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OCR for page 119
FiniteDifference Simulation of a Viscous Flow
about a Ship of Arbitrary Configuration
M. Zhu, H. Miyata and H. Kajitani
University of Tokyo
Tokyo, Japan
Abstract
The improved version of the WISDAMII
method, a finitedifference solution method
for a threedimensional viscous flow about a
ship of arbitrary configuration, is described.
A zonal method is used for the boundaryfitted
coordinate system so that the boundary layer
is sufficiently resolved with proper boundary
conditions on the body surface. The robustness
and the accuracy are improved by the
introduction of a new difference scheme.
Computations are performed for a flow about a
Wigley hull at Re=108 and the appropriateness
of the zeroequation and the SGS turbulence
models are examined.
l. In troduction
A great deal of efforts have been focused
for the development of theoretical or
numerical methods of solving the whole
features of the flow about a ship advancing
steadily in the deep water t5] [6] [8]. Since
the difficulties arise from the high Reynolds
number viscous flow, its separation, its
interaction with the freesurface waves and so
forth, we are still far away from the
completion of the method.
However, methods so far developed have
already turned out to be useful for the
partial explanation of the flow about a ship.
One example is the TUMMACIV method by Miyata
et al., which is currently used for the design
of the forepart of the hull, see Miyata et al
tl] [2] [3] [4]. The success of this method is
mostly due to the fact that the wave length
of the ship waves is sufficiently long and can
be resolved by the available grid system.
For the numerical solution of a viscous flow
about a ship, many research activities are
known. Larsson and his coworkers are
developing a method of designing hull forms by
use of the integral method for the boundary
layer, see Kim and Larsson t5]. Chen and Patel
have developed a partialparabolic and a
fullyelliptic method for the viscous flow
about the afterbody of a ship t7] t8].
However, it is still difficult to have
satisfactory solution of the separated flow,
the streamwise vortices and the viscous flow
119
under the influence of the freesurface.
For the elucidation of the details of a
turbulent flow a numerical approach by so
called large eddy simulation (LES) technique
is often employed, see Moin and Kim, et al
[11] [12] [13]. Since a turbulent flow at high
Reynolds number is composed of vertical
motions of wideranged spectrum and small
soaled motions may also play an important
role, the resolution of viscous motions of
high frequency is very important. They must be
directly solved or appropriately
approximated in the numerical solution method.
With the aim of developing a LESlike
technique for a flow about a body of complex
geometry with freesurface, a new finite
difference method called WISDAMII is
developed, see Miyata et al t6]. A boundary
f itted coordinate system, which moves at each
time step owing to the deformation of the
freesurface caused by waves, is employed and
the subgridscale (SOS) turbulence model is
incorporated following Deardorff ill]. This
method seems to be very promising since it is
very close to the direct solution of the
NavierStokes equation, and both the viscous
motion and freesurface motion are
simultaneously solved. However. the
improvement of the robustness and
is postponed to the f uture study.
The objective of this paper is
is to show the improved version of
II method and the other is to
appropriateness of the turbulence
order to obtain sufficiently fine
the accuracy
twofold, one
the WISDAM
examine the
models. In
spacing in
the boundary layer a zonal method is employed.
To attain sufficient robustness as well as
accuracy the fourthorder accurate
differencing scheme combined with the
artificial dissipation of the fourth
derivative term is used. Both the zero
equation model and the SGS model are used and
compared. The movement of the freesurface is
not considered in this paper.
2. Grid system for a zonal method
Elliptic partial differential grid
generation system proposed by Thompson et
al. [9] is adopted to construct a boundary
OCR for page 119
fitted curvilinear coordinate system similar
to the previous study t6] both in topology
and in numerical process. T he three
dimensional grid system has a HH type
topology. Figure 2.1 illustrates the
transformation from the physical region D
(X ~ ,X2,X3 ) to the imaginary transformed region
R (E ~ ,$ 2,$ 3 ), where the streamwise direction
is approximately parallel to the ~ ~ direction,
the lateral grid lines are approximately
perpendicular to the ship hull surface are in
the ~ 2 direction, and the grid lines parallel
to the girth line of ship hull is the ~ 3
direction. The grid generation is conducted in
a welldocumented grid generation procedure
t6] by solving the transformed Poisson
equation with Richardson relaxation method,
that is
gij ;) r + pk fir = 0
where for convenience, the notation of the
geometric coefficients is defined as
gij = TkT58k'
g = det~gij)
gij= 1 leitil,feJpy
2 g gap go
where T'i is the transformation matrix, giJ
the covariant metric tensor, and g t j the
contravariant metric tensor. ~ is the
Kronecker delta and e the Eddington
permutation symbol.
Since the elliptic grid generation system
used in the present study works for smoothing
the grid distribution rather than for
clustering grid lines in the regions of
interest, a great number of grid points are
required when a single grid system is employed
for a high Reynolds number flow. In the
present study, a grid system, which is
generated in the elliptic method with more
than 250,000 (170 x 30 x 50) grid points,
provides satisfactory resolution for the
viscous flow only at the Reynolds number 105.
In order to alleviate the gridrefinement
problems in the vicinity of the ship hull,
a zonal method is adopted in this study so
that sufficient grid resolution is achieved
in the turbulent boundary layer of ship at the
Reynolds number lO8. The zonal method is
applied only in the ~ 2 direction. Therefore,
the inner zone with finer spacing is located
in the vicinity of the hull surface and
centerplane. In this study, the original grid
system generated by the elliptic grid
generation procedure is called "coarse grid",
while a finer grid system in the vicinity of
the hull surface is called "fine grid". In the
fine grid system about 12 grid points of the
coarse grid system are subdevided into 40 grid
points overlapping a region with the thickness
about 2y/B=0.4 from the ship hull. The
locations of the fine grid points is set so
that they may accord with the grid points of
the coarse grid system. Therefore, at the
boundary of two zonal regions called "zonal
boundary" only metric discontinuity exists.
Figure 2.2 illustrates the methodology of the
grid refinement along the ~ 2grid lines and
shows that the coarse grid points are at the
same locations with the correspondent fine
grid points. Figure 2.3 shows a pair of the
coarse grid system and the fine grid system of
a cross section of the ship. The flux
conservation across the zonalboundary is on
the satisfactory degree. The details of the
boundary condition at the zonalboundary will
be described in the subsequent section.
3. Computational Procedure and algorithm
Timedependent NavierStokes equations in
rotational form and the continuity equation
are the governing equations. [6] [10]
8~' = —grad(P + 2 u u) + u X ~—veal (lo)
+ R
div(u) = 0
where u is the velocity vector, t is the time,
P is the pressure divided by the density, v is
the kinematic viscosity, and R is the net
contribution of the turbulent fluxes described
in the following section. All of the physical
values are defined in the regular grid system
of the general curvilinear coordinates. Since
the vectors are expressed with
contravariant components, the governing
equations are written by using the notation of
metric tensors as follows.
at g b: ~ ( ~ ~gk/~ I' ~ + g iCiklUkw!
—VE j 0< j (gk`~) ) + R
g _ i/2 ~ (gl/2~`i) = 0
where ~ i is the contravariant component of
vortici ty vector,
wi _ Eii' d: j ( g'.,u )
and aid k
120
is a permutation thirdorder tensor
ink  I/2cijk
OCR for page 119
The wellknown MAC method is employed as
the computational procedure. The time
derivative term of NavierStokes equations is
approximated explicitly by the forward
difference. From the mass conservation
condition, a Poisson equation is derived for
the Bernoullilike scalar field [Gig After
solving the Poisson equation iteratively by
the Richardson relaxation method, a
correspondent Bernoullilike field is given.
The details of the computational procedure is
described in E61
Since the tonal method is used in the
present study, the time increment for the fine
grid system is set at one fifth of that of the
coarse grid system for the safety of the
computational stability. Therefore after one
step of timemarching is conducted in the
coarse grid system, five steps of time
marching are conducted in the fine grid
system. The zonalboundary conditions seem
to be of crucial importance in this algorithm
and w111 be discussed in the subsequent
section.
Tile calculation is started with the f low
field of uniform velocity and constant
pressure.
4.Differencing scheme
The accuracy of the differencing scheme is
very important in the finitedifference
method, especially in the calculation of a
turbulent flow at a high Reynolds number.
Since the dominant equations are written in
the form with conjugate components of the
transformed coordinates, it is possible to
adopt various high order differencing
schemes so far known for the Cartesian
coordinates. The authors have employed the
thirdorder upwind difference scheme for
variable mesh system and suggested that one
may change the factor of the fourthorder
velocity differential derivative depending on
the mesh size and the Reynolds number to
compromise the accuracy with the stability
since the thirdorder upwind scheme is
composed of the fourthorder centered scheme
and the artificial dissipation of the fourth
order derivative of velocity [21] E22].
In this study, the above scheme is used in
the transformed coordinates. Although the
convection term is in the rotational form,
the dissipation term is derived from the
convection term in the gradient form. In the
curvilinear coordinates, the covariant
derivative of the contravariant velocity
component is written by using Christoffel
symbol as
of =/ + pf~t
'a 8~ ~
and the convection term in the gradient form
becomes
"J"~ = ~j~ + at
then using the differencing scheme recommended
by Baba and Miyata [23], the dissipation term
is obtained as follows
~ age'
Eli; lag
where ~ is the factor for the artificial
dissipation term Further details of the
derivation are referred to Miyata, Zhu et al.
[21] [22] and Baba and Miyata [231 The
derivatives of the convection term at and
near the boundaries where sufficient grid
points are not available are approximated by
onesided upwind diffarencing scheme and
secondorder centered differencing scheme
with the artificial diffusion term.
The other . derivatives are approximated by
the secondorder centered differencing scheme.
The time derivative term is approximated by
the forward differencing scheme.
S.Turbulence models
In the numerical simulation it is
considered that because of the machine
ability of the temporary computer it is
impossible to calculate the turbulent flow of
high Reynolds number without turbulence model
except few cases of direct simulation of a
flow with very simple geometry. The choice
of the turbulence model as well as the
computational procedure depends on the purpose
of the simulation. Although the turbulent flow
is substantially unsteady , only averaged
steady flow field is required in some of the
engineering problems. However, for scientific
purposes and in some engineering problems the
detailed unsteady flow should be simulated.
Many simulations of turbulent flow around ship
hull conducted so far use Reynoldsaveraged
NavierStokes equations and turbulence models
such as algebraic turbulence model, ~
equation, K E model or their combination [8]
E14] E151 It is a general approach in this
area that when the solution of simulation
converges, the results are compared with the
experimental ones which are the averaged
data of the real physical values. Some
simulations have shown excellent agreement
with the averaged experimental results. But
nobody so far answered several fundamental
questions how the flow of the boundary layer
is deformed to develops into wake and how is
the transition from laminar to turbulent flow
on the surface of the forepart of a hulL
In order to investigate into the
fundamental physical features of the
turbulent flow around a ship, the authors
employ a zonal method near the ship hull
surface, and on the other hand adopt a
computational procedure with SGS turbulence
model, the latter of which is similar to
large eddy simulation and is called LESlike
procedure by the authors. Contrary to the
methods for an averaged flow this method is
121
OCR for page 119
supposed to resolve unsteady turbulent fluid
motions of smaller scale and it will provide
useful information for the understanding of
the fundamental features of the turbulence
structure of the ship boundary layer. The
present study from abovementioned standpoint
will hopefully to throw a light to the
research of this area.
In this study two turbulence models are
used, one is the algebraic turbulence
model and the other is subgridscale
turbulence model. In order to exmamine the
possibility of applying the SGS turbulence
model to the turbulent flow around ship by
comparing the computational results of the two
turbulence models with the experimental data.
Then the details of the physical features from
the computation are discussed. The
formulation of the subgridseale turbulence
model is essentially same with the previous
study by Deadrofftype ill] and is based on
the eddy viscosity concept, that is to say,
the subgridseale stresses used in this study
are isotropic ones. The SGS eddy viscosity is
defined as
Vs = (Co ~)2 (2`,jj em
and the SGS stresses R;i are written as
Rii— 2`i'ui' = ~gij "us" g,,,,  2v' eiJ
However, using this formulation of SGS
turbulence model without any special treatment
near the ship hull surface, the turbulent
production may be insufficient and be diffused
out in the vicinity region inside the
laminar viscous sublayer [12] tl3]. This is
because the essential turbulence generation
near the hull surface is due to the
inhomogeneous wall turbulence, which is
characterized by a mixing length of the
scale of sublayer thickness and of
boundary layer thickness. The effects of the
curvature of a wall as well as the pressure
gradient should also be considered in the
turbulence models. However, they are postponed
to the future study.
In order to take into account the
inhomogeneous effect of the wall turbulence in
the subgridscale turbulence model, the
Prandtlvan Driest formulation is introduced
for the reduction of the turbulence scale near
the hull surface by multiplying the
subgridseale in the Smagorinsky eddy
viscosity by the exponential damping
function, that is,
a = a [ I  eY'/~ ]
122
where A is set constant at 26.0. The details
of the formulation of subgridscale turbulence
model are described in t6].
The algebraic turbulence model used in this
study is a modified CebeeiSmith type tl7]
[18]. For the inner region of the boundary
layer the Prandtlvan Driest formulation is
used as
~ _ 12ldUI
I=ky[l—e~Y+~ ]
and for the outer region the Clauser's
formulation together with Klebanoff's
intermitteney function is applied as follows.
~ l + 5.5( ye/) 63
The correspondent boundary layer displacement
thickness ~ and boundary layer thickness ~ in
the ease of zeropressure gradient are
determined by ye 8x of the maximum point of the
root of the shear stress where the velocity
is defined from the law of the wall of the
Coles formulation [19]
~=Yleyl[ley+/A ]
and the boundary layer thickness ~ and the
dispalcement thickness ~ are obtained as
~ — 1 .93 6ymnX
Also both the accelerated and the decelerated
flows including separated flows are considered
in this formulation of the modified Cebeci
Smith model, see Stock and Hasse [17].
6.Boundary Condition
Noslip velocity condition is implemented
at the ship hull surface. In this study the
first grid point in the fluid region is set
inside the viscous sublayer, and the velocity
at this point is interpolated by the velocity
profile of van Driest [18] given as
OCR for page 119
+2 it+
I+=KY+11—e Y / ]
where us = u/u~ and y. = y u~/v. As shown in
Figure 6.1, point A is the nearest point and
point B is the second point. At each time
step the friction velocity on the ship hull
surface is calculated so that the velocity at
point B satisfies the above equation. And
then the velocity at point A is interpolated
by the following equation.
t`A t`A qA
= =
UB tiB qB
where u ~ is the component of velocity along
~ ~ grid line and U3 iS along ~ 3 grid line,
both of the lines are parallel to the ship
hull surface. And q is the velocity magnitude
at the grid point, which is calculate] by the
ordinary procedure at point B and by the
equation of van Driest's velocity profile at
point A.
At the other boundaries, the uniform stream
velocity is set at the inflow boundary and
zero normalgradient condition is set at the
side and outflow boundaries.
The pressure is fixed at the bottom boundary
and zero normalgradient condition of
pressure are set at the other boundaries.
The zonalboundary that connects two zones
described in the previous section is placed
along the ~ 2 grid line. As shown in figure
6.2, the velocities and momentum terms of the
NavierStokes equation in the overlapping
region of the coarse grid system are set at
the same values with those of the fine grid
system for the mass and momentum conservation.
The algorithm of the calculation with the
zonalboundary is as follows.
1. Calculation in the coarse grid (zone
2):
a. Calculate the momentum terms of the
NavierStokes equation while they are
interpolated from zone 1 in the overlapping
region by using the updated velocity in zone
2.
b. Calculate the source term of the Poisson
equation in zone 2 and iterate the pressure
solution loop under the zonalboundary
condition, which the pressure at the point
A' (figure 6.2) is set at the same values
with the pressure at point A of the fine grid
(zone 2).
c. Update the velocity in the corse grid
(zone 2).
123
2. Calculation in the fine grid (zone
1):
a. Calculate the momentum terms of the
NavierStokes equation by using the velocity
of the f ine grid (zone 1).
b. Compose the Poisson equation for the fine
grid ozone 1) and solve it under the zonal
boundary condition that the pressure at point
B (f igure.6.2) is set at the same value with
that at point B' in the coarse grid (zone 1).
c. Update the velocity at the fine grid
(zone 1) and update the velocities in the
overlapping region of the coarse grid system.
7. Computed results
Computations are performed for a flow about
a Wigley hull at the Reynolds number (Re) 108
with the algebraic turbulence model (modified
CebeciSmith model) (Case 1) or the subgrid
scale model (Case 2). It is noted that in the
computed results the viscous flow about a
hull is not wholly developed but it is on the
transition stage, since the computations are
continued only for T=1.2 Dimensionless time,
T=tU0 /L, Us is uniform flow velocity and L is
ship length) in the Case 1 and for T=0.8 in
the Case 2.
The grid system shown in Fig.2.3 is used and
the number of grid points is 255,000 for the
coarse grid system and 340,000 for the fine
grid system. The smallest grid
spacing in the ~ 2 direction about 0.005% of
the ship length. The time increment At is
0.00005 for the coarse grid system and 0.00001
for the fine grid system, respectively. The
factor of the artificial dissipation term a is
set at 6.0. The computations are conducted on
HITAC S820/80 supercomputer with almost 20
hours of CPU time. The vectorization ratio of
CPU time is 98% for both cases.
The pressure distribution on the ship hull
surface (x3=o.O) computed with the algebraic
turbulence model is compared in Fig.7.1. The
agreement with the measured results by Sarda
[20] is not very satisfactory since the flow
is not f ully developed and f urthermore in the
algebraic turbulence model used in this study
the displacement thickness of the boundary
layer is determined by the wellknown Coles
velocity profiles for the zeropressure
gradient [17] while the decelerated flow near
the after end of ship is involves large
pressure gradient. However it is noted the
overall flow field is approximately realistic.
In order to examine the detailed flow field
comparison is made of the f low variables at
two longitudinal location x ~ /L=0.8012
(E ~ =110) and x ~ /L=0.9218 (E ~ =120>, where the
viscous flow along the hull surface may
gradually develops and threedimensional
motions may become important. The data are
illustrated in Fig.7.3 and 7.4 for Case 1, and
in Fig.7.5 and 7.6 for Case 2. All variables
are made dimensionless following the
equations described in the previous sections.
The distribution of velocity components,
OCR for page 119
vorticity components, eddy viscosity
coefficient and Reynolds shear stresses along
the lateral ~ 2 grid line are presented at two
vertical location x3=0.0463 (~=30) and X3=
0.0341 (E 3 =35) while the water plane is at
X 3 =0.0 and the keel is at x 3 =0.0625.
The contour maps of u ~ and c~ ~ indicate that
the boundary layer is still developing and the
streamwise vortex is going to be formed in it.
The thickness of boundary layer in Case 1 is
much thinner than Case 2. This is mostly due
to the small magnitude of the eddy viscosity,
which may deteriorate the diffusive effect of
turbulent flow. It is approximately one
seventh of the case of the subgridscale
model. At the bottom of Fig.7.3 to 6 the
computed Reynolds shear stresses are shown.
Since their magnitude reaches the maximum
value of 10 3 according to the measured results
t20], the turbulent flow is not wholly
developed in the computations. However it is
shown that two components of the shear
stresses which are not considered in Case 1
are important in this flow field and hence the
use of the algebraic zeroequation model is
questionable.
The relation between the location of the
maximum shear and the boundary layer thickness
is shown in Fig.7.2. According to Stock and
Haase [17], the relation should be given by
the linear equation 6=1.936Y~ .x . This
indicates that the boundary layer is
excessively enlarged by the subgridscale
model and on the contrary it is supressed by
the zeroequation model. This tendency is
amplified at ~ ~ =120 more than at ~ ~ =110. It is
supposed that the subgridscale turbulence
model overestimates the turbulence stresses
and the zero equation model underestimates in
this region.
The figures for vorticlty components
indicates that all three components play some
important roles and their interactions may be
significant. Notwithstanding the difference in
the thickness of the boundary layer it is
common to Case 1 and 2 that the inflection
of vorticity profile appears at ~ ~ =120.
8. Concluding remarks
The improved version of the WISDAMII method
is still under development and comprehensive
comparison with the measured results or
interesting elucidation of the complicated
turbulent motions are beyond the scope of the
paper. Since the present method avoid
approximation as far as possible, the accuracy
seems to be on a high level but it is of ten
accompanied by the extremely long CPU time
even by the supercomputer.
The purpose of the present method is to
solve both wave and viscous motions
simultaneously. It is already demonstrated
that this is achieved by use of the moving
grid system. t6] However the difference of
wave length between freesurface waves and
viscous turbulent motions is tremendous. The
use of zonal method described here will be one
124
of the promising approach.
For the numerical simulation of the detailed
viscous flows on the hull surface we must be
very careful as suggested by the present test
computations The zeroequation model ignores
some of the Reynolds stresses which may not be
sufficiently small in the real flow. The two
equation model is said to be insufficient for
the separated flow. The subgridscale model
may give excessive turbulence stresses when it
is used in the grid system of which spacing is
not sufficiently small. The subgridscale
model will be useful not only for the large
eddy simulation but also for the flow
simulation of engineering purposes. However,
the coefficients and scales for the model must
be carefully chosen.
This reseach is supported partly by the
GrantinAid for Cooperative Reseach of the
Ministry of Education, Science, and Culture
and partly by the LINEC group of shipbuilders
in Japan.
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method", J. Kansai Soc. N.A., Japan, No.207,
pp.1123 (1987~.
23.Baba, N. and Miyata, H., "Effect of the
form of NavierStokes equation on a separated
flow simulation", J. Computational Phys.
C1989) Csubmitted).
125
OCR for page 119
aG' aGs
Ir.:.~ .....,~~.. .
l Y~ :.:. :::. :.:.2:..~1:
~ ~ l ~ .:: :.~ :.::. :::.: ..~
a`7, lo I l ... .. ~ ^
, ~ ~ oQ
~W _ .
aG6 ,~ ~ /
,, / ; /Physical Region
aG7 ~ aG2
Transformation
~ al,
/Hs
1 ~
~ ~ ::. /
. . , I .: :.:::: . :. : ::.: :.
ail, , ~ , ...........
l A ~—ahoy
~ ::ir.:::
1 ~
j3 ~ eH6 ~ __
i / ~ /Transformed Region
~H' aH2
Fig.2.1 Coordinate transformation.
OVERLAPPING REGION
O
ll
Fig.2.2 Definition sketch for the interfacing
region
Fig.2.3 Grid system for the zonal method
(transverse section), coarse (left)
and fine (right) grids.
126
OCR for page 119
~2
\ _
;~ \ ~ w e\\
/ ~\\N'''
0.4 _
0 2
V
o
_
o.< ,,,, 1
0 0.2
Fig.6.1 Definition sketch for the treatments
on the wall.
I 1 1 ~ 1 1 ~ I ~ 1 ~ ' 1 ~ 1 ' ' ' ' 1 ' ' ' ' 
· °~. ~ __
i
,~ it.,.
 ° !  °
s
t
,,, 1,,,, 1
0.4
2x/L
o o
,, ,_' 1_, .,,
0.6 0.8 1
Fig.7.1 Longitudinal distribution of pressure on the hull surface, ; potential theory,
O; experiment by Sarda {20] at Re=4.5x1 O8 ,     ; present computation with
zeroequation model at T=0.6, x ; do at T=1.2.
0.~4
2 SIB 0 t2
0.10
0.08
0.~S
0.04
0.02
120, 35)
__ /=1~ 938Ymax
/ MU,/
/ /, /
/ 'l'0 ~~
0~ / (120,35)
(120,30) / ~
/,,,110,35tt(120,30)
/
/ ,,
/ ~(1~0,30)
0.02 0.03 0.04 B.05 0.06 0.07
2Ymax /B
Fig.7.2 Relation between the boundary layer thickness (S ) and the location of maximum
Reynolds stress (Y max), blank and black marks are for the SGS and the
zeroequation model,respectively, and numbers in parentheses indicate location.
127
OCR for page 119
CElNTClUR I NTERVf3L" 0.1000
1
(a)
CONT1IUR I NlERVRL~ 20.0000
1
(b)
0.4~
0.3 _
m
\
C\2
0.2
0.1 _
O l
O O
..
., ~
O O
0l 0
O. 0
O
0, 0 0
~ 0 0
<, O O
0 0
O 0 0
4 0 o
o O
· O o
· o O
~ O O
O o
· o o
~ o o
~ o O ° ° °
o 0.25 0.5 0.75 1
Velocity
0.3 
0.2:
0.1~
O ~
(d) t ~ ~ ol~
O I f ~—~A  ~ I l l l l
500 0
Vorticity
(e)
(f)
0.4 _
0 0.25 0.5 0.75 1
Velocity
500 o~ l l 1 ~l l l l 1
500 0 500
Vorticity
o ~=~! ~ ~ I ~ ~ ~ ~ I ~ ~ ~ ~ I ~ ~ ~ ~ ~ I , , , , I , , , , I , , , ,~
O 0 05 0.1 0.15 0.2 0.25 0.3 n
Eddy Viscosity Vt#105
n 1 L
~ ~
o ,,,, 1,,,,
400 200 0 200 400
Reynolds Stresses
l~,, 1,,,, 1
o l
0 0.05 O. 1 O. 15 0.2 0.25 0.3
Eddy viscosity Vt#105
O  , , , , t , , , , :, , I
400 200 o 200
Reynolds Stresses
Fig.7.3 Computed results with the zeroequation model at ~ 1=110, T=1.2.
(a) contour map of ul, (b) contour map of ~1, (c) velocity profiles,
5;U3 (left;: 3=30, right;? 3=35), (d) vorticity profiles, O;@l, 0;~)2,
(e) turbulent eddy viscosity, (f) Reynolds shear stress  R 1 2/ U r2
128
. . . .
400
O;u:, O;U2,
2;03 (d_),
OCR for page 119
(a)
C9NTOUR INlERV8L. O. 1000
r
/
03~
.
m
\
:^
C\2
0.2 _
0.1 _
0 ,
(d)
n , , I
0.1 ~
 500
(e) ~ ~~,,, I ,,,, I,,, , ~
0 0.05 0.1 0.1 5 0.2 0.25 0.3
Eddy Viscosity Vt~105
01 _
0 _
400  200 0 200 400
Reynolds Stresses
(f)
OONTOUR ] NlERVRL 20.0000
(b)
0.4 · .
O.
O
· .
· . O
., O
· . O
·! o
·t o
· o
·. o
· 1 0
·' o
· o
o, o
o: o
o. o
o o
o: 9
1 ~
I ~ o t°~°~ °]
0 0.25
O
° 1
O
O
O
o
O
O
o
O
o
O
o
O _
o
oo
O O
, 1,,,
0.75 1
o o °
,,_1,,,,1,,,
0.5
Velocity
o.b
0.4
0.3
\
C\2
0.2
O.]
O _
. · .
· . o
· . o
· , o
· . o
~ ~ o
· ! o
· . o
o, o
. ', o
·, o
. o
· . o
o . g
 I ° °°OIO,,, ,_1_,,,, I
011 ~ ~ It ~
Vorticity ~500 Vorticity
500 snn
0.1 ~

0.25 0.5 0.75 1
Velocity
1,, 1
500
o ~ , 1 , , , , 1 , , , ,'1 , , , , 1 , , , , 1 , , , ,~
0 0.05 o. 1 o. 15 0.2 0.25 0.3
Eddy Viscosity Vt~105
0.1 _
o ,,,, 1, . . .
400  ~nn
Fig. 7.4 same as Fig. 7.3 at ~ 1 =120, T=1 .2.
129
]
___ 0 200 400
Reynolds Stresses
OCR for page 119
(a)
CONTOUR INlERV~L" 0.1000
1  1
0.3
m
\
o.:
o
0.5 ~ !
1
 o
.,
. I
O ;V °°OIO , I I I
0 0.25 0.5 0.75 1
Velocity
o o °
o o
(d) n,:
f ~
1i ~\
l 11 ~ ~ ~ _
0 500
Vorticity
(e)
(f)
0.1 _ ~
oL , , ~ I ~ I ~ 1 1 , , , , 1 , , , ,~
Eddy Viscosity VL*104 2
0.~
o 1,,,, 1,...
200  100 0
Reynolds Stresses
O
b o
t ~ o
°
g ~
P 1
D
D r
t; jr~
'Oo ~
0 _
o
, 1 , , , ~
1 00 200
n ~
0 ~
n
o
i
0.1:
(b)
CONTOUR ~ NlERV9L. 20.0000

o
o o o °
0.25 0.5 0.75 1
Velocity
_~0
Or I I I I I ~ (! I I I I I I ~
500 0 500
Vorticity
0.1~ _
oW~ 1 , , , , 1 , , , ,~
Eddy Viscosity Vt*103 2
0.1 ~
n I
v ~ ~ ~ J 1 ~ ~ ~
200  100 0 100 200
Reynolds Stresses
F:g.7.5 Computed results with the SGS model at ~ ~=110, T=0.8.
(a) contour map of u~, (b) contour map of ~, (c) velocity profiles, O;u:, O;u2,
3;U3 (left;? 3=30, right;? 3=35), (d) vorticity profiles, 0;~, 0;02, 0;03 (do),
(e) turbulent eddy viscosity, (f) Reynolds shear stresses, 0; R 23/~ ~2 ,
0; R 3~/ ~ r2 , 0; R t2/ ~ r2
130
OCR for page 119
(a)
(c)
(e)
CONTOUR IN7ERVRL O. 1000
n
no
n
n
n
CONTOUR I N7ERVRLe 20.0000
n A
no
no
500 o
Vorticity
I I i I I 1 of
500
Jo°c
n ~ ~ I,' O p
~ · jO
1,,
500 o
Vorticity
01 ~
_ ~ n.C

o ~,, 1,,,, l , 1 1~
Eddy Viscosity Vt~105 2
~ WoO 1 °lt
(f) ~ ~
 200  1 00 0 1 00
Reynolds Stresses
500
1,
01 _ ~
o A, 1,, 1 1 1 1 1 1 1 1,, ,
o 05 1 1 5
Eddy Viscosity Vat l 0~
o
o o
o
o
200
Fig. 7.6 same as Fig. 7.5 at ~ 1 =l 20/ T=0 .8.
131
1
2
OCR for page 119