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OCR for page 588
24~ Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Calculations of Flows over Underwater Appended Bodies
with High Resolution ENO Schemes
Zhen-yu Huang, Hong-rong Cheng, Lian-di Zhou
(China Ship Scientific research Center, Wu Xi, 214082, China)
ABSTRACT
The numerical method based on flux
difference splitting LU decomposition; implicit high-
resolution third order Essentially Non-oscillatory
(ENO) scheme is constructed for efficient
computations of steady-state solution to three-
dimensional, incompressible Navier-Stokes equations
in general coordinates. The flowfields over
underwater axisymmetric bodies, full-appended
axisymmetric bodies and axisymetric bodies with a
ring-wing duct are simulated. The method has been
shown to be capable of predicting the
circumferential-mean velocity distribution at model
scale to accuracies of around 3% of measured values,
and of predicting some details of flow feature, for
example the wake harmonics.
KEYWORDS
Essentially Non-Oscillatory (ENO) Schemes,
Flux Splitting, LU Decomposition, Computational
Fluid Dynamics (CFD)
INTRODUCTION
The flow over an underwater appended
vessel during level flight is characterized by the
development of thick boundary layers, flow
separation, flow into and through the propulsor,
vortices formed at the root and tip of appendages and
appendage turbulent wake. The spatial nonuniformity
and temporal fluctuations of flow into vessel
propulsor significantly affect propulsor noise. Whilst
the fluid flow can be measured by using a scale
model of the vessel in a towing tank or wind tunnel,
there are errors associated with applying results
obtained at model scale to full scale. Furthermore,
the experiment requires many detailed measurements
to be taken in the regions of interest. This is costly
and time consuming. Recent advances in CFD have
been developed, which have been shown to be
capable of predicting flow over full-appended
underwater bodiesEHuang, 20011. This paper presents
a high-resolution numerical method, which have
been used to calculate the flowfields over appended
underwater bodies. The appendages include a
fairwater, four-wing stern appendages and a ring-
wing duct.
The numerical scheme based on flux-
difference splitting, LU decomposition and implicit
high resolution third-order ENO achieved through
flux reconstruction, are constructed for efficient
simulation of steady-state solution to three-
dimensional, incompressible conservative Navier-
Stokes equations in general coordinates. The
schemes have been shown to be capable of predicting
the circumferential-mean velocity distributions at the
propulsor plane of full appended submarine-like
bodies to accuracies of around 3% of measured
values in wind tunnel, and of predicting some details
of flow feature. The computer code has been applied
to evaluate and optimize the flow characteristics over
single axisymmetric bodies and underwater full-
appended bodies and to simulate axisymmetric
bodies with a ring-wing duct.
GOVERNING EQUATIONS
.
The governing equations, which describe the
motion of viscous, incompressible fluid, are Navier-
Stokes equations and the equation of continuity.
Artificial compressibility method, which adds a time-
derivative of the pressure to the continuity equation,
can couple the equations of motion with the
. . ,
OCR for page 589
continuity equation. Then the most efficient implicit
time-dependent methods can be applied to the
incompressible Navier-Stokes equations, the
complete set of governing equations can be solved
simultaneously~Peter, 19881.
The Navier-Stokes equations in conservation
law form for an incompressible, three-dimensional
flow are written as
Q'+(E -EV)X +(F -FV)Y +(G -GV)Z =0
With the inviscid flux vectors
E = ( pu, u + pi us, uw)
F = ( pV,Uv, V2 + petit (2b)
G = (~,uw,~,w + p) (2c)
And the shear flux vectors
EV = Re~~0~2ux~uy +vx~uz + WX)T
FV = Rem (0, uy + Ax, ,2vy, vz + wy )T
GV = Rem (0, uz + wx, vz + wy,2wz AT
Where Re is Reynolds number. Following the
artificial compressibility method, the dependent
vector Q in Eq. (1) is defined as
Q=(p,u,v,w) (4)
Considering a coordinate transformation of the form
~ =~(x,y,z), 77 =r1(x,y,z), and 5 =5(x,y,z),
Equal) can be rewritten in strong conversation law
form.
(QIJ)t +(E-EV): +(F-FV)n +(G-GV)5 =0
(5)
The flux vectors E, F. G are linear combination of
E ,F ,G in Eq. (11. For example, E can be
written as
E = (4X I J)E + (;Y I J)F + (¢Z I J)G (6)
Where J is the Jacobean of the coordinate transfor-
mation.
HIGH RESOLUTION SCHEMES FOR
INVISCID FLUX
Because of the complicity of flowfield
structure around the underwater bodies with full
appendages, the Essentially Non-oscillatory (ENO)
schemes [Harten,1987], which were developed by
Harten et al, are applied in discretization of inviscid
flux of three-dimensional incompressible Navier-
Stokes equations. The ENO schemes, which use
adapted stencil, are uniformly high-order accuracy
throughout even at critical points.
Following Yang [Yang 1992,Huang 1999,
2000], third-order nonoscillatory schemes are given
`2ay below. TakedE`:in direction ~ as an example, let
A = bE/3Q, (A E ji+l/2 = (71, /2, 73, /4 ~ iS
the eigenvalue diagonal matrix of A. R and L is right
and left eigenvector matrices of AK. Then one can
get A=RAEL. The spatial difference of E`: can
be reached by using finite volume method
(3a) (FVM)[Huang 1999, 20001.
(3b)
(3c) And
E`.EN,°32 = 2 (Ei + Em + Ri+~,2O ~EN1O,32 ) (8)
E: = EEN03 _ EEN03 (7)
Let ~i+l/2 = L(Qi+~ - Qi), the components of
~~.E+l03 can be defined as
~+l/2 = ~~/i+l/2~i + fit)
(/i+l/2 tori + ~i+
- ~~;i+l/2 + ii+l/2 + ii+l/2 ~ai+l/2'~ai-1/2 ~ < ~ai+l/
+
A
;i+l/2 tori + ~i+1 ~
- ~~/i+l/2 + ii+l/2 + ti+l/2 ~ai+ll2,1ai ll2 | > |ai+ll2 |
Where 6, 6, and ~ functions are given by:
6(z) = 2 (#r(z) - BZ2 ) (10)
(Z) = 6 (2|z| - 3BIzl2 + B2lzl3 )
~(Z) = ~ (B2 1Z13 - 1Zl) (12)
(11)
{ ( Z + £ 2 ~ / £ k At' £
OCR for page 590
His a small positive constant. B is the value of
At I A; . One can find that the flux OiE+N/ 3 iS self-
adapted with the distribution of the flowfield physics.
And
pi = min mod[(xi+ll2 '(7i-l/2 ] (14)
~ =m[^ Hi lid, /~+ai lit] lai lit| <|ai+l,2| (15)
pi=m[~-ai+ll2' ^+`xi+l/2] lail,2|>|ai+l,21 (16)
Ill/ 2 = ~ (ai+l/ 2 )~(§i+1 ~ i) / ~i+l/2 ai+ll2 ~ O
~ O otherwise
_ _
~i+l/2 = ~ (ai+l/2 ) ('i+1 -§ i) /~i+112 ~i+l/2 ~ O
I O otherwise
(18)
A ~
i+l/2 = ~ (ai+l/2 ) (§i+1 -I i) /ai+ll2 ai+l/2 ~ O
otherwise
(19)
H = it + jH(~) + kH(5
H(~)=E-R Ev
H(~) = F- R Fv
H( ) =G- ~ G
(22)
If Eq. (21) is applied over a hexahedral cell
(/~=/~=~\5=1), temporal derivative discretiza-
tion uses the first-order-accurate differencing,
inviscid terms use implicit differencing, viscous
(17) terms use explicit central differencing, Eq. (4) will be
discretized as
J ~ (Qn+~ _ Qn ~ + Att [E.n+~/2—E'n+~,
+ fF,.n++,.,2—F,.n+~,2 ~ + EGkn++~,2—Gkn+i,
= ~ ~ LEV .n+l,2—EVin I,2 ~ + EFvj+~/2—Fv~-~/2
+ [GVk+~/2 GVk-~/2 ~ ~
(23)
{Z eye> ~Z~ (20) The definition of (A~)i+~,2 and
A = (dE I aQ)i+~,2 are same as the last section. Let
The remaining fluxes of three-dimensional
Navier-Stokes equations (Eq.(5)) can be defined with
the similar way.
LU DECOMPOSITION FOR TEMPORAL
DERIVATIVE
(AK )i+l/2 = [(AK )i+l/2 + |(AE )i+l/2 |] / 2 (24)
If LE ~ RE are the left and right eigenvector
matrices of Jocob matrix A, one can get
By using Finite Volume Method (FVM), the AN = ReAi LE (25a)
integrated form of Eq. (5) can be rewritten as[Huang
and Zhou. 2000,2001] Similarly
—[— -IQ V]+— H.dS =— D V Bi =RFA-FLF (25b)
~tV; Al Vl
Where S is the surface around the cell, do is
the normal vector of each surface, H is the tensor
whose vector components in three directions are
(21) C = RGAGLG (25c)
Let Qn+1 _ On = i\Qn LU decomposition
formula of Eq.(5) caI1 be expressed as
OCR for page 591
J-l[I+AtJ(A+ +B+ +C+ )]x
[I—~J(A- + B- + C- )]~`Qin;
=—Ad{ [El+ll2—El-ll2) + (FJ+l/2—FJ-l/2 )
+ (Gk+l,2—Gk l,2 )] } + R { [EVI+1I2—Evl-l/2 ]
+ [Fv)+l/2—Fv) l/2 ] + [Gvk+l/2—Gvk_l/2 ] }
TURBULENCE MODEL
The effects of turbulence fluctuations, which
can be solved on the computational grid, are
approximated with simple Smagorinsky model.
A, = (CsL)2lSijI (27)
Sip = 2 (Oxj + Oxi )
In which the model coefficient
Cs = 0~05 ~ 0.1 and length scale is geometric mean
of the grid spacing L = (~c/\yAz)l/3 .
RESULTS AND DISCUSSIONS
The flowfields around the appended
underwater axisymmetric bodies are simulated as
steady-state solutions to the incompressible Navier-
Stokes equations. The appendages include a fairwater,
four-wing stern appendages and a ring-wing duct.
Three kinds of flowfields are included in the
numerical simulation. They are the flowfield over the
axisymmetric bodies, the axisymmetric bodies
appended with a fairwater and four-wing stern
appendages and the axisymmetric bodies with a ring
wing duct.
The main bodies of most modern underwater
vessels are axisymmetric or cylinder-like shapes,
such as submarine, torpedo or other deep-sea
vehicles. For validation of numerical method, the
flowfield around the axisymmetric bodies are
simulated firstly. Fig.1 shows the comparison of the
numerical velocity distributions at difference
positions with the experimental data, which are
measured in wind tunnel. The numerical results agree
with the measured very well, which indicates that the
third-order ENO schemes are high resolution, and
can be used in the numerical simulation of three-
dimensional incompressible Navier-Stokes equations.
In general, the geometry of underwater
vehicles are very complex. There are various
appendages attached to the main body. Take
submarines as an example, the appendages include a
fairwater, rudder, several wing-like stern appendages,
and even a ring-wing-like duct.
The flowfields around underwater axisymm-
etric bodies with a fairwater and four-wing stern
appendages are simulated with the third-order high-
resolution ENO schemes. The complete calculations
(26) are carried out by two steps. The flowfields around
the body and a fariwater are numerically simulated
firstly, which can provide the inlet boundary
condition for the following fine simulation of
flowfield around stern part of body and four-wing
stern appendages.
The numerical dimensionless circumferential
-mean velocity along the radius of the propeller are
presented in Fig. 2, which are in good agreement
with the experimental data. The details of numerical
dimensionless circumferential-mean velocity and the
experimental data can also been found in Table 1.
28 Except one points, the relative error between the
numerical results and the experimental data is less
than 3%, their average relative error is only 2.107%,
the accurate numerical nominal wake at propeller can
be used as input data of vehicle propulsor blade
design. Also the numerical dimensionless
circumferential velocity distributions at different
radius station are showed in Fig. 3(a)-Fig. 3(e). There
are difference between the numerical results and the
experimental data, but their phases are similar, so the
calculated circumferential velocity can be applied to
the optimization and evaluation of hydrodynamics
noise of vehicle propulsor. The same accuracy has
been reached from the other numerical simulations
over different models. The code developed in this
paper has been used in the design and optimization of
new underwater bodies with full appendages.
Table 1. Comparison of Numerical circumfere-
ntial velocity with experimental data
r/R
0.217
0.300
0.400
0.518
0.678
0.840
1.000
Numerical
results
0. 4866
0. 5177
0. 5471
0. 5935
0. 6681
0. 7387
0. 7933
Experiment
data
0. 4833
0. 4911
0. 5319
0. 5993
T o. 6814
0. 7479
_ 0. 8064
Error
(%)
.
0.68
.
5.42
2.88
-0.97
-1.95
-1.23
-1.62
OCR for page 592
Ducted propulsors are known to offer
significant advantages for particular marine
applications, such as increases in efficiency for high
propeller loading with flow-accelerating ducts, or
alternatively smaller propeller sizes; reduction of
inflow velocity and, consequently, of cavitation and
noise with flow-decelerating ducts; better control
over the inflow to the propeller; improvement of
maneuverability and position-keeping abilities of
vessels; protection from damage to the propeller, etc.
The ducted propeller has been installed in underwater
vehicle, such as submarine and torpedo, to increase
the vehicle speed and reduce the propeller
hydrodynamics noise. The inflow velocity, which is
related to the geometry of ducts and the interaction
between duct and main body, is very important to the
design of propulsors' blade. For the first step, the
flowfields around the axisymmetric bodies with a
ring-wing duct has been simulated. The numerical
results are presented in Fig.4, which is compared
with the experimental data measured in wind tunnel.
Because the geometry used in the calculation is little
different from the real model, which attached more
appendages. Meanwhile the experimental data is very
limited. The numerical results do not agree with the
experimental very well. But, by the innovation of
numerical schemes and multi-block grid system, the
internal and external flows of the combinations of
full-appended unterwater bodies and ring-wing duct
will be simulated more accurately. It is sure that the
numerical nominal wake in the duct would be used in
the design of duct-propeller.
CONCLUSION
Based on flux splitting, implicit high-
resolution schemes have been constructed for
efficient calculations of steady-state solutions to the
three dimensional, incompressible Navier-Stokes
equations in curvilinear coordinates. The third-order-
accurate efficient ENO has been applied in the
calculations, which can capture the details of the
flowfield around underwater bodies with full
appendages. The numerical results agree quite well
with the experimental data. The schemes and code
developed in this paper can be applied in the design
of underwater vehicle propulsor and in the
optimization and evaluation of its hydrodynamics
noise. Also the code can be used in the optimization
and design of shapes of vehicle body and its
appendages.
ACKNOWLEDGE
The project was supported by Shanghai
Natural Science Fund (No.OOZF14065) and National
Key Laboratory on Hydrodynamics (No. H9958~.
REFERENCE
Peter M. Hartwich, Chung Hao Hsu, "High -
resolution upwind schemes for the three-dimensional
incompressible Navier-Stokes equations", AILS J.
Vol. 26,No.11,1988,pp.1321-1326
Harten A, Osher S. "Uniformly High-order Accurate
Nonoscillatory Schemes I", STAM J. on Numer.
Analysis, Vol. 24, No. 2, 1997, pp.279-309
Huang Zhenyu, "Numerical simulation of jet flows
and nozzle flows", Doctorial dissertation, Beijing
Institute of Technology~in Chinese), 1999
Yang J. Y. "High-resolution, nonoscillatory schemes
for unsteady compressible flows", AI4A J. Vol.30,
No.6, 1992,pp.570-1575
Huan~ ZhenYu. "Flowfield calculation with high
resolution ENO", The proceedings of eighth
international space conference of pacific-basin
societies (8~ ISCOPS), Xitan China, l 999,pp.691-
697
Huang Zhenyu, Xu Wencan, "Flowfield calculation
with high resolution ENO", ACTA Aerodynarnica
SINICA, Vol.18, No.1, 2000, pp.14-21 (in Chinese)
Huang Zhenyu, Xu Wencan, Numerical simulation of
turbulent jet, Journal of Reijing, Institute of
Technology, Vol.19, No.6, 1999, pp.691-695 (in
Chinese)
Huang Zhenyu, Zhou Liandi and Zhao Feng "High-
resolution schemes in simulation of flowfield around
submarines", Technical Report 00867, CSSRC, 2000
(in Chinese)
Huang Zhenyu, Zhou Liandi, "Numerical Simulation
of Flows over Underwater Axisymmetric Bodies
with Full Appendages, .Shipbuilding of China, Vol.42,
No.4, 2001, pp.6- 11 (in Chinese)
Zhenyu Huang, Liandi Zhou, "Numerical Simulation
of Flows over Underwater Axisymmetric Bodies
with Full Appendages", proceedings of Fighth
International Symposium on Practical Design of
Sllips and Other Floating, Structure, Sept. 16-21,
Shanghai, China, pp. 429-436
OCR for page 593
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.2
0.1
o
0.8 _
0.6
0.4
0.2 _
1 1 1 ~ ~ I , , , , I
0.15 0.2 0.25 ut
R
~ig. l (a) x=-0.4333m
_~ ~ — ~ ' ;1
Exp.
Cal.
. ~ 1 1 1 ~ I I I I,,,, I,,,, I
0.1 0.15 0.2 0.25
R
Figl.(c) x=-O. 130
Figl. The comparison of velocity distribution at
difference x-axis stationsk the x=0 is at the tail of
body)
0.8t
0.75
0.7
.65
/,....
r
Cal.
/'
/
/K' ~
06
0.5
0.4 0.6 0.8 1
Fig. 2 Comparison of numerical circumferential-
mean velocity at propeller plane with experimental
data
— Exp.
~ C~.
~ , " ~
, ~/,
n ,,,, .,,,, ',,,, ',,,, I
45 90 1 35 180
deg
Fig.3(a) r/R=0.30U
0.8
0.6~
0.4L
0.~
nQ
~/
— Exp.
— C~.
/,
/
,,
1 1,, ,, 1, ,, , 1
90 135 180
deg
Fig.3(b) r/R=0.518
'''~ WN W~
3.6~l''
0.4
0.2
Exp.
C~.
1 , , , 1 1
45 90 1 35 . ~
Fig.3(c) r/R=0.678
OCR for page 594
1
0.8
0.4
o
1
0.8
0.6
0.4
0.2
,
. . . . . . . . .
~ ~ 45 90 1 35
deg
Fig.3(d) r/~=0.840
Ol . I ,
- Exp.
— — — Cal
l
180
i_ i=
· Exp.
_ _ _ — — Cal.
l
135 180
deg
Fig.3(e) r/R=l.OOO
Fig.3 Circumferential variation of axial velocity at
difference radius station of propeller plane
0.6
0.4
x
0.2
o
0.6
0.4
0.2
,ii
_ ~!
/
/f' '
—~
/ . ~
/ '
_ /,
At'
, . . . . , . . . . ,
0.06 0.08 0.
R
Fig4.(a) In duct(x=-0.165)
0.8
X- _
,, 1
0 O.05
,
it,
, . . . . ,
R
. . . . . ,
0.15 0.2
Fig4.(b) Behind the duct(x=0.03)
Fig. 4 Comparison the velocity distributions internal
and external of the ring-wing duct around
axisymmetric bodies.
Representative terms from entire chapter:
stern appendages
