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OCR for page 553
Computation of Viscous Flow around a Propeller-Shaft
Configuration with Infinite-Pitch Rectangular Blades
F. Stern and H. T. Kim
The University of Iowa
Iowa, USA
Abstract
A viscous-solution method is set forth for
calculating marine-propeller flow fields. An
overview of the computational method is giv-
en, and some example results for both laminar
and turbulent flow are presented and dis-
cussed with regard to the flow physics, for
the idealized geometry of a propeller-shaft
configuration with infinite-pitch rectangular
blades. It is shown that the flow exhibits
many of the distinctive features of interest,
including the development and evolution of
the shaft and blade boundary layers and
wakes, and tip, passage, and hub vortices.
Comparisons are made with results from a
lifting-surface propeller-performance pro-
gram, to aid in evaluating the present
method, which show that the present method
accurately predicts the blade loading,
including viscous effects, and clearly dis-
plays the ability to resolve the viscous
regions in distinction from the inviscid-flow
approach.
Introduction
-
Propeller-type flow fields are encountered
in a wide variety of engineering problems,
e.g., in the propulsion of marine vehicles,
airplanes, and helicopters, in turbomachine-
ry, and in inert and reacting swirl-flow sys-
tems. The present study concerns the de-
velopment of a viscous-solution method for
the analysis of incompressible propeller
flows. Of particular interest are marine
propellers which are unique because they
operate in the thick stern boundary layer and
wake such that the flow field is interactive,
i.e., the propeller-induced flow is dependent
on the hull flow which is itself altered by
the presence of the propeller. More specifi-
cally, here we are primarily concerned with
the propeller-induced flow; however, the pre-
sent study is an outgrowth of a larger pro-
ject concerning propeller-hull interaction
and, upon extension, is expected ultimately
to handle entire configurations.
553
Presently, only potential-flow methods are
available for calculating practical marine-
propeller flow fields (for a recent review
see Kerwin [13). Lifting-surface methods are
available for both steady and unsteady flows
(e.g., Kerwin and Lee [2]). Also, surface-
panel methods have been developed for steady
flow (e.g., Hess and Valarezo [3]). These
methods suffer from two major problems:
first, they rely on the incorrect assumption
that the propeller operates in an infinite
ideal fluid, but with a specified spatially
varying inflow which represents the hull
boundary layer and wake; and second, the
results, including the propeller thrust and
torque, are very sensitive to the specifica-
tion of the geometry of the trailing-vortex
wake sheet which requires a viscous-flow
analysis for its prediction. Consistent with
the first problem, the agreement with exper-
imental thrust and torque data for nonuniform
inflow has not been satisfactory. Also, the
predicted pressure distributions, even for
uniform inflow, do not show overall good
agreement with experimental data (ITTC
[43). A complete evaluation of the theory
has been hampered by the lack of knowledge of
the effective inflow which is usually assumed
to be the nominal wake of the bare hull.
Relatively little work has been done con-
cerning viscous effects for rotating propel-
ler blades. Most of the studies pertain to
boundary-layer development and are restricted
to laminar flow and idealized geometries
(Morris [5]). Only one study has considered
practical geometries and flow conditions
(Groves and Chang [6]). In general, these
methods suffer due to the inaccuracy of the
pressure distributions predicted by inviscid-
flow methods and are not easily expendable
into the wake. Similar difficulties with
this approach have been encountered in turbo-
machinery applications. Viscous effects have
also been studied with regard to the tip-
vortex generation process utilizing the par-
abolized Navier-Stokes equations (most
recently, de Jong et al. [7]).
OCR for page 554
Most work on propeller-hull interaction
assumes that the interaction is inviscid in
nature and has focused separately on either
propeller influence on hull resistance
(thrust deduction) or on hull boundary layer
and wake (effective wake). Recently, Stern
et al. [8,9] have developed a comprehensive
viscous-flow approach to propeller-hull
interaction in which a viscous-flow method
for calculating ship stern flow (Chen and
Patel [10]) is coupled with a propeller-
performance program in an interactive and
iterative manner to predict the combined flow
field; hereafter referred to as the interac-
tive approach. A body-force distribution is
used to represent the propeller in the vis-
cous-flow method. The steady-flow results
show good agreement with experimental data
and indicate that such an approach can accur-
ately simulate the steady part of the com-
bined propeller-hull flow field. Although
the unsteady-flow results generally follow
the trends of available data, these indicate
the limitations of this approach for simulat-
ing the complex blade-to-blade flow. The
work of Stern et al. [8,9] is precursory to
the present work.
Most of the relevant work from related ap-
plications is for high-speed flow in which
shock waves have a dominating influence;
therefore, the focus of these studies is, in
general, quite different from that of the
marine-propeller application. The most
closely related work is that done to develop
energy efficient turboprops and for turbo-
machinery applications (see Kim [11] for a
more complete discussion, including refer-
ences). Although advanced inviscid- and vis-
cous-flow methods are under development, in
most cases, incompressible-flow calculations
are either not possible without major modifi-
cations or require the use of the pseudo-
compressibility concept. In its usual form,
the latter precludes time-accurate unsteady-
flow calculations, although some recent stud-
ies have shown promising results for such
extensions through the use of subiter-
ations. Lastly, concerning related applica-
tions, the helicopter and swirl-flow calcula-
tions are helpful with regard to tip vortices
and swirling jets and wakes, respectively;
but, here again, involve large differences in
both flow conditions and geometry.
It is apparent from the foregoing that
present methods for calculating marine-pro-
peller flow fields are inadequate for analyz-
ing the detailed flow structures such as the
development and evolution of the unsteady
blade boundary layers and wakes, blade-to-
blade flow, hub and tip vortices, and overall
propeller wake. Furthermore, even the most
advanced computational fluid dynamics methods
from related applications are either inap-
plicable or require major modifications to
handle marine propellers. This overall situ-
ation motivated the present study.
554
In the following, propeller-flow phenomena
are described to aid in understanding the
nature of the flow as well as the differences
between the present and interactive
approaches. Also, the rationale for select-
ing the present geometry, i.e., a propeller-
shaft configuration with infinite-pitch rec-
tangular blades (figure 1) is discussed,
including its advantages and shortcomings.
Next, an overview of the computational method
is provided. Then, some example results are
presented and discussed with regard to the
flow physics, including the computational
grid and conditions and calculations for both
laminar and turbulent flow. Subsequently,
comparisons are made with results from a
lifting-surface propeller-performance pro-
gram, to aid in evaluating the present me-
thod. Finally, some concluding remarks are
made. The details of the computational
method and the complete results, including
additional calculations to study the influ-
ences of a thick-inlet boundary layer, the
propeller angular velocity, and the blade
number, as well as comparisons with some
additional relevant experimental and computa-
tional studies are provided by Kim [11].
Propeller-Flow Phenomena
Figure 2 displays sketches of both the
circumferential-average and blade-to-blade
flow for the relatively simple case of a pro-
peller-shaft configuration. The circumferen-
tial-average flow (figure 2a) clearly dis-
plays the expected features based on physical
considerations, i. e., axial-velocity U
increase (overshoot ) and negative radial-
velocity V ( contraction) associated with the
propeller thrust, and propeller-induced swirl
W. including hub vortex, associated with the
propeller torque. Also, there is a jump in
pressure p across the propeller plane and a
large decrease in pressure along the wake
centerline due to the propeller thrust and
-induced swi rl, respectively, and a large
increase in turbulent kinetic energy k,
including two peaks, one near the wake cen-
terline and one corresponding to the tip of
the propeller blades. As discussed above,
the interactive approach is able to predict
accurately many details of the circumf eren-
tial-average ( steady) f low; however, in order
to predict the complex blade-to-blade f low a
more detailed representation of the propeller
than the body force is required.
In comparison with the situation for the
ci rcumf Brent ial-average f low, relet ively li t-
tle is known concerning the complex blade-to-
blade flow due, no doubt, to difficulties in
perf arming such experiments and calculations
for this type of geometry. Figure 2b dis-
plays some of the expected f low structures,
including the leading-edge horseshoe, pas-
sage, tip, and hub vortices and the blade
boundary layers and wakes. It should be
emphasized that figure 2b is speculative. It
is based on the results f rom the present
OCR for page 555
study to be discussed later as well as on
information from other studies both for mar-
ine propellers and turboprops and for rele-
vant geometries such as turbomachinery and
simple tip and juncture flows. Here it is
sufficient to point out that there are numer-
ous fundamental issues associated with these
flow structures which have yet to be expli-
cated (see Kim [11] for a partial list).
Although some of these issues are in common
with other flows and applications, some are
unique, and, therefore, must be addressed
within the context of the marine-propeller
problem.
As mentioned earlier, the goal of the pre-
sent work is to develop a viscous-flow method
for marine propellers, which can analyze the
detailed flow structures described above. It
is appropriate to initiate such an effort
with as simplified a geometry as possible
without sacrificing the essential physics of
the flow under consideration. The geometry
chosen for this purpose is a propeller-shaft
configuration with infinite-pitch rectangular
blades (figure 1). This geometry has the
following important advantages: the grid gen-
eration is relatively simple so that the fo-
cus of attention can be given to the more
basic aspects of the numerics, which is im-
portant for the initial development; fine-
grid solutions are possible with the avail-
able supercomputer resources; the laminar-
and turbulent-flow solutions exhibit similar
flow patterns such that meaningful compariso-
ns can be made between the two flows; and its
simplicity facilitates the diagnosis of the
important features of the blade-to-blade
flow. Also, as will be shown below, the flow
field exhibits most of the distinctive fea-
tures of interest. However, it should be
recognized that this geometry also has short-
comings, such as the lack of blade section
geometry and, most importantly, thrust.
Issues related to these aspects will be ad-
dressed in future extensions for practical
geometries.
Overview of the Computational Method
-
Consider the viscous flow around a propel-
ler-shaft configuration rotating at constant
angular velocity ~ in an infinite uniform
stream with velocity UO (figure 1). It is
assumed that the Mach and cavitation numbers
are, respectively, sufficiently small and
large such that the fluid is incompressible
and noncavitating. Under these conditions,
the flow is cyclic in both space and time.
Moreover, the flow is steady and spatially
cyclic at blade-to-blade intervals in nonin-
ertial coordinates, which rotate with the
propeller. The situation is similar for
propeller-driven axisymmetric bodies; how-
ever, for the more general circumstance of
propeller-driven three-dimensional bodies the
flow is unsteady, even in noninertial coor-
dinates. For straight-ahead performance it
is cyclic with angular velocity a, whereas
for maneuvering,it is noncyclic. 555
As mentioned earlier, the present overall
computational method is based on that used
previously for calculating propeller-hull
interaction (Stern et al. [8,9]) in which a
viscous-flow method for calculating ship-
stern flow (Chen and Patel [10]) is coupled
with a propeller-performance program in an
interactive and iterative manner to predict
the combined flow field. This is expected to
facilitate future extensions for entire con-
figurations.
In order to extend this approach for the
present purpose, a number of major modifica-
tions were required, including the following:
use of a noninertial coordinate system, which
rotates with the propeller, and solution of
the corresponding equations; implementation
of boundary conditions, including periodic
boundary conditions for the blade-to-blade
region; adaptation of an ADI scheme at each
crossplane; and a complete restructuring of
the program for propeller geometries, includ-
ing calculations for both laminar and turbu-
lent flow. Also, during the time period that
the present work was in progress, the basic
viscous-flow method of Chen and Patel [10]
was upgraded for fully-elliptic calculations
of the complete Reynolds-averaged Navier-
Stokes equations (Patel et al. [12]). Sim-
ilar modifications were made for the present
work. Lastly, modifications were required to
execute the program efficiently on a super-
computer.
Below, an overview of the computational
method is given. A complete description is
p rovi deaf by Kim [ 1 1 ] . Als a, further detai Is
of the basic viscous-f low method are provided
by Chen and Patel [10] and Patel et al. [ 12 ] .
Equations and Coordinate System
The Reynolds-averaged Navier-Stokes equa-
tions are written in the physical domain
(figure 3a) using noninertial cylindrical
coordinates (x,r, 6) rotating with constant
angular velocity a= (~,0,0) as follows:
aU+i a (rV)+{ aW~=0
Dt = ~ aX (P + Uu) + fx ~ ear (uv)
_ 1 a (Uw) _ Uv + _ ~7 U
Dt r 2~ - ~ r = ~ aa (uv)
(2)
ar (P + vv) + fr ~ r a ~ ( vw ) - ~ ( vv
- ww) + ~ (V2V _ 22 aW~ _ V2 ) (3)
r r
Dt + r + 2~V = ~ aax (uw) ~ as (vw)
~ r as (P + WW) + f ~3 ~ r (vw)
OCR for page 556
+ie(V2W+22 aV_
r r
with
DDt = at + u aaX + v ear + W aid
(4)
and v2 = a2 + a2 + 1 a ~ a2
t is the time; U. V, W are, respectively, the
longitudinal, radial, and circumferential
components of mean velocity; p is the pres-
sure; uu, uv, etc. are the Reynolds stress-
es; f , f , f are, respectively, the longi-
tudinal, radical, and circumferential com-
ponents of the body force; and Re = U L/v is
the Reynolds number defined in terms of a
characteristic velocity U and length L,
which are used along with the density p to
nondimensionalize all variables, and molecu-
lar kinematic viscosity v. For laminar flow,
equations (1) through (4) reduce to the
Navier-Stokes equations by simply deleting
the Reynolds-stress terms and interpreting
(U,V,W) and p as instantaneous values.
Closure of the Reynolds equations is at-
tained through the use of the standard k- £
turbulence model. Each Reynolds stress is
related to the corresponding mean rate of
strain by the isotropic eddy viscosity vim as
follows:
=vt(ar+ax)
1 au am
- uw = It (r as + ax)
1 av aw w
- vw = It (r as + ar r)
- uu = At (2 aU) _ 3 k
- vv = v (2 aV) 2 k
- ww = At (r a9 + 2 r) ~ 3 k
v is defined in terms of the turbulent kine-
t~c energy k and its rate of dissipation £ by
2
v = C k
t ~ £
where C is a model constant and k and £ are
governed by the modeled transport equations
Dk a 1 ak 1 a 1 ak
Dt ax (R. aX) + r ar (R. r a
k k
+ 2 as (R. an) + G -
r k
Dt = ax ( - aX) + r ear ( - r aa£)
£ ~
(8)
+ (aU + aV)2 + (r am + aWx)2
+ (l aV + aW _ W)2
The effective Reynolds number R is defined
as
1 1 at
_ = + .
R Re ~
(10)
in which ~ = k for the k-equation (7) and
~ = £ for the £-equation (8). The model
constants are:
C = .09, Cal = 1.44, C ~ = 1~92,
MU ~ = ~ = ~ = 1, o~ = 1.3
The governing equations (1) through (10)
are transformed into nonorthogonal curvilin-
ear coordinates such that the computational
domain (figure 3b) forms a simple rectangular
parallelepiped with equal grid spacing. The
transformation is a partial one since it
involves the coordinates only and not the
velocity components (U,V,W). The transforma-
tion is accomplished through the use of the
expression for the divergence and "chain-
rule" definitions of the gradient and Lapla-
cian operators, which relate the orthogonal
curvilinear coordinates x~ = (x,r,8) to ithe
nonorthogonal curvilinear coordinates ~ =
(t,n, a). In this manner, the governing
equations (1) through (10) can be rewritten
in the following form of the continuity and
convective-transport equations
a ~ (blU + b2V + b3W) + a (blU + b2V + b3W)
+ a: (blU + b2V + b3W) = 0
all ~ + g22 ~ + 33
a: an a:
( 1 1 )
= 2A~ 2~ + 2B~ 2i + 2C~ 2~ + Rib an + So (12)
where ~ = (U,V,W,k, £)-
(6) Discretization and Velocity-Pressure Coupling
2 a ~ (R a3) + C£1 k G - C 2 k
r is the turbulence generation term
G = v {2 [(au)2 + (av)2 + (1 aW + V)2] in the form
556
The convective transport equations (12)
are reduced to algebraic form through the use
of a revised and simplified version of the
finite-analytic method (Patel et al. [12]).
In this method, equations (12) are linearized
in each local rectangular numerical element
65 = An = 6` = 1 , by evaluating the coef-
ficients and source functions at the interior
node P and transformed again into a normal-
ized form by a simple coordinate stretch-
ing. An analytic solution is derived by
decomposing the normalized equation into one-
and two-dimensional partial-differential
equations. The solution to the former is
readily obtained. The solution to the latter
is obtained by the method of separation of
variables with specified boundary func-
tions. As a result, a twelve-point finite-
analytic formula for unsteady, three-
dimensional elliptic emanations is obtained
OCR for page 557
up
1 + Cp[C + C + R] ~ nb~nb
+ Cp(Culu+ CD¢D+ ~ up - S)}
(13)
where the subscript nb denotes neighboring
nodes (NE: northeast, NW:northwest, etc.). It
is seen that ~ depends on all eight neigh-
boring nodal values in the crossplane as well
as the values at the upstream and downstream
nodes MU and ~D' nanny the values at the pre-
vious time step up . For large values of
the cell Reynolds number, equation (13) re-
duces to the partially-parabolic formulation
used previously (Stern et al. [8,9]). Since
equations (13) are implicit, both in space
and time, at the current crossplane of calcu-
lation, their assembly for all elements re-
sults in a set of simultaneous algebraic
equations. If the pressure field is known,
these equations can be solved by the method
of lines. However, since the pressure field
is unknown, it must be determined such that
the continuity equation is also satisfied.
The coupling of the velocity and pressure
fields is accomplished through the use of a
two-step iterative procedure involving the
continuity equation based on the SIMPLER
algorithm. In the first step, the solution
to the momentum equations for a guessed pres-
sure field is corrected at each crossplane
such that continuity is satisfied. However,
in general, the corrected velocities are no
longer a consistent solution to the momentum
equations for the guessed p. Thus, the pres-
sure field must also be corrected. In the
second step, the pressure field is updated
again through the use of the continuity equa-
tion. This is done after a complete solution
to the velocity field has been obtained for
all crossplanes. Repeated global iterations
are thus required in order to obtain a con-
verged solution. The procedure is facili-
tated through the use of a staggered grid.
Both the pressure-correction and pressure
equations are derived in a similar manner by
substituting equation (13) for (U,V,W) into
the the discretized form of the continuity
equation (11) and representing the pressure-
gradient terms by finite differences.
_olution Domain and Boundary Conditions
The physical and computational solution
domains are shown in figure 3. It is seen
that the solution domain is bounded by the
inlet plane Si; the shaft surface Ss; the
suction and pressure sides of the blade sur-
face S. s and Sb , respectively; the exit
plane ~e; the periodic symmetry planes S
and Spp; the symmetry axis Ls; and the outer
boundary SO.
The boundary conditions on each of the
aforementioned boundaries are as follows: on
the inlet plane Si, the initial conditions
for ~ are specified from simple flat-plate
solutions, initial conditions for p and p'
are not required; on the shaft Ss and blade
surfaces Sbs and Sb , for laminar flow, the
solution is carriedP out up to the actual
surface where the no-slip condition is ap-
plied, for turbulent flow, a two-point wall-
function approach is used; on the exit plane
Se, axial diffusion is negligible so that the
exit conditions used are ~ ¢/~: = 0, a zero-
gradient condition is used for p; on the
periodic symmetry planes S and S , an
explicit periodicity condition is imPpPosed,
i.e., ¢(5,n,() = ¢(g,n,( + ~ ), P(5,N,()
= p(5,n,: + ~ ), where ~ corresponds to the
blade-to-bladePinterval; in the symmetry axis
L , the conditions imposed are V = W = 0
5(U,k,c,p)/an = 0; on the outer boundary SO,
the uniform-flow condition is applied, i.e.,
U = 1, W = arts, p = a(k, c)/an = Be
o
Grid Generation
The computational grid is obtained using
the technique of generating body-fitted coor-
dinates through the solution of elliptic
partial differential equations, i.e., the
nonorthogonal coordinates ~ are related to
the orthogonal coordinates x by the set of
equations
V2xi = h h i ( 2 )
where
V2 = gi; a + fi a
a: aft ail
and fi = 1 a (Jai;
a:
In the present context, fit are called control
functions since their specification controls
the concentration of coordinate surfaces.
For specified boundary conditions and control
functions, equations (14) can be solved nu-
merically to obtain the coordinates of each
grid point in the physical domain.
(14)
i = 1,2,3 (15)
Because of the simplicity of the present
propeller geometry (figure 1), it is possible
to specify the transverse and longitudinal
sections of the computational domain as sur-
faces of constant ~ and A, respectively, and
moreover, the three-dimensional grid is ob-
tained by simply rotating the two-dimensional
grid f or the longi tudinal plane. Under these
conditions, equations ( 14 ) reduce substanti-
ally and can be readily solved once the con-
trol functions are specif led. The control
functions f1 are determined by the specified
grid distributions of axial stations, radial
distributions at the inlet and exit, and
girthwise distributions at the inlet and on
the outer boundary, respectively. These
control funcltions 1 were derived under the
con3ditions f = f ( i), f2 = f ( i, A), and f
= f ( :) only, which are of suff icient gener-
ality f or the present application.
557
OCR for page 558
Results
In the following, first, the computational
grid and conditions are described. Then,
some example results for laminar flow are
discussed to point out the essential features
of the solutions. These are followed by a
brief presentation of the results for turbu-
lent flow to highlight the differences. This
order and emphasis of discussion is selected
since the former represent solutions to the
exact governing equations, whereas the latter
are dependent on the choice of turbulence
model.
Computational Grids and Conditions
The geometry of the propeller-shaft con-
figuration (figure 1; see Tables 1 and 2 of
Kim [11]) was specified based on a config-
uration for which calculations had been pre-
viously performed, i.e., P4660 (Stern et al.
[9] ).
Partial views of the grid used in the
calculations are shown in figures 4a,b for a
longitudinal plane and a typical body tross-
plane, respectively. The shaft and blade
surface grid is shown in figure 1. Similar
grids are used for both the laminar and tur-
bulent calculations, but, in the latter case,
the near-wall grid lines (y < 30) are
deleted in order to implement the two-point
wall-function approach.
The inlet, exit, and outer boundaries are
located at x = (.54,6) and r = .9, respec-
tively; for laminar flow, the first grid
points off the body and blade surfaces are
located at .4 < y < 8 and 1 < x , y , or z
< 14, respectively; for turbulent flow, the
first grid points off the bo+dy and blade
surfaces+ are l+cated at 30 < y < 230 and 40
< x , y , or z < 190, respectively; 62 axial
grid points were used, with 18 over the up-
stream portion of the shaft up to the blade
leading edge, 11 over the blade, 14 over the
remainder of the shaft from the blade trail-
ing edge to the hub apex, and 19 over the
wake; for laminar flow, 40 radial grid points
were used with 22 over the blade span and 18
from the tip to the outer boundary; for tur-
bulent flow, 36 radial grid points were used
with 19 over the blade span and 17 from the
tip to the outer boundary; 30 and 26 angular
grid points were used for laminar and turbu-
lent flow, respectively. In summary, the
total number of grid points for the laminar
and turbulent calculations are 74,400 and
58,032, respectively.
The conditions for the calculations are as
follows: characteristic (shaft) length L = 1;
characteristic (uniform-stream) velocity U =
1; for la5minar flow, ReL = 2.02 x 10 and iec
= 1 x 10 , where ReL and Rec are the shaft-
(= U L/v) and chord-length (= U c/v) Reynolds
numbers, respectively; for turbid ent flow,
ReL = 6.08 x 10 and Rec = 3 x 10 ; the pro-
558
pelter angular velocity ~ = .3n (= 9 rpm)
(the blade section angle of attack varies
from 1.2 deg at the root to 4 deg at the
tip); for laminar flow, on the inlet plane,
6/Rh = .111 (where ~ is the boundary layer
thickness and Rh the hub radius) and there is
no inviscid-flow overshoot; and for turbulent
flow, on the inlet plane, 6/Rh = .489, U =
.04, and the inviscid-flow overshoot is
1.01. The ~ values are based on simple flat-
plate solutions and the selected Re. For
laminar flow, the Re value was selected based
on the fact that many investigators have
performed two-dimensional flat-plate bound-
ary-layer and wake calculations for this same
value. For turbulent flow, a reasonable
value of Re was selected for which fully tur-
bulent flow over the shaft and blades is
probable. The propeller angular velocity was
taken to be sufficiently low such that no
separation occurs over the blades.
For the nonrotating condition, the calcu-
lations were begun with a zero-pressure ini-
tial condition for the pressure field. For
the rotating condition, the complete nonro-
tating solution was used as the initial con-
dition. The values of the time at and pres-
sure a underrelaxation factors and total
number of global iterations used in obtaining
the solutions are .02-.1, .03-.1, and 70-100,
respectively. The calculations were per-
formed on a CRAY X-MP/48 supercomputer. The
central processor unit (CPU) and storage
(words) that were required for each of the
solutions are about 30min. and 1-1.7M words,
respectively. Note that the computer codes
were 23% vectorized and optimized to achieve
a 65% reduction in CPU, and that the maximum
normal system storage limit is 2M words.
Laminar Flow
The laminar-flow results for both the
nonrotating and rotating conditions are shown
in figures 5 through 13. Figures 5, 6, and 7
show the variation of some properties in the
longitudinal direction, i.e., the shaft and
blade surfaces and wake pressure, the wall-
shear (magnitude and angle for inertial coor-
dinates), and the wake centerline and maximum
swirl velocities, respectively. Figures 8
through 11 show the detailed results for some
representative axial stations in the form of
velocity and pressure profiles (i.e., ~ vs. Y
= r/Rp, where Rp is the propeller radius),
axial-velocity contours, crossplane-velocity
vectors, and axial-vorticity ~ contours,
respectively. Lastly, figures 12 and 13 show
close-up views of the tip vortex and the tip-
vortex trajectory, respectively. Note that,
in figure 8, the ordinate is Y such that the
distance from the plate is larger near the
tip than near the root. Also, the labeling
of each of the curves corresponds to the
angular grid lines shown in figure 4b.
First, consideration is given to the
results for the nonrotating condition. The
OCR for page 559
shaft and blade surfaces and wake pressure
variations (figure 5) for the mid-blade plane
indicate a minimal influence of the blades
and are typical of trailing-edge flow in the
presence of a thin boundary layer; however,
at this relatively low Re (laminar flow), the
adverse axial-pressure gradient associated
with the closing of the body is sufficient to
cause a small separation region in the vicin-
ity of the hub apex, .96 < x < 1.01. Note
the rapid rate of recovery of pressure in the
radial direction. The pressure variations
for the blade plane are similar, but clearly
show the effects of the blade leading and
trailing edges as well as a small displace-
ment effect of the blade boundary layer.
The wall-shear velocity magnitude U var-
iations (figure 6a) are consistent with those
just described for the pressure. For the
blade plane, there is a large reduction of
U in the juncture region due to the flow
retardation and also in conjunction with the
relatively large boundary-layer thickness
there, and a downstream shift of the region
of low U associated with the flow separ-
ation asp compared to the mid-blade plane.
The latter is consistent with the differences
in separation patterns for the blade and mid-
blade planes. On the blade, initially U is
larger at the mid-span than at the tipT in
response to the larger leading-edge pressure
peak at mid-span (i.e., more favorable pres-
sure gradient), then the trend reverses. The
wall-shear velocity vector (figure 6c) is
generally aligned with the axial direction
except near the blade leading edge where the
blade displacement effects are evident and in
the separation region where the complex topo-
logical nature of three-dimensional separ-
ation is displayed.
The wake centerline velocity Uc (figure
7a) displays the extent of the separation
region and the recovery of the wake. The
maximum swirl velocity Nmax is, of course,
nearly zero for the nonrotating condition and
not shown in figure 7b. The asymptotic forms
(figure not shown) display the details of the
recovery of the wake. Although the exit
plane is 34 diameters downstream of the pro-
peller plane (equivalently 5 shaft lengths
downstream of the hub apex), the slope of the
velocity defect of the shaft wake has not yet
reached its asymptotic value. This is con-
sistent with our previous turbulent-flow
calculations. In contrast, the slope of the
blade wake velocity defect is close to the
asymptotic value. The exit plane is 103
chord lengths downstream from the blade
trailing edge.
Lastly, for the nonrotating condition, the
detailed results are discussed. The discus-
sion to follow is based on the complete
results, which include the solution profiles
at all the stations designated in figure 4a;
however, for brevity of presentation, only
the near blade wake station is shown in fig-
559
ure 8. At the near-inlet station, the solu-
tion display the characteristics of the inlet
conditions, i.e., an axisymmetric, thin,
laminar boundary layer. At the leading-edge
and mid-chord stations, the solution shows
the initiation of the blade boundary layer,
including leading-edge (stagnation-point) and
displacement effects. Also, the juncture
flow indicates a weak leading-edge horseshoe
vortex. At the trailing-edge station, the
trailing-edge effects of both the blade and
the shaft are predominate, including a rever-
sal of the juncture flow. At the near blade
wake station and hub apex, the solution shows
the initial development of the blade wake.
Here again, the effects of the shaft trailing
edge are quite large. Two corner vortices
are apparent near the shaft axis which are an
indication of the nature of the flow within
the separation region. At the near, interme-
diate and far shaft-blade wake stations the
, ,
solution shows the recovery of shaft and
blade wakes. The crossplane flow and pres-
sure recover more rapidly than the axial
velocity component.
Next, consideration is given to the
results for the rotating condition. Referr-
ing to figure 5, in the vicinity of the hub
apex and in the near wake there is a decrease
in pressure due to the propeller-induced
swirl. The lifting effects due to the angle
of attack of the blade section are clearly
evident. Note that the pressure peak is at
the blade leading edge such that just up-
stream of the leading edge very large adverse
and favorable pressure gradients occur f or
the pressure and suction sides of the blade,
respectively, whereas just downstream of the
leading edge the reverse holds t rue.
The wall-shear velocity magnitude U (fig-
ure 6b) shows slightly increased valuers over
the spinning portion of the shaf t and greater
uniformity between the blade and mid-blade
planes in the separation region as compared
to the nonrotating condition. For the pre-
sent conditions, the rotation parameter R
= ~ Rh/Uo is quite small, i.e., R = .02,
which explains the only slight increase in
U as compared to the previous calculations
of ~ Stern et al. [9] ~ On the blades, U is
smaller on the suction than on the pressure
side, in conjunction with the relatively
thicker boundary layer on the suction as
compared to the pressure side. Consistent
with the results for the nonrotating condi-
tion, U is larger at the tip than at mid-
span except near the leading edge. On the
rotating section, the wall-shear velocity
vector (figure 6d) shows large effects due to
rotation, i.e., the flow is turned towards
the direction of rotation. In the blade
region, the passage vortex is evident,
including its helical nature. In the sep-
aration region, the f low is completely turned
in the direction of rotation which results in
the aforementioned greater uniformity in the
separation patterns between the blade and
OCR for page 560
mid-blade planes. Over the blade, the wall-
shear velocity vector is in the axial direc-
tion, except near the tip, where the flow is
outward, especially on the pressure side.
Figure 7a shows that the recovery of the
wake centerline velocity U is slower for the
rotating than the nonrotating condition.
This is due to the adverse axial-pressure
gradient induced by the hub vortex. Also
shown is the decay of the maximum swirl ve-
locity W in the wake (figure 7b), which is
associated with the intensity and decay rate
of the hub vortex. Finally, the asymptotic
forms (figure not shown) indicate that the
shaft wake is unaffected, the blade-wake
slope is increased, and the swirl decay is
relatively faster than that of the axial-
velocity defect.
The detailed results vividly display the
complexity of the flow for the rotating con-
dition. Here again, the discussion to follow
is based on the complete results, although
only representative stations are displayed in
figures 8 through 11. At the near-inlet
station, the solution is similar to that for
the nonrotating condition, except for the W
velocity component which shows a linear in-
crease due to the use of noninertial coor-
dinates. At the leading edge, the solution
shows the initiation of the blade boundary
layer, in this case, with significant differ-
ences between the suction and pressure sides
of the blade due to the influences of the
aforementioned abrupt changes in the pressure
gradients. Interestingly, the boundary lay-
ers on both sides of the blade are thicker
for the rotating than the nonrotating condi-
tion. The tip-vortex formation initiates
with flow around the tip from the pressure to
the suction side. The vertical flow is asym-
metric such that the tangential velocity
component is larger on the suction than the
pressure side, whereas the situation is re-
versed for the radial velocity component.
The passage-vortex formation also initiates
and dominates the juncture flow. At the mid-
chord station and trailing edge, the effects
of the pressure gradient changes are clearly
displayed, i.e., on the suction and pressure
sides, the flow is decelerated and acceler-
ated, respectively. On the suction side, the
boundary-layer thickness varies considerably
across the span. The tip vortex has lifted
off the suction-side surface such that the
radial velocity component is positive on both
sides. Braiding of the fluid from both the
suction and pressure sides is apparent, but
particle trajectories were not traced to
display this phenomenon. The pressure is
surprisingly uniform in view of the cross-
plane flow, however, very low values are
observed in the tip-vortex core. The passage
vortex increases in size and its core moves
towards the suction side. The axial-velocity
and -vorticity contours are hook shaped near
the tip due to the influences of the tip vor-
tex. At the near blade wake station and hub
560
apex, the solution shows the development of
the blade wakes, which indicate the charac-
teristics of the complex mixing of the suc-
tion and pressure side three-dimensional
boundary layers, including significant ef-
fects of the tip, passage, and hub vortices
and the hub-induced pressure gradients. The
minimum velocity in the wake migrates towards
the suction side. There is a rapid recovery
of the pressure-side wake such that the ve-
locity-defect region is mainly behind the
shaft and off the suction side of the
blade. The blade wake becomes quite thick as
it merges with the wake of the shaft and the
tip vortex. There is a region of backward
flow near the wake axis associated with the
flow separation. The tip vortex reduces in
intensity and the passage vortex merges into
a large asymmetric hub vortex. Finally, at
the near, intermediate, and far shaft-blade
wake stations, the nature of the recovery of
the wake is displayed. It is clear that the
circumferential mixing is faster for the
rotating than the nonrotating condition which
is also the case for swirling jets.
The close-up views of the tip vortex shown
in figure 12 clearly display its initiation
at the blade leading edge, subsequent migra-
tion off the surface along the blade chord,
and decay as it is convected and diffuses
into the wake. Also, they reveal the mechan-
ism of the tip-vortex formation. At the
leading edge, nearly all of the fluid forming
the tip vortex originates f ram the pressure
side, whereas further downstream the suction
side f luid is "pumped" into the tip vortex.
This indicates a "braiding" process, which is
of ten ref erred to as the tip-vortex roll-
up. The tip-vortex trajectory is shown in
figure 13.
Turbulent Flow
Some limited turbulent-f low results are
shown in f igures 14 and 15. The turbulent-
f low results are consistent with and very
similar to those for laminar flow. In gen-
eral, the differences are as expected based
on physical reasoning, i.e., viscous effects
are conf ined to narrower regions and the
three-dimensionality of the f low is consider-
ably reduced f or turbulent as compared to
laminar flow. Also, quite apparent for tur-
bulent f low is the reduced resolution near
solid surf aces and the wake centerplane due
to the present wall-function approach.
The overall trends described above with
regard to the shaf t and blade surf aces and
wake pressure, wall-shear velocity, and wake
centerline and maximum swirl velocities are
quite similar; however, the pressure peak at
the hub apex is considerably larger and there
are some dif f erences in the wall-shear veloc-
ity behavi or due to the absence of separ-
ation. The detailed results are also quite
s imi far. However, f or the nonrotating condi-
tion the juncture effects are minimal and the
OCR for page 561
crossplane flow and pressure variations are
reduced, whereas, for the rotating condition,
the tip and passage vortices are larger and
persist longer, the latter merges into a
larger hub vortex, lower pressures are
observed in the tip-vortex core, and the
recovery of the wake is considerably
faster. The turbulent kinetic-energy pro-
files show two peaks, one near the wake cen-
terline and one corresponding to the tips of
the blades.
Comparison With Results from a Lifting-
Surface Propeller-Performance Program
Unfortunately, no experimental information
is available for the present geometry.
Therefore, to aid in evaluating the present
work, comparisons have been made with some
relevant experimental and computational stud-
ies, including the following topics: juncture
flow which is related to the present flow in
the blade-hub juncture region for the nonro-
tating condition; tip flow which is related
to the present flow in the tip region for the
rotating condition; turbomachinery flow which
is related to the present blade boundary-
layer and wake development and blade-to-blade
flow; and propeller flow which is, of course,
the topic and goal of the present study.
Although in most cases, the comparisons are
only qualitative due to the large differences
between the topic and present geometries,
they support the present results in that the
predicted flow structures are similar and
consistent with the results from these stud-
ies. The complete comparisons are lengthy
and beyond the scope of the present paper
(see Kim [11]). Herein, only the direct
comparisons between the present turbulent-
flow results and those from a lifting-surface
propeller-performance program, i.e., PUF-2
(Kerwin and Lee [2]) will be presented.
Special modifications of PUF-2 for the
present idealized geometry were not deemed
necessary, and, therefore, not done. A con-
stant pitch ratio PlDp = 105 was used to
represent the infinite pitch of the present
geometry. All other geometry input data was
given the same values as those for the pre-
sent turbulent-flow calculations. Also, the
open-water condition value was used for the
advance coefficient J = 44.44, i.e., the
effective wake due to the interaction between
the propeller and the shaft boundary layer
was neglected. For the wake-model param-
eters, the standard values for the wake pitch
and zero contraction were used. A value of
.005 was used for the section-drag coeffici-
ent which is based on the present calcula-
tions.
Figures 16a,b show a comparison of the
chordwise and spanwise distributions of the
blade loading in terms of the pressure jump
(figure lea) and section-lift coefficient
(figure 16b), respectively. A large differ-
ence in the pressure jump is observed near
561
the leading edge. Differences are also seen
in the section-lift coefficient. The viscous
results show considerably larger values near
the root and the tip, but smaller values for
the mid-span region. The higher root loading
for the viscous flow is, no doubt, a result
of the increased effective angle of attack
due to the oncoming shaft boundary layer.
However, a part of the difference may be due
to the lack of hub effects in PUF-2. The
lower mid-span loading is consistent with the
aforementioned differences in chordwise load-
ing near the leading edge. The higher tip
loading may be due to the reduced pressure on
the suction side due to the tip vortex.
Interestingly, in spite of these differences
in the loading distributions, the total for-
ces and moments show remarkably close agree-
ment.
Figures 16c,d show a comparison of the
propeller-induced velocities just upstream
and downstream of the propeller at the mid-
span radius. For the viscous-flow solution,
the propeller-induced velocity (u,v,w) is
defined as the total velocity, (U,V,W) minus
the freestream (UO,O,O) value. Results are
shown using the blade angle coordinate
~ = at as the abscissa for the entire blade-
to-blade region from the suction (0 = 0 deg)
to the pressure side (D = 90 de").
The velocity components just upstream of
the propeller (figure 16c) clearly show the
effects of the leading-edge stagnation
point. The u velocity components show sim-
ilar trends, i.e., the point of the minimum
velocity shifts to the pressure side which
suggests that the stagnation point also
shifts to the pressure side. The increased
magnitude f or the viscous solution may be due
to the prescribed overshoot for the oncoming
shaf t boundary layer. The v velocity compon-
ent is nearly zero f or both results. The w
velocity components also show similar trends;
however, the inviscid solution indicates a
stronger local effect of the leading-edge
stagnation point than the viscous solution
such that the circumf erential-average is zero
for the inviscid but not the viscous solu-
tion, i.e., the viscous solution indicates
small negative preswirl.
The velocity components just downstream of
the propeller (figure led) highlight the
differences between the viscous and inviscid
solutions. The inviscid u velocity component
shows very small positive values f ram the
suction to the pressure side, whereas the
viscous u velocity component shows a large
change f ram the suction to the pressure side,
i. e., the vi scous blade wake appears as a
sharp drop on both the suction and pressure
s ides and the ef f ects of the retarded suc-
tion- and accelerated pressure-side boundary
layers are clearly evident. The v velocity
components show similar trends, but with
somewhat larger variations f or the viscous
solution. The w velocity components also
OCR for page 562
show similar trends, but with larger swirl
for the viscous solution in spite of the 2.
smaller loading.
Concludi ng Rema rks
The present work was motivated by the
limitations of the interactive approach for
s imulating the complex blade-to-blade f low.
This has certainly been accomplished by the
present viscous-solution method, albeit for
an idealized geometry. In fact, the present
work provides, for the first time, a very
detailed documentation of the viscous f low
around a propeller for both laminar and tur-
bulent f low. It is concluded that the pre-
sent approach is capable of simulating
marine-propeller f low f ields, including both
the propeller loading and the complex blade-
to-blade f low, and should be extended f or
practical geometries. It is also concluded,
based on the comparison of the laminar and
turbulent results, that, although most
aspects of the f low are governed by pressure-
gradient ef f ects, improvements in turbulence-
modeling procedures, especially near-wall
treatment, are important to resolve certain
flow features, including transition, separ-
ation, and small-scale vertical structures
such as leading-edge horseshoe and secondary
vortices.
Of course, much more work needs to be done
to extend the method to realistic propeller
and body geometries. Some of the issues that
need to be addressed are as f allows. Optimum
coordinates, including investigations of
inertial and helical systems. Optimum grid-
generation techniques f or complex, three-
dimensional, propeller-driven bodies, includ-
ing investigations of moving, adaptive, and
multi-block grids. As already mentioned,
improved turbulence-modeling procedures are
essential and possibly a pacesetting issue.
Also, further development of solution algor-
ithms is a necessity in order to perform the
required large-scale computations even on the
most advanced available supercomputers. It
should be recognized, that none of these
issues are trivial, on the contrary, all
require substantial effort so that it is
expected that the present problem will remain
a challenge f or many years to come.
Acknowledgements
This research was sponsored by the Of f ice
of Naval Research, Accelerated Research Ini-
tiative Program in Propulsor-Body Hydrody-
namic Interactions, under Contract N00014-85-
K-0347. The Graduate College of The Univer-
sity of Iowa and the National Center for
Supercomputing Applications Academic Affil-
iates Program provided a large share of the
computer funds.
Ref erences
1. Kerwin, J.E., (1986), "Marine Propel-
lers, " Ann. Rev. Fluid Mechanics, Vol.
18, pp. 367-403.
562
3.
4.
Kerwin, J.E. and Lee, C.S., (1978), "Pre-
diction of Steady and Unsteady Marine
Propeller Performance by Numerical Lift-
ing-Surface Theory," Trans. SNAME, Vol.
86, pp. 218-253.
Hess, J.L. and Valarezo, W.O., (1985),
"Calculation of Steady Flow about Propel-
lers using Surface Panel Method, " J.
Propulsion, Vol. 1, pp. 470-476.
ITTC, (1984), "Report of the Propeller
Committee," Proc. 17th Int. Towing Tank
Conf ., pp. 139-194.
5. Morris, P.J., (1981) , "The Three-Dimen-
sional Boundary Layer on a Rotating Heli-
cal Blade," J. of Fluid Mech., Vol. 112,
pp. 283-296.
6. Groves, N.C. and Chang, M., (1984), "A
Differential Prediction Method for Three-
Dimensional Laminar and Turbulent Bound-
ary Layers of Rotating Propeller Blades, "
Proc. 15th ONR Symp. on Naval Hydro., pp.
429-444.
7. deJ ong, F. J. ., Govi ndan, T. R., Levy, R.
and Shamroth, S.J., (1988), "Validation
of a Forward Marching Procedure to Com-
pute the Tip Vortex Generation Process
for Ship Propeller Blades, " Proc. 17th
ONR Symp. on Naval Hydro., Hague, The
Netherlands.
8. Stern, F., Kim, H.T., Patel, V.C. and
Chen, H.C., (1988), "A Viscous-Flow Ap-
proach to the Computation of Propeller-
Hull Interaction," J. Ship Research, Vol.
32, No. 4, pp.246-262.
9. Stern, F., Kim, H.T., Patel, V.C. and
Chen, H.C., (1988), "Computation of Vis-
cous Flow Around Propeller-Shaft Config-
urations, " J. Ship Research, Vol. 32, No.
4, pp. 263-284.
10. Chen, H.C. and Patel, V.C., (1985), "Cal-
culation of Trailing-Edge, Stern and Wake
Flows by a Time-Marching Solution of the
Partially-Parabolic Equations, " Iowa
Ins t i tute of Hydraul i c Resear~ch, The
University of Iowa, IIHR Report No. 285.
11. Kim, H.T., (1989), Computation of Viscous
Flow Around a Propeller-Shaf t Conf igur-
ation with Infinite-Pitch Rectangular
Blades, " Ph. D. Thesis, The University of
lowa, Iowa City, IA. -
12. Patel, V.C., Chen, H.C. and Ju, S.,
(1988), "Ship Stern and Wake Flows: Solu-
tions of the Fully-Elliptic Reynolds-
Averaged Navier-Stokes Equations and
Comparisons with Experiments, " Iowa
Institute of Hydraulic Research, The
University of Iowa, IIHR Report No. 323.
OCR for page 563
Figure 1. Propeller-shaft configuration with
inf inite-pitch rectangular blades.
(b) blade-to-blade f low
Figure 2. Propeller-f low phenomena.
L
\ ~~mm~:lry Antis ,~ \ ~
SOD
(a) physical domain
- 1
(a) circumferential-average flow
~//J ~
so
Exit Plans
1~ ~
_ ~ . ~
~~~
Figure 3. Solution domain.
563
(b) computational domain
OCR for page 564
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Figure 5. Shaf t and blade surfaces and wake
pressure: laminar flow.
. oa ~ NNRoTAT~NG
ur
. 06
ur
. 04
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Figure 6. Wa '1-shear velocity: laminar flow
564
1 0
OCR for page 565
Qel
0.6 .
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O-
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NONROTATI~==~
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Figure 7 . Wake velocit ies: laminar f low .
w,o
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Figure S. Velocity and pressure profiles: laminar flow.
565
OCR for page 566
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Figu re 8 . ( cont inued ) .
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Figure 9. Axial-velocity contours: laminar
:,,~,: :,~..
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- -. 01
~e
32 //~ 09S
31 W.
— — J l ~ ~
.05 .07 -.01 .01 .03 .OS .07
ZIL
f low, rotas ing.
`, .12 ~
X,L-I. Rsl
.08
.06
Y'L
. 04
. D2
1 o.
_ n~
Figure 10. Crossplane-velocity vectors: laminar flow, rotating.
X/L-. 97O2
oOO:
so 50 ,00 073
- - .072
. _. .03 .05
Z/L
n7`
. 08
. 07
.06
. 05
.n4
Y/L
. 03
. 02
. 0 1
O.
n.
~>lo X/L- 1.
.07 -. 001 0. .001 .002 .003 . nn4 -. 01 .01 Z/L -. 01 .01 . 03
Figure 11. Axial-vorticity contours: laminar flow, rotating.
566
X/L - ~ . S32
.
.05 . 07
OCR for page 567
.085 ~ .085 .
.083 _ _ .083 . '
. 031 ~ ~ . 081 .
. 079 ; _ _ _ ~ . 079 ,
Y/L,07s ~ '~ ~ ~ ~ - ~ ~ ~ Y/~.075 1 ! i. ~ ,` - , i ~ i
. 069 _ ~ ~ _ _ _ . 069 - ~ ~ ~~ _ ,
.067 _ X/L~. 8455 SCAL£: UO-. 016 .067 ~ X/L-. 942B - ~ ~~ ~uD~. 016
.065 1 - . 06S . ' ~
-. 02 ~. 01 0. .01 .02 -. 02 -. 01 0. .01 .02
Z/L Z/L
.085 _ _ _ _ .08 . , ... ~.
.083 ~ ~ ~ .078 .
.081 _ _ ~~_ _ _ .076 , , .... _
. 079 ; _ ~ ~ . 074 . . '
'C77g 5 ~ - i ~ ~ 06SI~ ' ~ ~~~—~~
. 069 ~ r te. ~ ~ ~ ~ ~ . 064 ~ _
. 067 ~ X/L-. 8702 SCALE: UD-. 016 ~ .062 ~ X/L 1. _~_ UO~. 016
.065 .06
-. 02 -. 01 0. .01 .02 -. 02 -. 01 0. .01 .02
Z/L Z/L
.085 _ _ _ _ _ .08 .
.083 ~ ~ .078 .
.081 _ _ ~~ . .076 , ,,,_ . .
. 079 ; ~1_ . ~ , 074 .
Y/L, O75~ ~ ~ ~ ~ ~ 05B , , ~ ~_~- _ ~ ~ _ _
. 069 ~ ~ ~ r ~ ~ ~t~ , , ~ . 064 ~ = =
. 067 ~ _ X~'L-. 895 SCALE: UO~. 016 ~ .062 - X/L-;. 05 SCA.LE : UO-. 016 ~ ~
.065 .06
-. 02 -. 01 0. .01 .02 -. 02 -. 01 0. .01 .02
Z/L Z/L
Figure 12. Close-up view of the tip vortex: laminar flow.
2.0 , , , , ,
1.0 . -10 , · · ·
_~ Helical Trace of T.E.
= -1 0~ ~ ~ -° 10~ ~ .-_~
_3 0t Turbulent(Large`) ~ ~ _ ~ Helicol T~cce of T.E. ~ ~ ,_
-40L + Laminar(Z=g) ~g 1 40~ .
-5.0 50
0 1 2 3 4 5 0 ~ 2 3 4 5
( X XLE)/ C LE )
Figure 13. Tip-vortex traj ectory.
toe .077 toe .oa
w-.: L~w'-~- Y~2~ :'"~
-.01 .01 .03 .0~ .07 . -.002 -.001 0. .001 .002 .003 -.01 .01 .03 .05 .07 -.01 .01 .03 .OS .07
Z/L 2'L Z'L Z'L
Figure 14. Axial-velocity contours: turbulent flow, rotating .
567
OCR for page 568
1.4 I .4 1.4
1.2 X..8702 1.2 X-1. t.2 X-1.532
r ~ := _: _, ~ ~ '; ~
0. 0. 0.
0 4 ~ 12 0 4 8 12 0 4 8 12
2 6 10 2 6 10 2 6 10
K ~ 1000 K ~ 1000 K ~ 1000
(a) nonrotating
1.4 1.4 ~ .4
1.2 X-.8702 1.2 X-1. 1.2 X-1.532
Y ~ ~ ~ ~ Y '0
0. 0. 0.
0 4 8 12 0 4 8 12 0 4 8 12
2 6 10 2 6 10 2 6 10
K ~ 1000 K ~ 1000 K ~ t000
(b) rotating: suction side
~ .4 ~ .4 1.4
1.2 X..8702 30 1.2 Xa1. 1.2 X-~.532
2 ~ ~ ~ ~[ Y ~ ~ ~ ~ 16
0. 0. 0.
0 4 8 12 0 4 8 12 0 4 8 12
2 6 10 2 6 10 2 6 10
K 41000 K 61000 K c 1000
(c) rotating: pressure side
Figure 15. Turbulent kinetic energy profiles.
0.5, . , . . . . . . . . 0.06 r ~
2 :~—==~1- ~
~ Viseous (Turbulent Flow)
-0.06 . ~ . . . . . . . .
0 10 20 30 40 50 60 70 80 90
~ (D.gr~o)
(c) upstream propeller-induced velocity
_
0.4
n
0.2
0.1
—Inviscid (PUF2) r/R. - 0.65
-. Viscous (Turbulent Flow) (laid—Span)
1't\
_ _~\
. ~__
0.0 _
0.2 0.4 0.6 0.8 1.0
X/C
(a ) chordwise load ing
0.9 ~ Inviocid (PUF2) ~~
. Cr ~ 0.~70. C4 - 0.116, '` \
0.8 - r/Rp — 0.680 '`
0 7 ~ Vlacous (Turbulont Flow) ,'
r ~ 0 162. Cux ~ 0 109. , /
0.6 r/RF ~ 0.674 ,' /
. ," /
0.5 _ ~
0.4 -
. ~ ,,
03 ~, . ~
0.00 0.02 0.04 0.06 0.08 a. 0
Cl
(b) spanwise loading
Figure 16. Comparison of turbulent-flow and lifting-surface
propeller-performance program results.
568
on~
0.02
~1 °
_n n,
—0.04
. , .
X/R~ — - 0.3606
r/R, - 0.648
0041-
0.02
,.1~- o
_nn'
° ::: r.'
—v.v~ ~
0 10 20 30 40 SO 60 70 80 90
~ (Dogreo)
(d) downstream propeller-induced velocity
OCR for page 569
DISCUSSION
by K. Mori
Although an explicit description about a
systematic accuracy analysis is requested by
the paper committee, no descriptions are found
in the paper. Because the accuracy analysis is
primarily important for the computational
fluid dynamics, it should have been mentioned,
although the procedures are not definite yet.
DISCUSSION
by S. Kinnas
I would like to congratulate the authors
for their interesting paper. I have however
two questions to raise.
1) Concerning the circulation distribution
that they show in Fig.16(b) as predicted by
the presented method: is it a convergent
result with respect the chordwise and spanwise
grid discretization on the propeller blade?
2) In the case of a realistic propeller,
with blade thickness included, what would they
expect to be a reasonable grid on the
propeller in order to capture the detailed
flow at the propeller leading edge and tip?
Author's Reply
We thank both the oral and written
discussers of our paper for their pertinent
remarks.
With regard to Prof. Mori's comments, we
apologize for not including an explicit
statement of accuracy in the paper, and, at
this time, offer the following. As stated in
the paper, the present overall computational
method is based on that used previously for
calculating propeller-hull interaction [8,9]
in which a viscous-flow method for calculating
ship-stern flow [10,12] is coupled with a
propeller-performance program in an
interactive and iterative manner to predict
the combined flow field. References [8,9] and
[10,12] provide numerous applications for
propeller-hull interaction and bare bodies,
respectively, including validation studies
through grid-dependency and convergence check
as well as comparisons with experimental data
and other analytic and numerical solutions.
Some limited grid dependency and convergence
check were also done for the present
application to test the extensions and
modifications for calculating marine-propeller
flow fields. That is, some preliminary
turbulent-flow calculations were performed
using a coarse grid, i.e. 36x22x16 (16,672).
The coarse-grid solutions converged more
rapidly (i.e. in about 40 global iterations)
than the fine-grid solutions. Qualitatively
the coarse-grid solutions were very similar to
the fine-grid solutions, but with considerably
reduced resolution. Also, as stated in the
paper, unfortunately, no experimental
information is available for the present
geometry; therefore, to aid in evaluating the
present work, comparisons were made with some
relevant experimental and computational
studies, including the direct comparisons
between the present turbulent-flow results and
those from a lifting-surface propeller-
performance program which are provided in
Fig.16 (see [11] for the complete
comparisons).
With regard to Dr. Kinnas's comments, the
solutions presented are fully converged for
the present grid. As discussed in the
Concluding Remarks, grid-generation for
complex geometries is an important issue which
must be considered in extending the present
method to realistic propeller and body
geometries. Presently, calculations are in
progress for the SR-7 turboprop using a
single-block, H-grid of somewhat higher
density than the present one (i.e.
64x46x36=105,984),but with x1=x1( ~,n,: ) in
order to have the grid conform to the three-
dimensional curved boundaries of the skewed
and twisted blades and the nacelle. The
results are very encouraging: however, it is
anticipated that in order to completely
resolve all the details of the flow field,
especially for marine propellers, multi-block
grids will be necessary, including H-, C, and
O-types.
569
OCR for page 570
Representative terms from entire chapter:
computational method
