In other words, details as
important to the designer as the gap flow at the tip of the propeller cannot be properly analyzed with these methods.
Recently, some hybrid models have been developed that combine viscous and potential theories mainly for the design
problem, for example, in Kerwin et al. (1994). More recently, calculations of the flow around a ducted thruster have been
made at Postdam Model Basin for Schottel Shipyard GmbH using either hybrid or fully viscous models. This work has
been outlined in Abdel-Maksoud (1999), but no validation data were released. In the present paper, the RANS equations
are solved for a ducted propeller configuration using the FINFLO code initially developed at the Laboratory of
Aerodynamics at Helsinki University of Technology (Siikonen, 1990). The flow around a ducted propeller of the NSMB
(now MARIN) Ka series is simulated and compared to experimental data from the cavitation tunnel of the Nagasaki
Experimental Tank (Kawakita, 1992). The experimental data reported by Kawakita are among the few available in the
open literature for the validation of ducted propellers.
FINFLO is a multiblock cell-centered finite-volume multigrid-structured computer code with sliding mesh, moving-
grid and free-surface capabilities. The code has been validated for a number of test cases including marine applications.
For propeller flows, validation work was carried out for conventional propeller geometries such as that of DTMB
propeller 4119 (Sánchez-Caja, 1998). Recently, the unsteady flow around a tractor thruster was simulated using a sliding
mesh technique and a comparison of some available experimental data to computed results was presented (Sánchez-Caja,
et al. 1999). The sliding mesh technique was found robust for the analysis of the time-dependent viscous flow. The
computations were performed in a quasi-steady and time-accurate manner. The former reduced the CPU time to about
1/10 relative to the latter. Its main merit consisted of decreasing the CPU time while maintaining a full representation of
the propeller geometry, i.e. without introducing simplified models for simulating the propeller action, such as actuator
disk or body force models.
In the present study, the flow around a ducted propeller unit is considered. Even though the unit consists of a rotating
part (the propeller) and a stationary part (the duct), a steady-state flow can be assumed provided that the inflow and duct
are axisymmetric. Consequently, there is no need to use special techniques based on overlapping or sliding meshes.
NUMERICAL METHOD
Governing Equations and Turbulence Closure
The flow simulation is based on the solution of the RANS equations. These can be written in the following
conservative form
(1)
where U is a vector of conservative variables (ρ, ρu, ρv, ρw, ρk, ρε)T; F, G and H are the inviscid fluxes; FV, GV and
HV are the viscous fluxes; u, v and w are the absolute velocity components; ρ is the density, k is the turbulent kinetic
energy and ε is the dissipation of k. The source term Q has non-zero components only for the turbulence equations. For
the steady-state propeller analysis, the equations are solved in a coordinate system that rotates around the x-axis with an
angular velocity Ω. In this case, Q has the additional component (0, 0, ρΩw, −ρΩv, 0, 0). For time-accurate simulations,
the source terms for the turbulence equations are retained, but there are no source terms in the momentum equations.
In the low-Reynolds number k-ε model, the solution is extended to the wall instead of using a wall-function approach
(Chien, 1982). The source term for Chien's model is given as
(2)
In Chien's model is solved instead of ε. The variable is defined so that it obtains a zero value at
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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 529
the wall and the true dissipation can be expressed as
The equations for k and ε contain empirical coefficients. In this study, the following coefficients are applied,
(3)
where the turbulence Reynolds number is defined as
Pseudo-compressibility
The FINFLO RANS solver utilizes a structured multiblock grid. The code was initially developed for compressible
flows (Siikonen, 1990) and it has been extended to incompressible flows using a pseudo-compressibility method (Chorin,
1967). In the pseudo-compressibility approach the continuity equation is replaced by
(4)
In the actual implementation of the present method, all the derivatives ∂ρ/∂ p are replaced by a pseudo-
compressibility factor β2 (Rahman et al., 1997). In this way, the characteristic speeds reduce to simple expressions of
λ1,2=u±β. The flux calculation is based on a simplification of Roe's method (Roe, 1981). The implicit stage uses an
approximate factorization and a multigrid method is applied for the acceleration of convergence.
Discretization
A finite-volume technique is used for solving the equations. The differential equations are integrated over a
computational cell
(5)
where
(6)
and the summation is extended over the faces of the computational cell. In a rotating frame, i.e. for propeller
calculations, the functional form of the flux equations is similar to the case without rotation. The difference is that in a
rotating frame the motion of the cell faces is taken into account in the evaluation of convective velocities (Siikonen and
Pan, 1992).
The inviscid flux is calculated with the help of a rotation matrix, which transforms the dependent variables to a local
system of coordinates normal to the cell surface (Siikonen, 1994). The interface values are evaluated by a MUSCL-type
formula.
Solution Algorithm
For steady-state flow simulations, the discretized equations are integrated in time by applying the DDADI
factorization (Lombard et al. 1983). This is based on splitting the Jacobians of the flux terms. The resulting approximately
factored implicit scheme consists of a backward and a forward sweep in every coordinate direction. The sweeps are based
on first-order upwind differencing. In order to accelerate convergence, local time-stepping and a multigrid method are
also implemented in FINFLO (Siikonen et al., 1990). In time-accurate simulations for unsteady flows, the above-
mentioned pseudo-time integration is performed inside a physical time-step (Sánchez et al., 1999). More detailed
descriptions of FINFLO can be found in Siikonen et al. (1990), Siikonen & Pan (1992) and Pitkänen & Siikonen (1995).
NUMERICAL RESULTS
Geometry, Mesh and Boundary Conditions
The case selected for analysis is the ducted propeller presented in Kawakita (1992). The propeller has five blades, a
diameter of 0.221 m and a pitch ratio of 0.9741. It belongs to the NSMB Ka series. The duct is NSMB nozzle no. 19A.
The clearance at the propeller tip is 0.72% of the propeller diameter. LDV measurements were reported at a rate of
rotation of 25 rps corresponding to an advance coefficient of
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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 530
0.5. The thrust and torque were measured as well as the velocities downstream of the propeller. Computations were
performed under the same conditions. Two additional computations were performed for advance coefficients of 0.35 and
0.65. The axial inflow was varied keeping the propeller rotational speed constant, as in the tests. Only thrust and torque
measurements were available for comparison for these last computations.
The grid used in the computations has over one million cells, as shown in Table I. Special emphasis was put on
modeling the propeller blades and their near-wakes accurately. The only noticeable difference in geometry from the
ducted propeller model was that the hub of the computational grid was extended downstream of the propeller, as is the
practice of MARIN, whereas the experimental model has it extended upstream. Only the portion between two contiguous
blades has been used in the computations due to the periodicity of the solution. Figure 1 illustrates the grid shape on the
duct and propeller surfaces, and Figure 2 shows the topology.
Table I. Number of cells in the mesh
Propeller Duct Rest Total Grid
562,688 154,112 440,832 1,157,632
Figure 1. Grid on the surface of the ducted propeller.
Detail of grid construction on the duct.
Figure 2. Grid topology.
The topology was II-type around the propeller blades with over a half-million cells inside the duct, and O-type
around the duct. The grid has the inlet boundary modeled by a spherical sector located at more than three diameters from
the propeller center. The outlet boundary is a plane located at an axial location between 3 and 4 propeller diameters from
the propeller center. Fine grid spacings are used in the vicinity of the leading and trailing edges of the propeller blades in
the chordwise direction, and near the blade and tip in the radial direction. The minimum grid spacing in the
circumferential direction for the resolution of the boundary layer was such that the y+ parameter was found to be about 1–
1.5 over most of the blade, and 1 on the duct surface. A total of 19 blocks was used in order to distribute the computing
load between 8 processors.
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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 531
Figure 3. Convergence history of the overall lift coefficient.
The hub and blade surfaces of the propeller are rotating solid walls with boundary conditions enforcing the velocity
field to match the propeller rotational speed. The velocities at the duct surface are set to zero in order to satisfy the non-
slip boundary condition. The lateral surfaces adjacent to the propeller blades and duct have cyclic or periodic boundary
conditions. The block boundaries where two adjacent block surfaces are coincident are defined as connectivities. A
uniform flow condition is applied to the inlet and peripherical surfaces. The streamwise gradients of the flow variables are
set to zero at the outlet.
Convergence
The computations were performed on an SGI Origin 2000 machine. Eight processors were used. The computation
time was 13 seconds per iteration cycle. For the second grid level, the computation time was 1/8 times that of the first grid
level. A satisfactory convergence was obtained with a Courant number of 0.5 using two multigrid levels.
The convergence histories of the overall lift and drag coefficients are presented in Figs. 3 and 4. After 3000
iterations, the overall drag coefficient converged within 1% of the final value, and the overall lift coefficient within 0.5%.
Figure 5 shows a magnification of the convergence history for the overall lift coefficient.
Figure 4. Convergence history of the overall drag coefficient.
Figure 5. Convergence history of the overall lift coefficient. Magnification.
Forces and Pressures
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The k-ε turbulence model gave a good correlation of flow patterns and performance coefficients with measurements.
Table II shows that the performance coefficients were calculated for advance number 0.5 within 4.5% of the
measurements. In particular, the thrust coefficient for the propeller (KTP) was predicted very accurately. The difference of
about 4% from measurements in the prediction of duct thrust (KTD)

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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 532
made a total difference in the total thrust coefficient (KT) of less than 1%. The torque coefficient (KQ) was overpredicted
by about 4.5%. For other advance numbers, the differences in thrust were a little higher but reasonable, although the
torque was better predicted. Figure 6 compares the experimental performance coefficients to the calculations for three
advance numbers.
Table II. Experimental and calculated performance coefficients for J=0.5
Experiment (*) Calculations
1st level 2nd level
KTP 0.197 0.197 0.220
KTD 0.048 0.046 0.042
KT 0.245 0.243 0.263
KQ 0.0345 0.0361 0.0418
(*) as read from the test diagram in Kawakita (1992)
Figure 6. Comparison of experimental and calculated thrust and torque coefficients for several advance numbers.
The calculated pressure distribution on the ducted propeller surfaces is shown for the suction and the pressure side of
the propeller blades in Figures 7 and 8, respectively. A low-pressure area can be identified in Figure 7 at the suction side
of the propeller tip extending to the duct surface.
Figure 8 shows a large area of moderate negative pressures at the pressure side of the propeller blades. The duct
accelerates the inflow to the propeller, which results in a large extent of low pressures on the propeller blades compared
with a corresponding open propeller.
Figure 7. Distribution of pressure difference on the suction side of ducted propeller NSMB 19A.
Figure 8. Distribution of pressure difference on the pressure side of ducted propeller NSMB 19A.
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Velocities and Hydrodynamic Pitch
Figure 9 illustrates the circumferential variations of the velocity components downstream of the propeller plane at r/
R=0.5, 0.9, 1.0 and 1.05, and at x/R=0.65 (just behind the duct) for both experiments and calculations. The velocities are
non-dimensionalized with the axial inflow. The advance number is 0.5. Each velocity fluctuates with the blade frequency.
The computations show the same trends as the experiments reported in Figure 3 by Kawakita (1992).

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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 533
Figure 9 Calculated circumferential variations of velocity components downstream of the propeller at x/R=0.65 and r/R=0.5,
0.9, 1.0, and 1.05.
Figure 10b. Calculated velocity contours downstream of
the ducted propeller (J=0.50, x/R=0.65).
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Figure 10a. Velocity contours downstream of the ducted
propeller (J=0.50, x/R=0.65). (Kawakita, 1992).
Figures 10a and 10b provide a comparison of experimental and calculated velocity contours at the same axial
location. The location and shape of the calculated trailing vortex sheet is apparent from the figures and coincides with that
of the experiments. The agreement is good. Figures 11a and 11b compare the experimental and calculated velocity vectors,

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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 534
respectively. The agreement is good except for the fact that the grid is not completely aligned with the propeller trailing
wake, which results in some lack of circumferential grid resolution and consequently in a prediction of tip vortex weaker
than in the experiments.
Figure 11a. Tangential and radial velocity field downstream of the ducted propeller (J=0.50, x/R=0.65). (Kawakita, 1992).
Figure 11b. Calculated tangential and radial velocity field downstream of the ducted propeller (J=0.50, x/R=0.65).
Figure 12a. Velocity contours downstream of the ducted propeller (J=0.50, x/R=1.00). (Kawakita, 1992).
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Figure 12b. Calculated velocity contours downstream of the ducted propeller (J=0.50, x/R=1.00).
Figures 12a and 12b show the velocity contours at x/R=1.00. The agreement is not so good due to numerical
dissipation. It should be mentioned that a third-order upwind was used in the radial and circumferential directions and a
second-order in the axial one for the calculation of the convective fluxes. Probably, the use of a third-order upwind in the
axial

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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 535
direction instead of a second-order discretization would have improved the results.
Figure 13. Comparison of calculated and experimental distribution of hydrodynamic pitch in the trailing vortex wake at x/
R=0.65 for J=0.5.
Figures 8 and 9 in Kawakita (1992) show the experimental radial distribution of hydrodynamic pitch angle and of
hydrodynamic pitch of the trailing vortex wake at x/R=0.65 for J=0.5. In this reference, the hydrodynamic pitch angle of
the trailing vortex wake of the ducted propeller was calculated in the experiments using the averaged axial and tangential
velocities vx and vt as
(7)
It seems apparent that a small error was made when the above formula was applied: the circumferential mean
tangential velocity was introduced in the formula with a positive sign instead of the correct negative one. The error
resulted in a low pitch and low pitch angle that is not easy to notice. If we calculate the hydrodynamic pitch for the
computational results in the same way, i.e. giving a positive sign to the vt, the curve labeled “unconnected pitch” in
Figure 13 is obtained, which can be directly compared to Figure 9 in Kawakita (1992). The experimental data from this
reference have been also included in Figure 13. The agreement is very good. On the other hand, the computed
hydrodynamic pitch with the correct negative sign is presented also in Figure 13 with the label “corrected pitch.” The
same process can be repeated for the experimental results presented in Figure 8 by Kawakita (1992). It can be compared
with Figure 14, where the computed hydrodynamic pitch angle has been calculated with and without the correction to the
tangential velocity sign. The agreement with experiments is also very good. Only at the duct wake located at about r/
R=1.05 did differences appear, i.e. the strength of the duct wake is a little stronger in the calculations.
Figure 14. Comparison of calculated and experimental distribution of hydrodynamic pitch angle in the trailing vortex wake at
x/R=0.65 for J=0.5.
CONCLUSIONS
The incompressible viscous flow around a Ka series propeller with NSMB nozzle 19A has been simulated by solving
the RANS equations with the k-ε turbulence model. The FINFLO code was used for the calculations. The grid contained
over one million cells. Good correlation with experiments is obtained in terms of force coefficients and velocity
distributions in the wake at locations not far away from the duct. The thrust coefficient has been calculated without
noticeable error for the design advance number; however, the torque coefficient differs from measurements by 4.5%. For
other advance numbers, the differences in thrust were a little higher but reasonable, although the torque was better
predicted. Important features of the flow, like the hydrodynamic pitch angle of the propeller wake and the propeller wake
itself, were accurately predicted. The calculation reveals areas of low pressure at the propeller tip and duct. This
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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 536
information is useful for improving a ducted propeller design from the standpoint of cavitation. The results of the
computations show that RANS solvers are mature enough to provide valuable information to the designer.
ACKNOWLEDGEMENTS
This work was funded by the Technology Development Centre (TEKES) of Finland. The computing time was
provided by the Centre for Scientific Computing of Finland. The authors wish to thank Dr. Jaakko Pylkkänen for the
valuable help provided during this research.
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ICAS Congress, pp. 2023–2034, Stockholm, Sept. 1990. ICAS Paper 90–6.10.3.
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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 538
DISCUSSION
C.Dai
NSWCCD
This paper is another example of proliferation of Reynolds Averaged Navier-Stokes Solvers (RANS) applications in
analyzing propulsor performances. Today, RANS analysis of propulsor design is part of a standard design procedure in
most design organizations. RANS simulation gives the designers greater confidence in their prediction. It also enables the
designers to expand their design envelopes and/or explore new concepts that the designers may feel too risky to do in the
past. Furthermore, the simulation results can also serve as a guide for experimentation. This paper showed excellent
agreements on thrust coefficient and 4.5% difference on torque coefficient between computation and experiment data.
There are several possible causes, both numerical and experimental, for the discrepancy. One potential cause is the
difficulty associated with the computation of integral functional such as thrust and torque. Active research [1] is
undergoing to address the issues of accuracy improvement in computing integral quantities. The flow field computations
were mostly qualitative and it did capture the general pattern especially at near field of x/R=0.65. In general, the designers
are interested in the details of flow field to a less degree of accuracy as compared with thrust and torque. There are
situations that the flow field parameters are important in the overall propulsor design. Examples include designs for multi-
blade rows ducted propulsor, and management of vortical flow structures near the tip gap region.
Despite big strides have been made in RANS technologies; there are still issues that designers have to be aware of,
when they use RANS codes in their work. In general, there are three major areas of concerns:
(1) Discretization schemes and numerical algorithms for robustness and accuracy:
There are a wide variety of techniques available. Some of them have been developed to a level for
production use while others are still undergoing development. Various tradeoffs between speed and accuracy,
ease of use and robustness etc need to be assessed before it can be used in the production mode.
(2) Turbulence models and their relevance to the real simulation:
RANS simulation makes use of phenomenological models that had its origin of Reynolds decomposition
and Prandlt eddy diffusivity conjectures. They rely on the calibrations of experimental data in order for the
models to be effective. An appreciation of their limitation in terms of regimes of applications is required in
order to conduct simulation credibly.
(3) Grid structures and their sensitivities.
Propulsor is a complex geometric artifact. Blade surface modeling needs to be robust and accurate.
Furthermore, the grid clustering and spacing in order to obtain proper accuracy needs experiences and skills
of an experienced modeler.
It must be said that verifications and validations are a must before any use of RANS codes in the propulsor design
process. I would like to conclude by mentioning a number of topics that may be useful for the future applications of
RANS to propulsor design and analysis.
(1) High fidelity design optimization
The time seems mature to consider propulsor design optimization at RANS level. High performance
computing techniques for optimization using high fidelity physics-based models have advanced through the
rapid development of adjoint formulation. The adjoint approach greatly reduces the number of flow
simulations for computing sensitivities of design variables in the optimization process and makes the design
optimization feasible at the detailed design stage.
(2) Ease of use and implementation.
Seamless integration of data transmission between the design geometry and the surface representation for
RANS simulation is highly desirable. An adaptive and versatile grid system that can offer users more
flexibility in grid layout is definitely needed. Unstructured grid approach may seem to be the way of gridding
for the future.
(3) High order physical modeling.
Ability to predict flow transition is of great interest, in particular, for the scaling problem. Approaches are
also needed to address change of flow nature from one part of flow domain to another. For example, from
boundary layer adjacent to the blade surface to free vortical flow in the wake.
Use of RANS in propulsor design and analysis will continue to have a significant role in the future. It will
compliment experimentation
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SIMULATION OF INCOMPRESSIBLE VISCOUS FLOW AROUND A DUCTED PROPELLER USING A RANS EQUATION SOLVER 539
nicely and aid designers in exploring more design options through optimizations.
[1] Pierce, N.A. and Giles, M.B. “Adjoint Recovery of Super convergent Functional from PDE Approximations,” SIAM REVIEW, Vol. 42, No. 2, 2000,
pp. 247–264.
AUTHORS' REPLY
We would like to thank Dr. Dai for his valuable comments on the present concerns and future trends of RANS
applications to propulsor design and analysis. We agree with his views and many of the topics he has quoted are subject
to continuous research in our institutions.
He has mentioned three major areas that should be addressed when using RANS codes as part of the design work.
With respect to grid generation we would like to point out that care should be taken for the definition of the grid shape.
The designer should define the grid shape in such a way that regions with strong gradients of flow quantities are correctly
captured by concentrating enough number of cells in such areas. The grid that we have used in the computations is
structured in such a way there is a high concentration of cells in the propeller wake for x/R<0.7, which explains the good
correlation of flow patterns at x/R=0.65. On the other hand at x/R=1.0 the correlation is not so good since the grid is not
following the wake anymore at this location, and the size of the cells is relatively large. Another topics that should be
mentioned are the criteria of convergence and the boundary conditions. In this calculation we get relatively fast a drop by
three orders of magnitude in pressure residuals, but this does not guarantee the convergence of overall quantities like
thrust and torque. Boundary conditions that reproduce in a natural way the physics underlying the hydrodynamic problem
are key to solving it in a fast and accurate way.
DISCUSSION
K.Nakatake
Kyushu University, Japan
I am impressed by your huge CFD calculations. In the region behind the propeller hub, there is a white region of
velocity field. Could you calculate the propeller hub vortex by your CFD scheme?
AUTHORS' REPLY
We have modeled the propeller shaft following the practice at MARIN, i.e. the shaft is extended downstream of the
propeller/duct. The experiments at the Nagasaki experimental tank were performed with the shaft extended upstream of
the propeller. For this reason there is such white region in our results at axial locations downstream of the propeller. For
modeling the hub vortex we only have to build the computational grid with the shaft pointing downstream instead of
upstream, which does not mean any further complication. In fact, we have used grids for podded propulsors with O-O
topology around the pod, which allows to capture hub vortices (see Sanchez-Caja et al. 1999). However, we have not
investigated the details of hub vortices.
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