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OCR for page 991
24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13, July 2002
Computational Design Optimization Using RANS
J.C. Newmar~ ID, R. Pankajakshar~, D.~. Whitfield, and L.K. Taylor
(Computational Simulation arid Design Center, Mississippi State University, USA)
ABSTRACT
Sensitivity analysis and computational design
capabilities have been developed and incorporated into
a Reynolds Averaged Navier-Stokes code referred to as
UNCLE. Sensitivity analysis techniques are reviewed
and a novel method for computing derivatives of real
functions using complex variable numerical
differentiation is presented. Three exploratory
hydrodynamic optimization examples are performed to
access the computational design methodology. These
optimizations include the redesign of a modern marine
propeller for improved cavitation performance, redesign
of a bow bulb on a surface combatant to reduce
downstream vortices, and the redesign of a rudder in the
wake of a propeller to reduce outboard surface
cavitation.
INTRODUCTION
Through relentless experimental and computational
field simulation (CFS) studies, resourceful design
engineers have produced near optimal (marine,
aeronautical, automotive, etc.) configurations. To
further improve these designs, where the margin for
improvement is small, designers will require additional
information such as sensitivity derivatives. Sensitivity
derivatives, which provide a measure of how the system
output will respond to a change in system input, are an
invaluable commodity to the designer and may be used
to make informed decisions about possible directions
for improving existing designs. This information may
additionally be used to expedite the design of new
engineering systems for which there is no vast
experimental or computational database, thus, reducing
design cycle time and cost. Sensitivity derivatives may
also be used in other fields such as parameter
estimation and-uncertainty analysis, as well as provide
additional understanding of the physical process at
hand. As an example, the additional understanding of
the physical phenomena may lead to new or improved
scaling laws and, thus, address issues between model
and full scale experiments.
These needs are the impetus for the development of
efficient and accurate sensitivity analysis procedures
based on well-validated CFS software. To maximize the
benefits of these procedures, they must have the
capability of resolving both the physics and the
geometric complexities of practical configurations (i.e.,
high-fidelity CFS). High-fidelity CFS may be used to
evaluate the performance of hydrodynamic components
such as propellers, as well as resolve interaction effects
between components (e.g., propeller and the ship hull).
Using the propeller and ship hull as an example, it is
expected that the efficiency and cavitation
characteristics of a propeller designed in open water
will differ from a propeller operating in the wake of a
ship hull, it may then be concluded that the interaction
must be considered. Numerous propeller and ship hull
geometry, at various operational conditions (e.g.,
advance ratio, Reynolds number, etc.), can be quickly
investigated with CFS (as compared to the time
required to fabricate and experimentally test the same
number of variations). From these candidate designs,
only the most promising will then require experimental
verification. Coupling high-fidelity CFS codes with
numerical optimization software can be used to produce
these promising designs, and may provide solutions to
current design deficiencies or shortcomings.
Computational design has become an active area of
research in the hydrodynamic community. To this end,
a brief review of the literature shall be included.
Although simulation has been used in the design of
hydrodynamic configurations in the past, only those
that utilize sensitivity derivatives and gradient-based
optimization methods will be considered in this review.
A detailed and concise overview of sensitivity analysis
methods and aerodynamic design optimization research
may be found in Newman et al. (19991.
Hino (1999) used the Navier-Stokes equations
coupled with a sequential quadratic programming
(SQP) method to minimize the wave and viscous drag
on ship hull forms. In that work, Hino adopted a
discrete-adjoint variable approach to compute
sensitivity derivatives and redesigned a hypothetical
tanker hull and a Series 60 hull for minimum resistance.
Subsequently, based on Jameson's (1988) work,
Cowles and Martinelli (2000) described a control-
theory (continuous) approach to sensitivity analysis for
incompressible, turbulent viscous flows and applied this
approach to match target pressure distributions (inverse
design) on finite span wings and sails. Tahara et al.
OCR for page 992
(2000) used a Navier-Stokes code, with finite-
difference gradients, for CFD-based design of the bow
bulb on a surface combatant. In Tahara no free surface
effects were considered, and the objective was to
minimize the downstream vorticity in the vicinity of the
bulbous bow. Ragab (2001 ) describes the use of a
continuous-adjoint approach in free-surface potential
flow and applies the approach to redesign a base-line
ship for minimum wave resistance, and to match target
pressure distributions. Soto and Lohner (2001)
proposed an incomplete-gradient adjoins formulation
based on the continuous approach to sensitivity analysis
where only the adjoins on the boundary of the domain is
computed. In their approach the derivatives of the cost
or objective function were computed using finite-
difference. Soto and Lohner, using the incompressible
Euler equations, redesigned 3D hydrofoils to maximize
the minimum pressure on pressure and suction surfaces,
and to redesign the bow bulb on a surface ship for
minimum wave drag. Dreyer and Martinelli (2001)
utilize a continuous-adjoint approach for target pressure
matching of propulsor configurations using the psuedo-
compressible Euler equations in a rotating frame.
In this work, the current computational design
methodology is demonstrated on three, nontrivial
hydrodynamic components. Presented herein is a
discussion of the flow analysis code, the methods for
computing sensitivity derivatives, the parameterization
of the design surfaces, and the exploratory
optimizations performed.
MOW ANALYSIS
The flow analysis code used to perform the
simulations, and into which sensitivity analysis and
design capabilities have been incorporated, is referred
to as UNCLE - UNsteady Computation of fieLD
Equations (Whitfield et al. 1994, Pankajakshan et al.
2002~. UNCLE is a parallel, multiblock, multigrid
structured grid code which solves the time dependent
Reynolds Averaged Navier-Stokes (RANS) equations
in either a rotating or an absolute reference frame.
Additionally this software has linear and nonlinear free-
surface capabilities, and incompressible and arbitrary
Mach number (Taylor et al., 2001) versions. UNCLE
has been extensively validated and is currently used by
the Navy and industry for hydrodynamic simulations.
More information on UNCLE may be found in the cited
literature.
SENSITIVITY DERIVATIVES
Background
For sensitivity analysis and design optimization
based on CFS the field state equation to be solved is
usually a system of partial differential equations (PDE).
In the current hydrodynamic design methodology this
system is represented by the RANS equations.
Differentiation of this system of PDE (i.e., sensitivity
analysis) can be performed at one of two levels. In the
first method, termed the continuous or variational
approach, the PDE are differentiated prior to
discretization, either directly or by introducing
Lagrange multipliers that are defined as a set of
continuous linear equations adjoins to the governing
PDE. Subsequently, these directly differentiated or
adjoins equations are discretized and solved. In the
second method, termed the discrete approach, the PDE
are differentiated after discretization. The discrete
approach may also be cast in either the direct or an
adjoins formulation. A more detailed discussion on the
discrete and the continuous formulation, as applied to
aerodynamic shape optimization, may be found in Hou
et al. (1994) and Jameson (1988), respectively.
With gradient-based optimization methods, the
search direction is determined using the first derivatives
of the objective and constraint functions with respect to
the vector of independent or design variables (i.e.,
sensitivity derivatives). This is not to say that the search
direction is solely based on first-derivative information;
it is possible to estimate higher-order derivatives using
the computed first derivatives. In general, the objective
and constraint functions may be expressed as
Fi(Q,X ,pk ~ . Here, Q represents the disciplinary state
vector which is produced from the CFS, X is the
computational mesh over which the PDE are
discretized, and ink the vector of design variables. The
sensitivity derivatives of these functions may be
obtained by direct differentiation with respect to
implicit and explicit dependencies as
aF (aF )T aQ (aF )r aX `1'
To compute the sensitivity derivatives in Eq. 1 the
sensitivity of the state vector, ac/~3k' and the grid
sensitivity terms, ax/ask ~ are needed. This approach
to sensitivity analysis is referred to as the direct
differentiation method. Since these derivatives require
the sensitivity of the state vector, the number of linear
systems needing to be solved will be equal to the
number of design variables. On the other hand, if in the
design problem under consideration the sum of the
objective and constraint functions is less than the
number of design variables, a more efficient alternative
approach may be formulated. This method is referred to
as the adjoins variable approach, and may be written as
T
VFi = at +(3X ) ask Fi
[
3
(2)
OCR for page 993
where OF are adjoins vectors defined in such a way as
to eliminate the dependence of the objective and
constraint functions on the sensitivity of the state
vector, and R represents the disciplinary state equation.
The number of linear systems needing to be solved is
equal to the number of functions required by the
problem formulation.
As previously discussed, two methods exist to
perform the sensitivity analysis - continuous and
discrete approaches. Either may be cast into a direct
(Eq.1) or an adjoins (Eq.2) formulation. From the fact
that the derivatives produced by a continuous approach
are not consistently discrete with the CFS solver,
optimization routines using these derivatives may fail to
converge or even diverge. Furthermore, derivation of
numerical boundary conditions for the continuous
adjoins have been found to be difficult and time
consuming. Thus, in the current work, the discrete
approach to sensitivity analysis is used.
Sensitivity Analysis
For the discrete-direct approach (Eq.1) the
sensitivity of the field variables ac/a'~k are required,
and for the discrete-adjoint approach (Eq.2) the adjoins
vectors OF are needed. To obtain these, the discrete
residual vector from the CFS for a steady-state solution
may be written as
R(Q~k ),X ark cook )= 0 ~ ~
where the explicit and implicit dependencies of the
residual on the state vector Q. the computational mesh
X, and the design variables ink are asserted.
In the discrete-direct approach, Eq.3 is directly
differentiated with respect to the design variables to
produce the following linear equation
dR = aR an + aR ax + aR =0 <4y
Ask I Opk OX S0k 3pk
or, rearranging
aR aQ ~aR ax aR~
_ = _- + use
aQ apk ax ark al3k
where aR/aQ and aR/ax are the Jacobian matrices
evaluated with-a converged (steady-state) solution, and
ax/apk are the grid sensitivity terms. The solution of
Eq.S poses the difficulty of solving a large linear
system of equations for each design variable.
Furthermore, because this equation is a linear system,
the linearizations of the residual wth respect to the
state vector and the grid (i.e., the Jacobian matrices)
must be exact. The detriments of using inexact
linearizations in Eq.S have been explored by Newman
et al. (1995~. Solving these systems, however, is made
more tractable when the above equations are recast into
what has been termed the incremental iterative form
(Korivi, 1994) as follows
A^n~&Q)=_~OR(3Q ~f+&R ax + aR ~ (bar
ink ) LaQ~pk ) ax ark ink ~
(afk ) (ark ) +/\n\) (6b)
where A may be any convenient approximation to the
higher-order Jacobian that converges the linear system.
An approximation is possible because the equations are
now cast in delta form, with the physics contained in
the right-hand-side vector. Is has been found that the
first-order Jacobian works well for use in the coefficient
matrix of Eq.6a; most CFS codes also use the first-order
Jacobian for this purpose.
Two particularly attractive features of the
incremental iterative strategy are that (i) a more
diagonally dominant matrix may be used to drive the
solution of the linear system (as opposed to the
sometimes ill-conditioned higher-order Jacobian), and
(ii) the higher-order Jacobian now resides on the right-
hand-side of the equations and may be dealt with in an
explicit manner. When in this form, only the k-vectors
resulting from the matrix-vector product of
(aR/~Q)(~Q/apk~ are of concern. Hence, CPU time and
memory efficient methods for constructing the exact
matrix-vector product can be utilized. To this end,
higher-order spatially accurate discrete-direct
sensitivity analysis procedures for aerodynamic shape
optimizations have been developed (Newman et al.,
1997~.
The discrete-adjoint variable formulation begins by
combining Eq.4 from the direct differentiation method
with the sensitivity derivatives in Eq.1. From this
adjoins vectors may be conveniently defined such that
the sensitivity of the field variables are no longer
needed. Nevertheless, the end result requires the
solution of the following linear systems for the adjoins
vectors
T(a3R ) Gil = aaFi (7)
The adjoins variable approach may also be recast in
incremental iterative form, however, for brevity these
details shall be omitted. It should be noted that all the
same linearizations required by the direct approach are
required by the adjoins variable method; they are simply
transposed and used at different stages in the
computation of the sensitivity derivatives. Hence, a
sensitivity analysis code which was developed for
OCR for page 994
either of the discrete approaches may be modified to
produce the other.
The task of constructing exactly or analytically all
of the required linearizations and derivatives by hand
for the discrete approaches, and then building the
software for evaluating these terms can be extremely
tedious. This problem is compounded by the inclusion
of even the most elementary turbulence model (for
viscous flow) or the use of sophisticated grid generation
packages for adapting (or regenerating) the
computational mesh to the latest design. One solution to
this problem has been found in the use of a technique
known as automatic differentiation. Application of this
technique to an existing source code, that evaluates
output functions, automatically generates another
source code that evaluates both output functions and
derivatives of those functions with respect to specified
code input or internal parameters. A pre-compiler
software tool, called ADIFOR (Automatic
Differentiation of FORtran, Bischof et al., 1992), has
been developed and utilized with much success to
obtain complicated derivatives from advanced CFS and
grid generation codes for use within aerodynamic
design optimization procedures (Green et al., 1993,
1996 and Taylor et al. 1997~. The use of ADIFOR
produces code that, when executed, evaluates these
derivatives via a discrete-direct approach, referred to as
forward-mode automatic differentiation. More recently,
automatic differentiation software has emerged that
enables the derivatives to be evaluated with a discrete-
adjoint approach (Mohammadi, 1997 and Carte et al.,
1998~. This type of automatic differentiation is known
as reverse-mode. A detailed and concise review on the
use of sensitivity analysis in aerodynamic shape
optimization has been reported by Newman et al.
(1999a); the reader is directed to this source for
discussion on the methods presented thus far.
Complex Taylor Series Expansion (CTSE) Method
The methods previously discussed will require
differentiation of the CFS software, either by hand or
with pre-compiler software. Other methods to obtain
sensitivity derivatives are based on numerical
techniques. The simplest numerical technique is the
finite-difference approximation. Another is a relatively
new numerical technique, developed by Newman et al.
(1998) for -performing disciplinary and multi-
disciplinary sensitivity analysis, Rich uses complex
variables to approximate derivatives of real functions.
This method is based on ideas that where explored over
three decades ago by Lyness and Moler (1967) and
Lyness (1967), and recently revisited by Squire and
Trapp (19981. Both numerical approaches will be
discussed below.
For a central finite-difference approximation to the
derivative, one may expand the function in a Taylor
series about a given point using a forward and a
backward step, and then subtracting to yield
df tf (x+h)- f (x-h~ _ h d f _ h d f _... `8
dx 2h 3! dx3 5 ! dx5
This expression for the derivative has a truncation error
of O(h2~. The advantage of the finite-difference
approximation to obtain sensitivity derivatives is that
any existing code may be used without modification.
The disadvantages of this method are the computational
time required and the possible inaccuracy of the
derivatives. The former is due to the fact that for each
design variable, two well-converged CFS solutions are
required to evaluate the central finite-difference. In the
case of nonlinear fluid flow, for example, these
solutions may become prohibitively expensive for a
large number of design variables. The latter is attributed
to the sensitivity of the derivatives to the choice of step
size. To minimize the truncation error one selects a
smaller step size, however, an exceedingly small step
size may produce significant subtractive cancellation
errors. The optimal choice for the step size is not
known a priori, and may vary from one function to
another, and from one design variable to the next.
Instead of the finite-difference approximation,
consider expanding the function in a Taylor series using
a complex step as
f(X +hi)= f~x)+h' df _ h d f
h3' d3f h4 d4f
_ + +
3! dX3 4! dx4
(9)
where ~ = `/~. Solving this expression for the
a imaginary part of the function yields
df hn[f (X + h ~ ~ + h d f _ h d f + . .. (10)
Tic h 3 ! dX3 5 ! d~c5
This expression for the derivative also has a truncation
error of O(h2~. By evaluating the function with a
complex argument, both the function and its derivative
are obtained, without subtractive terms, and thus
cancellation errors are avoided. The real part is the
function value to second order.
The advantages of the complex variable
approximation, referred to as the Complex Taylor
Series Expansion (CTSE) method, are numerous. First,
like the f~nite-difference approximation to the
derivatives, very little modification to the software is
required. All the original features and capabilities of the
software are retained. Thus, user experience is not lost
and ongoing advancements and enhancements can be
readily introduced into subsequent versions without
extensive modifications or re-differentiation. This is in
OCR for page 995
direct contrast to hand or automatically differentiated
sensitivity analysis codes where any modification to the
original software will require re-differentiation. This
advantage is extremely useful in the problem
formulation stages of the design process when new
objective and constraint functions are being explored.
Second, this method is equivalent to a discrete-direct
approach, either from automatic differentiation or hand
differentiated codes solved in incremental iterative
form, in the way that the state vector and its derivatives
are being solved for simultaneously. When solving the
state equation, the state vector resides in the real part
and the derivatives in the imaginary part. Unlike the
finite-difference approximation, fully converged flow
solutions are not required to obtain derivatives of
sufficient accuracy for design. Finally, the CTSE
method is not sensitive to step size selection and only
requires step sizes that avoid excessive truncation error;
thus, it has been shown that this method demonstrates
true second-order accuracy (Newman et al., 1998 and
Anderson et al., 2000~. Additionally, the CTSE
technique can be used to compute second derivative
information using available data, but these
computations are subject to cancellation errors. The
only disadvantage of the CTSE method is the increased
runtime required by the evaluating routines when run
with complex arguments.
RESULTS
Three
examples
analysis
exploratory hydrodynamic optimization
were performed to access the sensitivity
and computational design capabilities
developed in UNCLE. Each is representative of typical
components that have been recently redesigned for the
DDG51 fleet. These examples include redesign of a
marine propeller, redesign of the bow bulb, and
redesign of the rudder.
Figure 1: P5 158 Geometry and tip vortices.
P5168 Propeller
For this study a propeller of typical, modern design
was selected, and is designated as Propeller 5168
(P5168~. P5168 is a five bladed controllable pitch
propeller, and was tested in the DTMB, CDNSWC 36"
water tunnel (Jessup, 1996~. This propeller was used by
ONR for validation purposes of CFS codes in
predicting tip vortex flows about Navy surface ship
propellers. The ultimate goal of the ONR was to
demonstrate the usefulness of CFS codes in ranking
various candidate blade tip geometries to suppress tip
vortex cavitation. The geometry and tip vortices, as
predicted via UNCLE, are shown in Fig. 1.
To reduce tip cavitation, the objective of the
redesign was to maximize the weighted sum of average
and minimum pressure on the suction and pressure
surfaces. The design surface was selected as the outer
most 10% of the blade span as shown in Fig. 2. The
ores sure and suction design surfaces where
parameterized with a Bezier surface that controlled the
normal thickness variation as
N M
In = ~ ~ Pij
i=0 j=0
N! ui (1-U)N-'
i! (N-i)!
| M! vj (1 - v)M -i 1 (11)
where Pij are the Bezier control points that were used
as the design variables, and u, v represent the
parametric variables in the chordwise and spanwise
directions, respectively. A two-dimensional example of
this parameterization is shown in Fig. 3 for an initially
symmetric hydrofoil. Geometric constraints were
enforced such that slope and curvature would remain
unchanged between the pressure and suction surfaces.
With these constraints, the total number of design
variables reduced to 8. Transfinite interpolation (TFI)
of surface deformations into the volume mesh was used
to modify the computational mesh to the latest design.
Figure 2: P5 168 design surface.
OCR for page 996
\
h4~n comb - LIn.
~~0itI" sud=~ /
Twins ~
~ 44
case 5~a''~ i: VAst] ~: I'
OF use -
Dig
4s
HE
.~
~ .
iiO.0 1
ND it: - t:~ ~
~~ W.~¢'4~S~
f ~
Hi
t
1. 07
Figure 3: Sample 2D normal thickness variation.
The analysis used the incompressible version of
UNCLE and the viscous simulation ms performed in
parallel with 10 blocks. The flow Reynolds number was
4.26M with an advance ratio of 1.1. Sensitivity
derivatives where obtained via the direct approach with
the CTSE method used to construct all linearizations.
Since this was an exploratory study, only design
improvement over the original P5168 propeller was
sought. To this end, one design cycle was performed
(i.e., only one search direction was computed and
traversed) with 23 line-search steps taken. The results
of this design cycle are shown in Fig. 4. As seen, the
minimum and average pressure on the suction design
surface has been increased by 5.44% and 0.40%,
respectively, and the minimum and average pressure on
the pressure design surface has been increased by
5.20% and 22.28%, respectively. The blade shape for
this design cycle became more bulbous on the pressure
side while introducing a cusp near the trailing edge on
the suction side. The normal thickness variation for the
pressure and suction surfaces are shown in Fig. Sa and
b, respectively. The pressure on the suction surface and
the change in pressure between optimized and original
are depicted in Fig. 6. As seen, the lowest surface
pressure occurs at the trailing edge tip and, thus, this is
the region of greatest increase.
It should be noted that as additional design cycles
are performed, the shape of the propeller will continue
to evolve as the optimum is approached. The propeller
shape after this first design cycle may or may not be
indicative of the final design. During each design cycle
further improvements will be observed. However, if the
shape is progressing towards a design that exhibits
undesirable features, or manufacturability difficulties,
the optimization problem can be reformulated with
additional constraints and/or objective functions.
.£ ~ ~, .
,i
s
.. .
s
~ ~ ~~ ~ ~i¢~~s.4 ~
,A< ~-i: ~~.:.:f'
I.
1;
~ ~0
US
~ 1~-(
.'
~ Pours S~rF~ ~ it
>
j' ~ Sew S~ M~u~
t! _, s=~u sU§
I._ ~C==
' ' ' ' ~ ' ' ' ' 1'O' ' ' ' 1t ' ' ' :'0 ' '
Let ~ ea~ Sew
Figure 4: Optimization results for the P5 168 redesign.
—4 U OFF p~ ~~ ~
NonnalTnic~n.Ss Vad~on
MOD 0~17 6~ 6~ 4~6 g~3
(a) Pressure surface.
Palm t - _
.
Normal Thickness V - anion
~ 4013 NEWS O~
O Suction surface.
Figure 5: Results of normal thickness variation (u=0 is
leading edge, v=1 is tip).
OCR for page 997
-
Pi~ureon Such SuH~
~20 0,St41
S - opt Path::
~.02g7: 0~. Of
Figure 6: Suction surface pressure and pressure
change.
Bow Bulb Redesign
The second exploratory study consisted of the
redesign of the bow bulb (or sonar dome) on a modified
5415 ship hull. Typical sonar arrays that reside in the
bulbous bow are illustrated in Fig. 7. The design
surface, which is shown in Fig. 8, was parameterized
with the normal thickness variation given in Eq. 11.
Volume constraints where enforced such that the largest
sonar array that could reside in the original geometry
would not be penetrated. The objective was to minimize
the swirl of the flow in the region around and
downstream of the bulbous bow. To this end, a
weighted sum of the average and maximum swirl
parameter (Remotigue, 1999) was minimized.
_
Figure 7: Typical bow bulb and sonar arrays.
Figure 8: Design surface for the bow bulb redesign.
The analysis used the arbitrary Mach number
version of UNCLE and the viscous simulation used the
~-e turbulence model. The flow Reynolds number
was 14M with a Froude number of zero (i.e., no free
surface effects were considered). Sensitivity derivatives
where obtained via the direct approach with the CTSE
method used to construct all linearizations.
Due to the fact that the ~rameterization of the
sonar dome resulted in 24 shape design variables, and
that the analysis and sensitivity analysis were
performed sequentially on a single block, only two
design cycles were performed. As seen via the width
contours for the original and modified bow shown in
Fig. 9, the resulting design after two cycles became
more bulbous in the rear portion of the sonar dome. The
resulting effect on the swirl parameter on a downstream
cutting plane is illustrated in Fig. 10. It can be observed
that the swirl has been reduced, and thus design
improvement has been achieved.
OCR for page 998
(a) Oricinal width contours
I. - ~ ~ ~ =l ¢~-1 ~,`4 '.~' {~ ~ ~
(b) (change (optimized-original) width contours.
Figure 9: Width contours for the original and change
in width contours for bow bulb redesign.
Twisted Rudder Redesign
The final optimization example was the redesign of
the twist distribution of a rudder placed in the wake of a
propeller. This design was originally performed and
patented by Shen (1997) and is considered a
tremendous success for cavitation reduction (Krueger,
2001~. The experimental apparatus used for testing this
design by Shen is shown in Fig. 11; additional details
on the propeller placement and the rudder geometry
may be found in the cited literature.
The original design by Shen was inspired by the
fact that during full scale trials to access the
hydroacoustic and hydrodynamic performance of
redesigned propellers, severe surface cavitation was
observed on the outboard surface of the rudders.
Subsequently, drydock inspections confirmed cavitation
erosion on the outboard rudder surface, while none was
present on the inboard surface (Jiang 19951. Shen then
conducted a computational and experimental design of
the rudder twist distribution and found that by aligning
the local rudder twist to the incoming propeller induced
flow angles, cavitation on the outboard surface could be
significantly reduced.
Since the rudder profile is symmetric, the objective
used in the current work was to design the rudder for
zero side force at zero rudder deflection. The analysis
and design was conducted using the arbitrary Mach
number version of UNCLE. The propeller was modeled
as a body force propulsor, and the fluid simulated as
inviscid flow. The twist distribution was parameterized
with a Bezier curve. Constraints were enforced to
allow no twist at the root or tip, and zero twist
derivative at the root. With these constraints, the
parameterization resulted in 4 design variables. Once
again, the CTSE technique was used to construct all
linearizations required by the direct method.
(a) Downstream cutting plane.
;~
~) Swirl parameter for original geometry.
(c) Swirl parameter for redesigned geometry.
Figure 10: Results of the bow bulb swirl
minimization on a downstream cutting
plane. Swirl parameter contours are
shown with the same scale for the original
and redesigned geometry.
OCR for page 999
(a) Side view.
(hi Front view.
Figure 11: Experimental apparatus for twisted rudder
of Shen (1997~.
_ c
—ot$;hen (~) ~
fit—,
:
h ~
.
S.
.
:
.
e
me,
~ i/
Norma sun
Figure 12: Results of the twisted rudder redesign.
(a) Front view of initial and final twisted rudders.
- ,,,,,, (b) Pressure for initial and final twisted rudders.
(c) Pressure for initial and final twisted rudders.
Figure 13: Pressure distributions on untwisted and
twisted rudders.
For this design, the analyses required in the design
cycles and the computation of the sensitivity derivatives
where performed in parallel over 16 blocks and 4
design variables. Hence the optimization was
performed using a total of 64 processors. After 3 design
cycles the side force was reduced by three orders of
magnitude. Results of this optimization are shown in
Fig. 12 and compared with the design produced by
Shen (19971. As seen, the developed computational
design procedure produced nearly the same twist
distribution. However, the current computational design
code took less than one day to produce this distribution.
The front view of the original and redesigned rudders,
shaded by pressure, illustrating the twist along the span
of the rudder, are shown in Fig. 1 3a. The pressure
change on the pressure (inboard) and suction (outboard)
surfaces are shown in Figs. 13b and c, respectively.
The minimum pressure on the twisted rudder rose
15.44% from the original.
OCR for page 1000
CONCLUSIONS
The primary objectives of this paper were to detail
the development of sensitivity analysis and
computational design capabilities incorporated into
UNCLE, and to introduce a novel technique for
computing derivatives of Hal functions using complex
variables to the hydrodynamic community.
Sensitivity analysis provides an additional level of
information about the flow physics and how such
phenomena will change with variations in geometric
and non-geometric parameters. In the current work, this
sensitivity analysis has been used to demonstrate design
enhancement for three physically and geometrically
different hydrodynamic configurations. Complete
design optimization was not performed, the objective
was to demonstrate that design improvements over the
baseline geometry could be achieved using numerical
optimization. For these design results to be meaningful,
experienced design engineers would need to formulate
the optimization problem. To this end, the redesign of
the twisted rudder represents the most realistic design
example. This optimization was formulated based on
the design conducted by Shen (1997) and, thus, very
close agreement was observed. Additionally, this
demonstrates that computational design may be a viable
tool to aid in the design process and ultimately reduce
design cycle time and costs.
ACKNOWLEDGE
This work was sponsored by Dr. L. Patrick Purtell
of the Office of Naval Research. This support is
gratefully acknowledged.
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Representative terms from entire chapter:
sensitivity derivatives
