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PROPULSOR DESIGN USING CLEBSCH FORMULATION 284
Propulsor Design Using Clebsch Formulation
C.Dai, R.Miller (Naval Surface Warfare Center, Carderock Division, USA),
M.Zengeneh, C.Yiu (University College London, United Kingdom)
ABSTRACT
A three dimensional inverse design method is presented for the design of marine propulsors. This method makes use
of the Clebsch representation of the rotational part of the velocity field for modeling both the bound vorticity of blade
surfaces and the free stream vortcity. The blade shape is determined by imposing the flow tangency condition for a given
loading specification. This technique is demonstrated here for the design of a ducted pod propulsor operating in a uniform
stream and a ducted propulsor mounted on the tail of an axisymmetric body operating in a shear onset flow. For the case
of ducted propulsor design, both mixed and axial flow configurations are presented to demonstrate the use of mixed flow
concept for cavitation performance improvement.
INTRODUCTION
The problem of propulsor design has aroused considerable theoretical interest for over half a century. For example,
the lifting line [1] and lifting surface theories [2], [3] have advanced to a stage that they are routinely used in propulsor
design in the last three decades. Particularly, in the area of open screw propeller design, the classical approach of lifting
line and lifting surface in conjunction with the notions of thrust deduction and wake fraction has been proven to be a very
reliable approach to the propulsor design problem. Recently, there has been a strong interest in the internal or ducted
types of propulsor. The internal type of propulsor is mainly referred to as a waterjet propulsor. Both waterjet and ducted
types of propulsors were conceived and developed in the latter part of 19th century. The resurgence of interest in the
internal and ducted propulsors is mainly due to the fact that there is better understanding of both physical phenomena and
design issues related to those types of propulsors. For example, the hull and propulsor interaction can contribute
positively to the waterjet hydrodynamic performance [4]. The ducted propulsor has long been recognized for its ability to
improve cavitation performance and sustain higher loading near the tip region. Historically, the ducted propulsor design is
based on the principle of potential flow [5]. In the last decade, efforts have been made to couple Euler or Navier-Stokes
solver with a lifting surface code for the ducted propulsor designs [6], [7] and [8]. The procedure has been used very
successfully in the situation, which involves simple throughflow geometry, or the onset shear is weak. Another approach
that has been taken in the last 3 decades in the design of internal or ducted propulsors is the use of streamline curvature
methods for throughflow analysis and a semi-empirical method of meanline design for blades [9]. Despite the fact that the
streamline curvature method can handle the throughflow with the blading effect more efficiently than the Euler/lifting
surface approach, the blading design is relatively weak and it has to rely on experience and the experimental database.
With increased demand for high performance propulsor, a design method, based on a high order physical model,
which can account for multiple stages, shear flow and fully three-dimensional effect, is needed. An inverse design method
based on the idea of Clebsch decomposition of the velocity field has been successfully developed in [10] for uniform flow
and vortex free loading conditions. Subsequently, the method has been applied successfully to turbomachinery designs
[11]. It has also been used to demonstrate its potential for ducted propulsor design [12], [13]. The work to further develop
the inverse design method as part of code enhancement and verification processes is described in this paper. Several
improvements and new design capabilities were incorporated in the existing code so it can be more effectively used for
propulsor design. The new features include addition of three dimensional boundary layer calculations, arbitrary blade
planform layout for accommodation of rake and skew, and a new
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 285
mixing plane formulation to account for close spacing between stages. This paper will address the fundamental
foundation of the present approach and three design examples will be presented as illustrations. The theoretical part of the
presentation is to identify the basic concept and the key features of the method. The idea of using mixed flow for
cavitation improvement will also be demonstrated in one of the examples.
THEORETICAL FORMULATION
The usual method of expressing the velocity vector field is to express it as the gradient of a scalar potential plus the
curl of a vector potential, which results in the Helmholtz's decomposition. An alternative way is to define all potentials as
scalar variables and Clebsch in 1859 introduced such definition for the case of isentropic fluid flow as follow:
(1)
where is the potential part of and is the rotational part of All vector qualities are in bold with
underscores. It is interesting to observe that is determined uniquely by a set of potentials but not vice versa. This unique
property of the Clebsch representation enables modeling of different vorticity types for a propulsor flow. In this work, the
Clebsch variables are used to model bound and trailing vortices associated with blade loading and the onset shear effect.
The vorticity vector is defined by taking the curl of velocity and it can be written as
(2)
It is clear from (1) that the vorticity lines lie on the surfaces of constant λ and χ. Furthermore, it satisfies Kelvin's
theorem exactly by taking the divergence of (2). One can also observe from (2) that both λ and χ have to satisfy the
conditions that
(3)
and
In the propulsor flow problem, the Clebsch representation of the bound and the trailing vortices can be identified
with the loading and the geometric parameters of the blading design. One logical choice for λ and χ is the blade
circulation and the normal vector to the blade surface. The bound circulation around the blade section is related to the
circumferential averaged swirl velocity around the axis of rotation and the first Clebsch potential can be defined as
(5)
where B is the blade number and Γ is the blade circulation. The blade surface can be represented by the wrap angle
of the blade geometry and the second Clebsch potential can be defined as
(6)
where f is the wrap angle and x, r and θ are the cylindrical coordinates of the problem. The bound and the trailing
vortices can then be defined as
(7)
where δ(α) is the periodic delta function about α and it can be written as [14]
(8)
The free stream vorticity can also be modeled by selecting a different set of Clebsch potentials. The momentum
equation for an inviscid and incompressible fluid can be expressed as
(9)
where H is the total head, is the relative velocity and ρ is the fluid density. It can be seen from (9) that the
vorticity vector lies on the surface of Bernoulli surface. Therefore the first Clebsch potential to model the free stream
shear can be identified as the total head H. The second Clebsch potential requires a time scale representation in order to
make the vorticity definition. The drift function [15] provides such representation and is consistent with the overall
scheme of the formulation. The drift function can be interpreted, as the time required for a fluid particle to travel from a
reference position to reach a particular point in the flow field and it can written as
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 286
(10)
where is the distance along the stream surface from the reference position. Taking the cross product of (9) with
and making use of a vector identity, it can be shown that
(11)
Appropriate Clebsch potentials for both blade effect and free stream shear effect have been identified and can be
applied to the propulsor flow modeling. The velocity field has to satisfy continuity condition by taking the divergence of
(1) and setting to zero,
(12)
If the circumferential component of the velocity field is represented by generalized Fourier series, then one can write
(13)
where is the zero harmonic or the circumferential mean velocity component and vn*(r, x) is the complex periodic
components of velocity field. Overbar and star represent mean and periodic components respectively. The above Fourier
series representation of circumferential components can also be applied to the potential functions as depicted in (12). The
result is a system of equations that governs each harmonic component of the velocity field that must satisfy continuity.
The circumferential mean component will be examined first before the periodic components will be evaluated. The mean
velocity can be represented better by considering the axisymmetric stream function instead of the potential functions
because of the convenience in implementing blockage and thickness effect in the formulation. Furthermore, the stream
function satisfies continuity by definition and it explicitly relates to the vorticity in the flow field. By definition, the
axisymmetric stream function ψ can be written as
(14)
where ∇ is the Laplacian operator in the r-x plane, eθ is the unit vector in the circumferential direction,
2 and
are the circumferential averaged components of the blade and free stream shear vorticities. is simply the mean of (11)
and it can be written as
(15)
and
is the mean of (7).
(16)
since the mean of the periodic delta function is 1 [14]. The periodic components of the velocity field can be
evaluated by solving the Fourier components of λ and χ. The periodic component of the blade vorticity can be written as
(17)
where(δ(α)−1) is S'(α), the first order derivative of saw tooth function [14]. Therefore, the periodic components of
velocity due to blade vorticity that satisfies (17) can be written as
(18)
and from [14], S(α) can be expressed as
(19)
The periodic velocity components due to free stream shear can be represented by the Clebsch potentials H and τ as
follow:
(20)
where H* and τ* are the complex Fourier components of H and τ. It can be seen from (20) that the periodic velocity
of free stream shear consists of cross product terms and they can be expanded as
(21)
(18) and (21) presents the Clebsch representation of complex periodic velocity components induced by the blade
vorticity and the free stream vorticity. The Clebsch potentials as specified in (18) and (21) can be substitute directly into
(12). The resultant equation for the nth order of the complex Fourier component can be written as
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 287
(22)
The present code only models the circumferential mean of shear effect and fully three-dimensional shear effect will
be investigated in the future. Two auxiliary equations are required to solve for H and τ. It can be seen from (9) that the
total pressure convects along the relative velocity, i.e.
(23)
and from (3) that
(24)
Both H and τ are decomposed into the circumferential mean and periodic components in (23) and (24) and the
Fourier components of both H and τ can be calculated from the resultant relative velocity. An additional condition is
required in order to solve for the blade geometry f for a given loading r The flow tangency condition is required and it
can be expressed as
(25)
where and and are the upper and lower surface velocities on the blade. (14), (22),
(23), (24) and (25) constitute the complete set of equations that are required to be solved for the design calculation of a
propulsor system. The boundary conditions for (14) are very straightforward since the inflow condition is specified and
the solid boundaries are simply lines of constant ψ. The boundary conditions for (22) are no penetration condition on the
solid boundary. For the multi-stage propulsor design problem, * is solved for individual blade row where inflow
boundary condition for downstream blade row is specified by the solution from the upstream blade row. The periodic
components are assumed to vanish far downstream. (22), (24) and (25) are hyperbolic set of equations and initial
conditions are required to start the integration. The equations need to be solved in an iterative manner subject to the blade
loading and boundary conditions specified. Another advantage of the present formulation is that the Kutta condition at the
blade trailing edge is satisfied exactly and the trailing wake geometry can be computed as part of the overall solution. The
pressure jump across the blade surface can be shown [10] to be
(26)
It is obvious from the above equation that the kutta condition can be imposed explicitly by specifying to be
zero at the trailing edge. Furthermore, the geometry of the trailing vortex sheet due to the force vortex loading distribution
can be computed as part of the overall solution. The set of equations and the boundary conditions are cast in body fitted
coordinates and solved by finite difference technique with a multigrid algorithm for quicker convergence.
ALGORITHM DESCRIPTIONS
The present code was originally developed at University College London (UCL). It was verified and further
enhanced at Naval Surface Warfare Center, Carderock Division (NSWCCD) so that it can be used for practical propulsor
design. The code can perform inverse blading design for a given body and duct geometry. Upstream inflow velocity and
static pressure distributions must be prescribed. The blade loading is specified by prescribing spanwise swirl velocity
distributions at the leading and trailing edges of the blade rows. The code has two modes of blade design with a given
duct and body geometry. The first mode is to design the blades according to the given loading and duct geometry. In this
case, the mass flow rate is a calculated parameter. The second mode is to design for a given mass flow rate and the load
distribution will be scaled accordingly to achieve the required mass flow rate for a fixed duct and body geometry. The
second mode also allows variation of duct geometry by increasing or decreasing duct radius for a fixed load distribution.
A generalized load distribution algorithm has been incorporated so that different chordwise load distributions can be
specified along different streamlines. The new capability enables fine-tuning in blade design, which is particularly useful
for blade section design near the endwall
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 288
region. In the original code, the duct can also be designed with an inverse mode by specifying the target surface velocity
distribution on the duct inner surface and the duct lower contour shape will be calculated. The solution may not always
guarantee a unique solution. A direct approach is taken in this work for the duct design. The lower surface of the duct was
determined through design iteration using the streamline curvature method for the throughflow calculations. The major
advantage of the streamline curvature approach is relative quick evaluation of the impact of duct shape on the
throughflow solution before the detailed blade design. A thickness form is added to the lower duct surface to define the
duct shape. The duct geometry can be perturbed during the blade design cycle to affect the duct pressure distribution. The
approach is extremely effective and robust for practical design applications. A brief discussion on the ducted propulsor
design will be given in the next section and followed by three design examples.
DESIGN CONSIDERATIONS
An internal or ducted propulsor consists of a rotating impeller operating inside a casing or duct with stationary vanes
for swirl cancellation so that loss of rotational energy in the slip stream can be reduced or eliminated. Stationary vanes
can also serve the purpose of supporting the duct. In contrast to an internal propulsor, such as a waterjet, which is inside
the hull, a ducted propulsor is usually external to the hull. An integrated hull/propulsor design can also lead to a potential
improvement in powering performance by making use of positive interaction effect between the hull and the propulsor.
The intent here is not to give a detailed account of propulsor design, but to highlight the important design considerations
in the process. In the ducted propulsor design, the domain of interest is the region of the hull where the propulsor will be
located. The shaping of the hull and the location of the propulsor relative to the hull are the first considerations in the
design cycle. Once they are settled, the flow path inside the propulsor needs to be defined. This is an extremely important
step in the whole design process. For an axisymmetric vehicle, the flow path will be defined by specifying the lower duct
surface and the shape of the hull where the rotor and stators are mounted. The “meridional shaping” of the internal
passage of a ducted propulsor defines the “environment” where the rotor operate. The next step is to perform a tradeoff
study in terms of different performance requirements. Mass flow rate and rpm are the first order design parameters that
will dictate the overall size and performance of the unit. Once the major size, design conditions, passage geometry are
decided through an iteration process, the final step will be the blade design. Two major global design decisions need to be
made in the beginning of the blade design process. They are the planform of the blade layout, which includes the skew,
rake, chord and thickness distributions and the load distribution. The specification of load distribution is critical in the
success of propulsor design. The prescribed load distribution should be such that all adverse effects i.e. flow separation
and cavitation can be avoided or delayed. Obviously, one will seek a load distribution that gives the most efficient blade
performance subject to the constraints. The issue can be more complex if the acoustic performance becomes an integral
part of the consideration. Flexibility in the load distribution is definitely a requirement for any advanced high
performance propulsor design.
There are other issues in the design that will affect loading prescriptions. For example, a mixed flow passage
geometry will induce an additional secondary flow component due to streamline curvature. The vorticity generated by the
boundary layer on the lower surface of the duct interact with the tip of the blade and can significantly affect the loading
characteristics near the tip region. These physical phenomena away from the surfaces may have a significant impact on
the blade design process. Presently, Reynolds Averaged Navier-Stokes Solver (RANS) can predict the above-mentioned
effect with fairly good results. It is still not very time and cost effective to incorporate RANS in the design iteration.
RANS is used only to analyze selected design candidates. Use of RANS as analysis tool in propulsor design has been
proven to be very effective. However, it is highly desirable to incorporate in the inverse design mode as much physical
phenomena as possible. The benefit is that the candidate design for RANS analysis should be very close to the best-
compromised design. The Clebsch approach provides a base for incorporation of other effects during the design
calculation. The secondary flow is intrinsically captured using the fully three-dimensional formulation. Theoretically, the
boundary layer developed on the lower surface of the duct can also be modeled in the
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 289
circumferential averaged sense. The tip gap effect can also be simulated using this model. The planform geometry can
also affect the loading distribution and if properly chosen, can help the loading distribution.
A word on design optimization may be appropriate to conclude this section. When one thinks about design, one must
consider the way that will give the “best design” within the given resources and constraints. The Clebsch based inverse
design method provides a theoretical background for the development of a toolbox for practical propulsor inverse design.
The ultimate goal is to find shapes, which perform “optimally” during the mission of the vehicle. This is truly a multi-
disciplinary optimization (MDO) problem. The propulsor not only has to satisfy hydrodynamic requirements, but also
structural and sometimes acoustic requirements. From the total ship system level, the propulsor also has to be compatible
with the power plant and the ship layout. Furthermore, the effect of propulsor on stability and maneuvering also needs to
be considered in the design cycle. It was noted earlier that there is strong interaction effect between propulsor and hull. It
is imperative that both propulsor and hull be considered as an integrated part of each other during the design process. This
makes the design, not only multi-disciplinary, but also multi-component. Despite major advances made in optimization
and computation techniques, human experience and knowledge cannot be overlooked. Optimization tools should be used
in the context of assisting designers in making design decisions during the design cycles. Sometimes they may even
reveal areas that the designer never considers before. Yet, a proper framework needs to be established to make
optimization part of a toolbox that complements designer's knowledge.
DESIGN EXAMPLE 1—DUCTED POD PROPULSOR
A ducted pod propulsor was designed by the use of inverse design technique. A grid structure of the computational
domain is shown in Figure 1. A grid sensitivity study was performed to ensure the grid size used in this computation is
sufficient for the convergence of the solution. It was found that thrust, torque and the maximum difference in wrap angle
are all within 0.03% by reducing the grid size by half and a quarter in both x and r directions respectively. The wrap angle
is defined as the angular position of the camber surface. The ducted pod is a post swirl propulsor operating in the uniform
stream with axial velocity equal to unity. A post swirl system is one with a rotating rotor forward of the stationary vanes.
All geometric dimensions are normalized by the inlet diameter of the duct. All velocities are normalized by the free
stream velocity. The pressure is non-dimensionalized by the free stream dynamic head. The code can design the whole
propulsor, which includes the duct, the hub and all the blade rows simultaneously. No blockage or thickness effects are
modeled in this example. No empirical loss models are incorporated at this time. The effect of blade load distribution was
illustrated in this example. The design was constrained by the overall size of the unit and the hub shape. The flow path is
very much restricted by the imposed geometric constraints. The result is a highly accelerated flow path that results in a jet
velocity ratio of approximately 1.8. The stacking positions for both rotor and stators are at the leading edge of the blades.
Only the zero harmonic or the circumferential averaged design calculations were performed to evaluate different blade
load distributions. The zero harmonic solution is equivalent to the infinite blade solution where the blade to blade effect is
ignored. The fully three dimensional blade design will be performed once the load distribution is selected. A free vortex
(constant r distribution along the span) loading distribution was specified initially as a baseline. The net pressure
distribution on the blade surface and the pressure distributions on both suction and pressure sides are shown in Figures 2,
3 and 4. It seems that the load near the rotor hub region is too high. The absolute tangential velocity is even greater than
the rotational speed of the rotor at the rotor hub.
Figure 1 Grid Structure for the Pod
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 290
This is undesirable not only for the possibility of flow separation, also for the potential of early cavitation on the suction
face near the hub. The r distribution was revised so that the hub was unloaded. In the second calculation, a force
vortex load distribution with a linear distributed r was prescribed. The hub r was set to two thirds of the tip r
The resultant pressure distributions are shown in Figures 5, 6 and 7.
Figure 2 Net Pressure Distribution with Free Vortex Figure 3 Pressure Distribution with Free Vortex Loading
Loading for the Pressure Side
Figure 5 Net Pressure Distribution with Force Vortex
Figure 4 Pressure Distribution with Free Vortex Loading
Loading
for the Suction Side
Figure 7 Pressure Distribution with Force Vortex
Figure 6 Pressure Distribution with Force Vortex
Loading for the Suction Side
Loading for the Pressure Side
The effect of hub unloading is clear by comparing the pressure distributions depicted in Figures 2 to 7. The negative
pressure peak and the adverse pressure gradient near the rotor root region have been greatly reduced. The effect on
powering is very small. The force vortex loading resulted in less than 1 percent reduction in propulsive efficiency as
compared that with free vortex loading. The force vortex load distributions for the entire blade is shown in Figure 8.
Figure 8 Blade Load Distributions for Rotor and Stator
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 291
The blade geometry was updated iteratively during the velocity computation cycles until convergence. The
convergence criterion was set to be the maximum relative change in wrap angle less than 0.05%. The convergence history
for this computation is shown in Figure 9. It can be seen from Figure 9 that the blade shapes converge very rapidly in the
first 4 iterations.
The full three-dimensional computation was performed for different numbers of harmonics to ensure convergence.
16 harmonics were computed for less than 0.5% relative change in averaged wrap angles. Pictures of the blade shapes for
the rotor and the stator are shown in Figures 10 and 11. The red is the mean and the blue is the three dimensional designed
blade shapes. A maximum change of 7% in wrap angel between the circumferential mean and the three dimensional
solution was noted.
Figure 10 Rotor Blade Shape
Figure 9 Blade Shape Convergence History
Figure 11 Stator Blade Shape
Figure 12 Pressure Distributions for Duct and Hub
The major differences between the circumferential mean and the three dimensional solution seems to occur near the
trailing edge of both rotor and stator.
The pressure distributions for the duct and the hub are shown in Figure 12. The net thrust contributions from the duct
and hub can be computed by integrating the pressure distributions. It is interesting to note that the contributions from the
hub and the duct lower surface very much cancel each other and the duct upper surface contributes approximately 16% of
the rotor thrust. The vanes also contribute 18% of the rotor thrust. The next two design examples show the effective use of
mixed flow concept for cavitation improvement if manipulation of loading is not adequate for desired improvement in
cavitation.
DESIGN EXAMPLE 2—AXIAL PRESWIRL
A preswirl propulsor is defined as a ducted propulsor with a set of preswirl stators in front of the rotor. The preswirl
stators ahead of
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 292
the rotor blades induce a swirl distribution as the onset loading to the rotor blades. This can effectively cancel the swirl
produced by the rotor and reduces the rotational energy loss in the slipstream. The penalty paid is the additional drag due
to negative thrust that the stators produce. In this and the next example, a preswirl propulsor is designed for a given
axisymmetric body. The stern for this example is a tapered cone and the flow path is straight conical. Despite the fact that
the flow is not strictly axial, this case is called axial preswirl. The incoming velocity at the inflow plane is not uniform
because of the boundary layer developed on the surface of the body. The boundary layer thickness can be of the order of
the duct inlet diameter because of rapid pressure recovery near the stern region. A notional body with a tapered stern is
defined for this design example. The length scale for this and the next design example is the radius of the body. The
velocity scale is the vehicle speed. Pressure is normalized by the free stream dynamic head. All design calculations
converged according to the criterion mentioned in the last example. The grid structure for this design computation is
shown in Figure 13.
Figure 13 Grid Structure for the Axial Preswirl
The inflow meridional velocity distribution is shown in Figures 14. There is no upstream tangential velocity
component at the inflow plane. The loading distributions for both the rotor and stator are shown in Figure 15. Notice that
the blades are approximately 70% more loaded at the tip as compared with that at the hub. The resultant pressure
distributions on suction and pressure sides and the net pressure distribution are shown in Figures 16, 17 and 18.
Figure 14 Inflow Meridional Velocity Distribution
Figure 15 Loading Distributions for the rotor and stator
Figure 16 Net Pressure Distribution for Axial Preswirl Figure 17 Pressure Side Pressure Distribution for Axial
Preswirl
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 293
The pressure distributions for the duct and the body are shown in Figure 19. Lowest pressure occurred on the suction
face of the rotor blade near the tip region. This design constitutes the baseline design and improvements are sought for
cavitation performance. Further improvements can be made to the axial design by further refinement in load distributions
and relaxing the design requirements. The intent here is not to optimize the present design, but to demonstrate the idea of
mixed flow in propulsor design for cavitation performance improvement under the same design requirements. Figure 19
shows the pressure distributions on the duct upper and lower surfaces and on the stern surface. The duct slightly
accelerates the flow and produces a net thrust equivalent to approximately 50% of the rotor thrust. The stators produce a
negative thrust approximately equal to 10% of the rotor thrust and part of the stern included in the computation produces
a drag of approximately 18% of the rotor thrust.
Figure 18 Suction Side Pressure Distribution for Axial
Preswirl
Figure 19 Pressure Distributions for the Duct and the
Body for the Axial Preswirl
Pictures of the rotor and stator blades are shown in Figures 20 and 21. The red and blue layouts represent the
circumferential mean and the full three dimensional design solutions. The three dimensional correction to the mean
solution tends to increase the pitch at the tip and to reduce the pitch near the hub for the rotor blade. The three
dimensional solution tends to reduce the camber and increase the pitch along the entire span.
Figure 20 Pictorial View of Rotor Blade for the Axial Figure 21 Pictorial View of Stator Blade for the Axial
Preswirl Preswirl
The next example is to explore the concept of mixed flow for improving cavitation performance and compare the
design results with the axial preswirl.
DESIGN EXAMPLE 3—MIXED FLOW PRESWIRL
The notion of mixed flow is not new. Mixed flow turbomachinery has been designed and used in various
applications for many years. Turbomachines can be categorized into three
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 294
major configurations and they are the axial flow, mixed flow and radial flow machines. The selection of different
configurations is mainly dictated by the operation requirements. For example, an axial machine is usually characterized
by high mass flow rate and low head. On the contrary, a radial flow machine is applicable for high head and low mass
flow rate application. The mixed flow configuration contains features from both axial and radial and it occupies the
middle of this design spectrum. Most of the marine propulsors can be characterized as axial flow machine. Use of the
mixed flow unit provides the flexibility in the design space to address the load limits of axial flow machine. The main
idea can be illustrated from the Euler's turbomachinery equation, which can be written in the differential form as follow:
(27)
where ω is the rotational speed. If one assumes that the total head across the blades for both axial and mixed flow
units are almost the same, then one can write
(28)
The superscripts a and m represents axial and mixed flow. Consider the change along a differential distance δs on the
stream surface, (28) becomes
(29)
For the moment, the rotational speed ratio is assumed to be unity. For an axial configuration, δra/δs is zero or in the
previous design example, it is negative. The mixed flow configuration has a positive δrm/δs and it can be concluded by
inspection of (29) that loading for the mixed flow is less as compared with that for the axial. (29) enables one to study the
effect of differences in rotational speeds between the axial and mixed flow on the relative changes in loadings and the
geometry from an axial configuration to a mixed flow configuration. A notional mixed flow propulsor design is presented
here to illustrate the use the present design code for such propulsor design and to show the potential gains in performance
as compared with that for an axial unit. The body configuration and the inflow conditions are the same as the previous
axial preswirl example. The stern shape has been modified to accommodate the mixed flow design. As it was mentioned,
the flow path design is an important part of the entire design process. It requires design iterations between flow path
design, and the blade design. In this case, the flow path design is mainly the design of the stern and the duct. This
example mainly demonstrates the mixed flow concept and no attempt is made to “optimize” the entire stern/propulsor
design. The computational grid for the design computation is shown in Figure 22.
Figure 22 Grid Structure for The Mixed Flow Preswirl
The mass flow rate and the rotational speed for the mixed flow design is same as that for the axial design. The load
distributions for the stator and the rotor are shown in Figure 23. The cavitation inception depth is improved by 50%
relative to that for the axial preswirl. The effect of mixed flow enables load reduction relative to the axial preswirl and
hence an improvement in cavitation performance.
Figure 23 Loading Distributions for Mixed Flow Propulsor Stator and Rotor
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 295
The computed pressure distributions on the pressure and suction surfaces of the blades and the net pressure
distributions are shown in Figures 24, 25 and 26
Figure 25 Pressure Side Pressure Distribution for Mixed
Figure 24 Net Pressure Distributions for Mixed Flow
Flow Preswirl
Preswirl
The pressure distributions for the duct and the stern are shown in Figure 27.
Figure 26 Suction Side Pressure Distribution for Mixed
Flow Preswirl Figure 27 Pressure Distributions for the Duct and Stern
The duct and the stern axial forces as a percentage of rotor thrust for the mixed flow preswirl is different from that
for the axial preswirl. The net force acting on the duct is almost zero. The part of the stern included in the computation
produces a positive thrust of approximately 25% of the rotor thrust and the stators produces a negative thrust of
approximately 25% of the rotor thrust. Pictures of the blade shapes are shown in Figures 28 and 29. The red and blue
layouts stand for the circumferential mean and the full three dimensional design solutions.
Figure 28 Pictorial View of Mixed Flow Preswirl Rotor Blade
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 296
Figure 29 Pictorial View of Mixed Flow Preswirl Stator Blade
The circumferential mean and the three dimensional solutions are surprisely close. There are some minor differences
in wrap angles near the tip and hub region. The two solutions agree with each other on most parts of the blade surfaces.
Computation was also performed for a rotational speed 30% less than the baseline. An improvement of 12% in cavitation
inception depth relative to the axial preswirl was obtained. Mixed flow concept as applied to propulsor offers a way of
improving cavitation performance. There are other issues that may require further study before mixed flow concept can be
fully developed. Three-dimensional effects are more prominent in mixed flow design as compared with that for axial flow
design. Flow passage geometry becomes more important in mixed flow design and the blade load prescriptions may
require new set of guidelines as compared with that for axial design.
CONCLUSIONS
A new approach to the three dimensional blade designs for marine propulsors has been presented. Traditionally, the
propulsor design relies on the use of experimental data and experience of a designer to compliment the use of analytical
tools. The present approach is of no exception. Experience and test data continue to play an important role. With the
advances in computational fluid dynamics, Reynolds Averaged Navier-Stokes solver (RANS) starts to play an important
role in the design cycle. Gradually, it may even replace some of the experiments for design validations. Presently, RANS
still has its own limitations in terms of turnaround time, gridding issues and turbulence modeling. It is extremely
important that a designer understand the limitations of the tools in terms of physical models and the numerical accuracy.
In general, design codes are much faster running as compared to RANS codes and they are easier to couple with
algorithms for design optimizations. Therefore it is highly desirable to have a design code based on a physical model that
can offer three-dimensional modeling and less assumptions. In the case of propulsor design, the ability to model free
stream shear and its interactions with the blading is important, the effect of multiple blade rows on each other and their
interactions with nearby boundaries such as duct and body are equally important. The use of Clebsch potentials in
modeling bound vorticity of blades and the free stream vorticity proves to be an effective way of approaching the
propulsor design problem. It addresses the consistency of a single formulation for the free stream shear and multi-stage
blade-rows interactions problem. The code can further be enhanced to capture additional flow physics such as distortion
of Bernoulli surfaces and tip gap flow. It also offers relatively quick turnaround time so it can be used for design
optimization problems. Three design examples are given in this paper. The pod propulsor design is for a heavily loaded
blade design. The effect of different loading distributions on blade design was demonstrated in the pod example. A
notional mixed flow preswirl propulsor is designed to demonstrate the use of Clebsch method. Comparisons were made to
a baseline axial preswirl design. Significant improvement in cavitation performance can be obtained. The mixed flow
concept offers opportunity for performance gains but it also poses a more difficult design problem. The tortuous flow path
as compared with that for an axial unit offers great design challenges to a designer. The stern/propulsor interactions can
be more complex. The effect of secondary flow on blade design is more important. The duct lower surface boundary layer
development is quite different as compared with that for an axial flow. Its interaction with the rotor tip gap and effect of
flow passage geometry on the tip gap physics is still unclear for mixed flow propulsor design. Consideration of propulsor
design can no longer be considered as a separate entity in the design of any vessel. The propulsor has to be part of a total
system in ship design where commonality in both design and performance parameters with other subsystems must be
identified. With more stringent design requirements for the future ship propulsors, it is
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PROPULSOR DESIGN USING CLEBSCH FORMULATION 297
imperative to use optimization at both system and detailed design levels.
ACKNOWLEDGEMENT:
The work described in this paper was performed by the Propulsion and Fuild Systems Department of the
Hydromechanics Directorate, Carderock Division, Naval Surface Warfare Center. The work was sponsored by the Office
of Naval Research, Mechanics & Energy Conversion S&T Division (ONR 333)], under the Advanced Propulsors Task of
the FY00 Submarine HM&E Technology Program (PE0602121N). Dr. Peter Majumdar initiated the project. Dr. Edwin
Rood has provided the support for the both inverse design technique development and the mixed flow concept study. The
authors would like to thank both Dr. Majumdar and Dr. Rood for their interests and support during the course of this
investigation.
REFERENCES:
1. Lerbs, H.W. “Moderately Loaded propellers with a Finite Number of Blades and an Arbitrary Distribution of Circulation”, Trans. SNAME, Vol. 60.
pp. 73–123, 1952.
2. Brockett, T.E. “Lifting Surface Hydrodynamics for Design of Rotating Blades”, Proceedings of the SNAME Propeller '81 Symposium, pp. 357–378.
Virginia Beach, VA., 1981.
3. Greeley, D.S. and Kerwin, J.E. “Numerical Method for Propeller Design and Analysis in Steady Flow”, Trans. SNAME, Vol. 90, pp415–453, 1982.
4. McMahon, J.F., et al. “VMP Waterjet Test Results”, Naval Surface Warfare Center, Carderock Division, Hydromechanics Directorate Research and
Development Report, NSWCCD-50-TR-1999/015, April 1999.
5. Morgan, W.B. “Theory of Annular Airfoil and Ducted Propeller” 4th Symposium on Naval Hydrodynamics, pp. 151–197, Washington, D.C. 1962.
6. Dai, C.M., Gorski, J.J. and Haussling H.J. “Computation of an Integrated Ducted Propulor/Stern Performance in Axisymmetric Flow”, Proceedings of
the SNAME Propeller/Shafting '91 Symposium, pp. 14.1–14.12, Virginia Beach, VA., 1991
7. Kerwin, J.E., Keenan, D.P., Black, S.D. and Diggs, J.G. “A coupled Viscous/Potential Flow Design Method for Wake-Adapted, Multi-Stage, Ducted
Propulsors Using Generalized Geometry”, Trans. SNAME, Vol. 102, pp2.1–2.28, 1994.
8. Renick, D.H. “An Analysis Procedure for Advanced Propulsor Design”, Master of Science Thesis, Massachusetts Institute of Technology, May 1999.
9. McBride, M.W. “The Design and Analysis of Turbomachinery In an Incompressible, Steady Flow Using the Streamline Curvature Method”,
Technical Memorandum TM 79– 33, Applied Research Laboratory, Pennsylvannia State University, February, 1979.
10. Tan, C.S., Hawthorne, W. R, McCune, J.E. and Wang, C. “Theory of Blade Design for Large Deflections”, Trans. ASME J. of Engineering for Gas
Turbines and Power, Vol. 106, pp. 354–365, 1984.
11. Zangeneh, M., Goto, M., and Takemura, T. “ Suppression of Secondary Flows in a Mixed Flow Pump Impeller by Application of 3D Inverse Design
Method. Part 1: Design and Numerical Validation”. Trans. ASME, J. of Turbomachinery, Vol. 118, pp. 536–543, 1996.
12. Zangeneh, M. and Roddis, M.E. “A Three-Dimensional Method for the Inverse Design of Marine Ducted Propeller Blding”, Proceedings of the
SNAME Propeller/Shafting '94 Symposium, pp. 7.1–7.11, Virginia Beach, VA., 1994.
13. Yiu, K.F.C. and Zangeneh, M. “On the Simultaneous Design of Blade and Duct Geometry of Marine Ducted Propulsors”, Journal of Ship Research,
Vol. 42, Number 4, pp. 274–296, 1998.
14. Lighthill, M.J. “An Introduction to Fourier Analysis and Generalised Functions”, Cambridge University Press, Cambridge, 1967.
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PROPULSOR DESIGN USING CLEBSCH FORMULATION
15. Lighthill, M.J. “Drift”, J. Fluid Mech. Vol. 1, pp. 31–53 1956.
298

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PROPULSOR DESIGN USING CLEBSCH FORMULATION 299
DISCUSSION
J.E.Kerwin
Massachusetss Institute of Technology
USA
The authors have provided an extremely interesting and valuable paper on the topic of propulsor hydrodynamics.
While the Clebsch formulation has been applied to a limited extent to the field of turbomachinery since the 1980's, it
represents a new and unfamiliar approach to many of us in the field of marine propulsor design.
Regardless of the specific hydrodynamic formulation which is used, I believe that it is essential to provide designers
with computational tools that will enable them to evaluate propulsor alternatives as rapidly as possible. The problem of
developing a truly optimum design for a single, open propeller is hard enough, but one now has to deal with unusual,
multiple blade-row, ducted propulsors of the types illustrated in the paper. In my opinion, the focus of the author's
research is therefore well directed.
The authors are aware of parallel research that I have been involved in, as indicated by their references [7] and [8].
Our objective has been to develop a fast and versatile design/analysis method by combining a lifting-surface
representation of the blades with either an axisymmetric RANS or Euler solver. The authors characterize this approach as
successful “in the situation which involves simple throughflow geometry, or the onset shear is weak”. That comment is
certainly valid in the case of earlier lifting-surface codes that were limited to cylindrical flow geometry, and for which no
provision was made for the interaction of the propulsor with a vortical inflow. However, neither of these limitations apply
to the methods cited in References [7] and [8]. Could the authors comment on this?
While the Clebsch decomposition results in a very different set of equations to be solved, I am curious as to whether
or not this approach is actually different from our coupled Lifting-Surface/Euler scheme [8] in terms of underlying flow
physics. In either case, the blade forces are introduced into the throughflow solver by means of the swirl velocity, as
indicated in the authors Equation (5). In the author's case, the fact that the blade forces are concentrated at discrete
angular locations is handled by a Fourier representation in the angular direction, while in our method, it is handled
explicitly in the lifting-surface computation.
While the latter approach adds complexity to the process, the local velocities on the blades are computed directly. I
note that the author's report that 16 harmonics in the Fourier decomposition of the blade flow is sufficient for convergence
for the cases presented. However, these were all for ducted propulsors with zero tip gap, where blade-to-blade variations
might be expected to be small. Have the authors examined the convergence of their method for an open propeller, where
the effect of blade number is greater? I also could not find the blade number reported for the author's examples. It would
be useful if this information could be provided.
One possibly fundamental difference between the methods is that we make the assumption at the outset that the
vortical interaction between the propulsor and an axisymmetric inflow remains axisymmetric, while the author's method
can treat the fully three-dimensional interactive problem through a Fourier representation in the angular coordinate.
I have been concerned about whether this assumption is justifiable, and was therefore very interested in the findings
of Kinnas and Choi, presented at this same conference. They developed a special unsteady, 3-D Euler solver, which was
coupled with a propeller lifting-surface code. While the latter code has the limitation of constant radius flow geometry, it
is suitable for testing the difference between the effective wake predicted by the steady version of their Euler solver and
the time-average of the results obtained with their unsteady solver. Such a comparison is shown in Section 5.5 of their
paper, with the conclusion that the differences between the two methods are extremely small.
The authors have presented results for an extremely interesting variety of propulsor types. I would be very interested
in making comparative calculations, and believe that others might have the same interest. Would the authors consider
posting the geometries, circulation distributions and pressure distributions on the web to enable quantitative comparisons?
AUTHOR'S REPLY
We want to thank Professor Kerwin and Dr. Sanchez-Caja few their cements. We would like to start by answering
Professor Kerwin's questions. Propulsor design is indeed a complex task that involves many performance requirements
and constraints. Different propulsor types and their interactions with bull further complicate the design problem. We
totally agree with Professor Kerwin's comment that design foods need to be fast enough so the designers can evaluate
different propulsar types and explore design space in an efficient meaner. The computational model based on the Clebsch
Formulation is extremely fast because of its numerical formulation of decomposing a three-dimensional problem into a
series of two-dimensional calculations.
Use of potential flow models in propulsor design requires the knowledge of effective wake and thrust deductions.
The use of either axisymmetric RAMS or Eider Salvers with a Lifting Surface Code that Professor Kerwin and his
colleagues have developed represents a hybrid approach in solving the give wake/thrust deduction problem. The
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propulsor is modeled as an actuator disk in the axisymmetric part of the computation. An iterative process is required to
relate the effective inflow to the design circulation. The procedure has been demonstrated to be very effective to modeling
flow passage that free stream vorticity is relatively weak and is mainly connected by the background potential flow. In the
situation where the flow passage may have important effect of redistributing the onset

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PROPULSOR DESIGN USING CLEBSCH FORMULATION 300
vorticity such that the vorticity may not be small in certain region of the flow passage. The small shear assumption may
not be valid and blading design has to take into account the distortion of the free stream vorticity due to blade deflection.
The Clebsch formulation provides a theoretical framework that is not restricted to the weak shear assumption.
In principle, the Clebscb formulation represents a design theory based on inviscid rotational theory model. It can be
degenerated to a special case that is equivalent to the Euder/Lifting Surface Model by only accounting for the shear in the
zero harmonic solution. We have not performed a simulation for an open propeller and are planning to perform such
simulation in the future. The convergence properties for the open propeller will definitely be evaluated.
In regard to Dr. Sanchez-Caja's comment on the large hub design. The Clebsch formulation does not make use of the
image model that potential flow models have to use. The use of angular momentum as design variable does not impose
any constraints on the large hub boundary. The challenge for the large hub design is merely the question of prescribing
good loading distribution for such design application.
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