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OCR for page 59
Numerical Simulation of Three-Dimensional Viscous Flow
arounct a Submersible Body
C-I Yang
David Taylor Research Center
Bethesda, USA
P-M. Hartwich and P. Sundaram
NASA Langley Research Center
Hampton, USA
Abst ract
A second-order accurate, implicit, high resolution up-
wind scheme has been used to solve the three-di~nensional
incompressible Navier-Stokes equations in general curvilin-
ear coordinates for the steady-state computation of the flow
field around axisym~netric hull geometries at high Reynolds
numbers. A hybrid algorithm with relaxation in the stream-
wise direction and approximate factorization in the cross-
flow plane is used to reduce the temporal splitting error.
Three axisymmetric bodies with different stern shapes have
been chosen for the present investigation to highlight the
boundary layer development and to pinpoint flow separa-
tion in the stern regions of the hull geometries. Three-
dimensional viscous grids of C-O topology have been gen-
erated around each body using a transfinite interpolation
technique. Turbulence is simulated using algebraic eddy
Hi I tv m Nazi cif Rn1H wi ', T.nm em
J ... CAM The computed re-
sults are compared with the available experimental data
for on-body pressure distribution and radial and axial ve-
locity profiles in the afterbody boundary layers with and
without propeller in operation. The computed results show
close agreement with the measurements in most cases.
I. Introduction
The flow in the stern region of a submersible is char-
acterized by the presence of a thick and possibly sepa-
rated turbulent boundary layer. There is an increase in
the stern pressure drag and skin-friction drag, called thrust
deduction, due to tl~e upstream suction produced by the
propeller when operating. Accurate numerical prediction
of the flow in the stern region of the hull both with and
without the propeller operating is important to evaluate
the afterbody thrust deduction, thus limiting the expen-
sive resistance and self-propulsion model experiments in a
towing tank. Although several numerical algorithms have
been so far developed for computing the two- and three-
dimensional incompressible Navier-Stokes equations, most
of then are prohibitively expensive for very high Reynolds
number flows typical in marine hydrodynamics.
Several panel methods such as VSAERO (1) ale used
In aircraft design and analysis practices. However, their
true application in marine flows, particularly for comput-
ing the flow field in the stern region, is limited because of
the presence of fairly thick boundary layers in the stern
.
region. A viscous-inviscid interaction procedure Night be
once again inadequate because of the possible presence of
flow separation in this region. Hence, an accurate compu-
tational method for Ellis problem should attempt to solve
the inco~npressible Navier-Stokes equations.
Aziz and IIellums (2) proposed a vector potential vor-
ticity formulation to solve the three-di~nensional inco~n-
pressible Navier-Stokes equations. Because of its large stor-
age requirements and the necessity to solve three Poisson
equations at each time level, the method is not very popu-
lar. The direct extension of time-dependellt methods, both
explicit and implicit, developed for the compressible Navier-
Stokes equations to incompressible flows is not possible be-
cause of the 'stiffness' of the physical problem associated
with low speed viscous flow. 'ho circumvent this problem,
Chorin (3) proposed the use of artificial compressibility
when solving the equation of continuity, thus introducing
an unsteady term to make the system hyperbolic as in flee
case of compressible flows.
In this paper, we compute complex flow fields around
various axisym~netric bodies by obtaining numerical so-
lutions of the incompressible Navier-Stokes equations in
primitive variable formulation. A hybrid second-order ac-
curate implicit high resolution upwind scheme is used to
solve the system of conservation laws in general curvilin-
ear coordinates. Tllree axisymmetric bodies with different
stern shapes to highlight specific flow details ill that region
have been chosen for the present study, primarily because of
the availability of well documented experimental results for
these bodies (4,5,G,7). Body fitted three-dimensional grid
systems of C-O topology have been generated around these
bodies using an algebraic grid generator based on transf~lnite
interpolation procedure. The computed on-body pressures
together with axial and radial velocity profiles are compared
with the experimental data. In addition, the flow around
the stern region of one of the bodies with a propeller in oper-
ation is computed. The propeller is simulated by imbedding
body forces in a disk located at propeller plane as suggested
by Stern et all. Finally, in order to illustrate the ability
of this scheme to simulate a separated flow, the results of
a low speed vertical flow about a 3.5 caliber tangent-ogive
cylinder at an angle of attack are presented.
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II. Governing Equations and the Jacobian of the coordinate transformation is given
by
Using Chorin's (3) artifical compressibility formula-
tion, flee incompressible Navier-Stokes equation are written
in conservation law form for three-dimensional flow as
Qua + (E* - Evict + (F* - Felt + (G*—Gall = 0 (1)
In Equation (1) the dependent variable vector Q is defined
as
Q = (p, a, v, W)T
and the inviscid flux vectors E*,F*,G* and the viscous
shear flux vectors EV,FV,GV are given by
E* = Ala, u2 + p, uv, uw)T
F = (pv, uv, v2 + p, VW)T
G = (pw, uw, vw, W2 + p)T
Ev = Re (0, As, sexy, ~z)T
Fv = Re~l(O,ty=~7yy'tyz)T
Gv = lle (O. The, Tzy'[zz)
(2)
The coordinates x, y, z are scaled with an appropriate char-
acteristic length scale L. In Eq. (2), the Cartesian velocity
components a, v, w are nondimensionalized with respect to
the free stream velocity TOO, while the normalized pressure
is defined as p = (P - poO)/pV2 . The kinematic viscosity
z, is assumed to be constant, and the Reynolds number is
defined as Re = vow. The artificial compressibility pa-
rameter,6 monitors the error associated with the addition
of the unsteady pressure terns If in the continuity equa-
tion which is needed for coupling the mass and momentum
equations in order to make the system hyperbolic.
For large values of,0 or when the solution of Eq. (1) leas
approached asymptotically a steady state, the continuity
equation is accurately satisfied. However, the choice of ,B
is also dictated by the important constraint of the stiffness
of the partial differential equation as discussed in Ref. 9.
Hence, ,5 = 1 has been chosen uniformly in the present
computations.
To develop the equations in a general curvilinear coor-
dinate system, a coordinate transformation of the form
~ = ((x,y,z), ~ = <(x,y,z) and ~ = y~x,y,z)
with
and
where
x; y; z:
J-i = det x~ ye z:
x~ ye zig
The Cartesian derivatives of the shear fluxes are obtained
by expanding them using chain rule expansions in the (, I, 71
directions.
The Jacobians of the inviscid fluxes E, F and G are
needed for the flux-difference splitting and for the implicit
algorithm. The Jacobian matrices in the different coordi-
nate directions are obtained as the linear combination of
the Cartesian Jacobian matrices as
D = a1A* + a2B* + a3C*
where D = A, El, or C with A = All, B = Ale, C = Ail
, and (a~,a2,a3) are the row vectors of the T matrix. For
the Jacobian matrix A, al = (x/J' a2 = ¢y/J, as = (z/J'
and so on. The Jacobians in the Cartesian coordinates
themselves are
-O
BE* 1
A* = =
am O
O
-a
B* = i7 = O
~Q 1
O
-o
* dG* O
C = act = O
The eigenvalues of D are
A = allay (~1,>27~3,>~)
2u
v
w
o
v
o
o
o
w
o
1 0
o
a
u
O u
a
a
n
U
U 0
2v 0
w u
O p~
O u
w v
0 2w
= diag (U - S,U + S,U,U)
where U is the contravariant velocity component in the cor-
responding coordinate direction given by
U = alu+a2v+a3w
has been considered. Eq. (1) is rewritten in strong conser-
vation law form as
(Q/J)'+ (E—Ev)< + (F—Fv)` + (G—Gv)?7 = 0 (3)
(E,F,G)T = [T] (E* F* G*)T
(E,F,G)T = [T] (E* F* G*)T
and
S = [U2 + p~a2 + a2 + at] / = S(U,V,W,ai,a2,a3)
The Jacobian matrices in different directions are diag-
onalized using a similarity transformation D = RAR-t to
obtain the eigenvalues of D. The rows of R-i and the
columns of R are computed such that they give an or-
thonormal set of left and right eigenvectors. The gener-
alized similarity matrices R and R-i are given by (Ref.
tT] = [be (y (z ~ R = ~ alp + Ail alp + UA2 -a2 -as
BY 71z at+ V>l a2,B + V>2 al + as —as
~3,0 + we) as p + W>2 —a2 al + a2
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and
—A2/p al a2 as
-1 1 —>1/~ al a2 as
R = S2 (~2d1 + Ald2)/p —2$2/k—aids —2S2/k—a2d5 —a3ds
L(~2d3 + gilds)/> —2S2/k - ald6 —a2d6 - 2S2/k—a3d6
where 1; = al + a2 + as al = a' _ etc and
d1 = (T31—r2l )/k, d2 = (r32 - r22)/1c, d3 = (r41 - r2l )/k,
d4 = (r42—r22)/k, d5 = d1 + d2, and do = d3 + d4
Quantities such as rid represent the ith row and jth column
of the R matrix and S = S(u,v,w,c`~,a2,a3).
In hilly Reynolds number flows, it is appropriate to
make use of the thin-layer approximation. The justification
for such a simplification can be found in Bullions and Steger
(10). Consequently, all viscous derivatives in the ~ and
direction (along the body) are neglected. The viscous
shear flux Gv (normal to the body) and its Jacobian can
be derived after Steger (11). That produces the coefficient
matrices
Gv (Re ]) : ~:lvr' + ¢277
¢1Wn + ¢29z
and
with
~O O O O ~
AGE - 1 0 Z22 Z32 Z42
Z=0Q=(Re'J) LOZ39Z33Z43~
O Z42 Z43 Z44 Using (G) we can write (5) as
Discretization of the Inviscid Fluxes
Consider a system of quasi one-dimensional, hyperbolic
partial differential equations
(Q/ J)t + He = 0 (4)
where ~ = (, (,or y. Defining computational cells with
their centroids at I = ,,8~ and their cell interfaces at I ~ 1/2,
a discrete approximation to (4) is written as,
( ~ )^Qn+/\lH=0 (5)
where At is the time step ,/~Qn = Qn+1 _ Qn and /\I( ) =
t( )I+~/2 - ( )I-1/24/~8 · Superscript denotes the time level
at which the variables are evaluated.
To construct an approximate Riemann solver for the
initial value problem in Eq. (5), each variable is regarded
as an averaged state in each cell so that the flux difference
is preserved in each cell and Eq. (5) can be regarded as an
integral rather than a differential law. According to Roe's
scheme (12), the flux at interface I it 1/2 call be expressed
in terms of the left and right travelling waves,
¢1 = 71~' + 71y + 71~
Z29 = 39= + By + hz Z32 = 39X9y
Z42 = 3r1=r/z Z33 = 71x + 371y + hz
Z43 = 3 rack Z44 = 71x + By + 3 id
¢2 = 3(~Un + hymn + flown)
The inviscid fluxes, the viscous shear fluxes and their Jaco-
bians just obtained in Ellis section are ready for discretiza-
tion later.
III. Numerical Flux Differencing
It is well known that upwind schemes possess an in-
herent solution-adaptive dissipation that eliminates the ad-
dition and fine tuning of artificial dissipation terns for nu-
merical stability and accuracy required in schemes based
on central diIferencing. In the present approach the invis-
cid fluxes are discretized by using Roe's flux-differencing
splitting concept (12) and the viscous fluxes are discletized
lay the central differencing technique. We shall discuss the
numerical disc~etization scheme in one din~e~sio~ arid as-
seml~le them together for three-`lin~ensions.
Hl+~/2 = Hi :t (OH F I/2) (6)
(~`J)~\Q + I\HI+1/2 + /\HI-1/2 = 0 (7)
Equation (7) relates the development of Q at the centroid
to the waves at the interfaces according to their propagation
directions. Defining a mean value matrix according to Roe
(12)
so that
Dl+l/~ = D(QI, Ql+1 )
Dl~l/~Al~l/2Q = AHI+1/2,
Using the above we can write (7) in delta form as
[(,Nt]) - (Dl+l/2) ~1+1/2 + (Dl+-1/2) ~1-1/2] AQ
= (Dl+l/2) /\I+1/2Q —(Dl+ l/2) /\I-1/2Q
where
and
with
61
Dl~l/2 = (RAR )I+l/~ = tR(A+—A-)R-l~l l/2'
(Dl+l/2 ) = (RA-R-1 )1+1/o
(Dl+-l/2) = (RA+R )-
A =(~A~+A)/2.
(8)
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Representative terms from entire chapter:
body forces
The construction of D' /2 is accomplished in two
steps. First, the analytic Jacobian D = dH/0Q is formed.
D is found to be a function of metric coefficients, of the
parameter ill, and of the state vector Q
D = D(a, b, c, id, Q )
where a,b,c are the metric quantities <2/J, (y/J, (z/J,
etc. Second, the local values aim, balm, cam, and
All ~/2 are computed at interfaces and fed into D to form
Do ~/2. In addition, in order to ensure that the flux di~er-
ences taken over the six bounding surfaces, (i at 1/2,j,~),
(i, j ~ 1/2, k) and (i, j, ~ it 1/2) of a three-dimensional cell
at (i,j,~-) cancel out completely so that the present finite
difference formulation essentially becomes a finite voulme
method, a special weighted averaging procedure has been
adopted (9), for example,
(¢~/J)iJk = t(~k~jY)~j{hZ) (~j{~Y)~{jZ)]i
where
with
ktjY = [({j§~+1 + ([jy)~_1]/2
fj( ~ = [( ~j+1 - ( )j-1]/2
The advantage of the delta form formulation is that
the steady-state solution is independent of the time step
size At. The justification for the accuracy of Eq. (8) as
the approximate Rie~nann solver for the present problem is
given in Ref. 9.
In order to ensure that the present scheme has a high
resolution capability, which is equivalent to preserving To-
tal Variation Diminishing (TVD) property for a nonlin-
ear equation, it is necessary to construct a quantity sub-
ject to TV requirement. For a linear problem such as the
one described above, the TV of the characteristic variable
W = R-iQ can be forced to diminish in tine. Assuming
R,R-i and A to be constant and using Else definition for
the characteristic variables, the scheme given in Eq. (8)
can be rewritten as
[aim ~ Arn^~/2] twin = ~ ~777~/214 In (9'
(at= 1,2,3,4)
The above equation comprises four scalar linear equations.
To enhance the accuracy up to third order, Else characteris-
tic variables lF,in at the cell interfaces are reconstructed by
using piecewise linear distributions as described in Ref. 9.
We then obtain a family of implicit TVD scenes
[( ^t Jo r7t iTI/2] Adorn
[~\7n ]~ l wj~r7I, + (~l + ~~>r+n)
Ghan ~+~ — ¢7n ~/4~ /\~+~/2W'n
[>m + {(~l In + (
Tilde Differencing and the Inexplicit Scheme
The backward Euler time di-fferencing of the three-
dimensional conserva,tio~ equations with the tl~in-layer a,p-
. . .
prox~n~at~on Is
/\ Q =—t/\~(En+i ~ + A`~(Fn+~ ~ + A71l(G7~+i—Gv+i )]
(13)
Linearizing it about time level n, we obtain
+~9En'~` + (0Fn)~\ + ¢~9Gn _ ~GV )A Amen
=—f/~(En) + /\~(Fn) + ,/\~(Gn _ Gn)]
The left hand side is the implicit part and the piglet
hand side is the explicit part of the for~nulatio~. The ex-
plicit part is the spatial derivatives in Eq. 3 evaluated at
the known time level n; its value diminishes as the steady
state solution is approached. Hence, it is also called the
residual . The L2 Noreen of the residual is often used as
a measure of convergence of a solution. Discretize the in-
viscid and viscous fluxes according to upwind difEerencing
scheme and central differencing scheme respectively in (,
and ~ coordinate direction independently according to Eq.
(8) and then assemble then together. Eq. (14) becomes
tf /\~J)—A:+~/27Ni+~/' + A+ i/~/\i-~/2
—BJ+~/2Ai+~/2 + B}_~/~Aj_~/o
(C + Z)~+~/2~+~/9 + (C + Z)k—I/~k—I,/2]n~Qn
= - Res(Qn)
(15)
where i, j, and ~ are spatial indices associate with the (, it
and ~ coordinate direction. A+,B+,C+, arid Z are 4 x4
block matrices ~ flux Jacobians) associated with implicit
spatial differencing in the coordinate directions by evalu-
ating the metric terms at cell interfaces in each direction.
Eq.~15) is solved by an implicit hybrid algorithm, where
a symmetric planar Gauss-Seidel relaxation is used in the
streamwise direction ~ in combination with approximate
factorization in the remaining two coordinate directions ~
and y. It is used to avoid the At3 spatial splitting error
incurred in fully three-dimensio~a,l approximate fa,ctoriza-
tion methods. The hybrid scheme is unconditionally stable
for linear systems and offers tl~e advantage of being con~-
pletely vectorizable like a conventional three-dimensional
approximate factorization algoritl~n. As a result, Eq. (15)
becomes
[M—(B )j+~/2~j+~/2 + (B+ )j-~/2^j-~/24~Q
=—Res(Qn,Qn+~)
iM—(C + Z)~+~/~+~/2 + (C + Z)k—1/2~k—I/2~Q
=Mi\Q
Qn+1 = Qn + Chin
with
M = t~J + (A ji+1/~ + (A+)i-l/2]
and the residual on the RHS indicates the nonlinear up-
dating of Else residual by using Q7~+~ whenever it becomes
availabe while sweeping in the ¢ direction back awl forth
through the computational do~na,i~.
Turbulence Modeling
For la~ninar flow computations Else coefficient of molec-
ular viscosity ,u = Ill is obtained front Sutherland's law.
Turbulence is simulated using the Baldwin-Lon~ax algebraic
turbulence model. For turbulent flow computations the
lan~inar flow coefficients are replaced by,
~ = pi +pt
(14) The turbulent viscosity coefficient lit iS computed by using
the isotropic, two-layer Cebeci type algebraic eddy-viscosity
model as reported by Baldwin-Lo~nax. In Ellis formulation
. .
At IS given by
~1 = { (tinner Y < Yc
(lit )O?tter lY > Yc
where y is the local distance measured normal to the lady
surface and Yc is tl~e smallest value of y at Chicle the val-
ues front the inner and outer region viscosities are equal.
Within the inner region, the Pra,ndtl-Van Driest for~nula-
tion is used
(let )inner = I ~~
where I = k-yt1- e-(Y /A )] and ~~ is the magnitude of
vorticity given by
/( Al dv )2 + ( do _ bW )2 + ( Ho _ 0ll )2
N/ By 0,~ fez by ()x 0z
and y+ = (rW/,tGw)Y ~
In the outer region, for attached and separated boundary
layers, the turbulent viscosity coefficient is given by
(lit )outer =It CcpFwa~eFI~ leb(Y)
Forage =~7lin(ilmaXFmax, CWk~lmaXUdif /I;37~2ax)
FI`leb(Y) =[1 + 5 5(C~leb/Yntax) ]
In the above equation k~ and cop are constants and
Unlay = 77la~[l~ly(l —e (A /A ))]
and Y7nax is the value of y at which F''~,aX occurs. Tile quaII-
tity Waif is Else difference between ~naxi~nu~n and minilDUTn
total velocity in Else profile
U4if = \/(U2 + V2 + W2)n7,ax— V(tG2 + V2 + W2)77tin
The various constants in the model are given in Ref. 15 as
( 16) and
63
A+ = 26, ~ = 0.4, It'= 0.0168,
Ccp = 1.6, Cat = 0.25, CI`leb = 0 3
IV. Grid Generation
An algebraic grid generation procedure based on trans-
finite interpolation technique has been used to generate
the viscous grid around the hull geometry y. The three-
dimensional grid generated around the body is of C-O type.
The C-O grid is particularly needed to adequately resolve
the wake region. Details of the theoretical aspects of the
transfinite interpolation ~ncthod and the various mapping
functions and their behavior in grid control is highlighted
in Ref. 10. The method is fundamentally a two body grid
generation procedure. The stern shape could be either open
or closed. A modified osculating interpolation Injunction has
been used in the present programs.
V. Results and Discussion
The hydrodynamic characteristic of the boundary layer
flow around the stern of a ship is quite different with and
without a propeller in operation. The action of a propeller
produces suction that accelerates the flow upstream. As a
result, the pressure and the skill friction drag around the
stern increase and the thicl;ness of the boundary layer de-
creases. The knowledge of the effective flow profile near the
propeller plane and the amount of the added drag is essen-
tial for designing an efficient propulsor. In the past, exten-
sive efforts have been made to study the interaction between
a propeller and a thick boundary layer experimentally and
computationally (4,5,C,7,8,17,184. For the reason of sim-
plicity, in most cases, axisymmetric bodies were chosen. At
present, we perforce the numerical simulations of flow over
DTRC Afterbodies 1, 2 and S without a propeller. In addi-
tion the flow over afterbody 1 with a propeller in operation
is analyzed. The results are then compared with available
experimental data. The purpose of the simulations is to
develop and validate a numerical scheme for analyzing the
complex interaction between boundary and propeller. The
details of the afterbodies are shown in Fig. 1, where rmaX
is the maximum radius of the body, r is the radial distance
measured from the body axis, ~ is the axial distance from
the nose and L is the total body length. The afterbody
length to ma:;i~num diameter ratios of all three afterbodies
are different; they are 4.308, 2.247 and 2.018 for Afterbodies
1 2 and 5, respectively. ITere, the afterbody length is de-
fined as the distance between the end of the parallel middle
body and the after perpendicular. Furthermore, Afterbody
5 has an inflected stern. The hubs of all three afterbod-
ies are identical at the position where the propeller can be
mounted. The different stern shapes generate a large data
base variation of stern flow suitable for the purpose of val-
idation of the computational scheme.
Flow over Axisy~n~netric Bodies
Flow variables such as pressure and velocity co~npo-
nents of an axisymmetric flow are independent of circumfer-
ential variation. They can be obtained through a. set of de-
generated equations based only on two spatial dimensions.
However, in order to validate the three dimensional for~nu-
lation we have just presented, full three dimensional com-
putations over three different segments of a body ale per-
formed. The sizes of the segments are 45°, 90° and 180° .
A viscous grid with C - O topology around tlte body is gen-
erated by a transfinite interpolation method. Grid domain
extends to three body lengths both upstream arid down-
stream of the body. Figure 2 shows a partial view of a
C - O grid around a generic body. The grid size used
for presented simulations is 91 x 25 x 49 in ~ (stream-
wise), ~1 (circumferential) and ~ (normal) direction. In
the ~ direction, the grid is clustered near the body with the
smallest grid spacing 5 x 10-4 of the body length. In the
direction, the grid is clustered near the bow and the stern.
Grid sizes used for all three segments (45°,90°and 180° ~
are identical. Consequently, we can perform computations
based on three different grid densities in the ~ direction.
The differences among the solutions are negligibly small,
therefore, only the results based on the 90° segment will
be presented here. In all our axisymmetric flow computa-
tions, the mixing length of the turbulence model has been
modified according to IIuang et al (5~.
A detailed analysis of Else measurement accuracies is
not available. Ilowever, else standard deviations of mea-
sured data were estimated from repeat runs. The standard
deviation of the measured static wall pressure was less than
5 percent of their mean values and the deviation of the mea-
sured velocities was less than 2 percent of the free -stream
velocity (19~.
The experiments of flow over Afterbodies 1, 2 and 5
were carried out at Reynolds numbers of 6.60 x 106 ,6.80 x
106 and 9.30 x 106, respectively (based on total body
length). Figures 3, 4 and 5 show the comparisons of the
computed and the measured pressure distributions over the
surfaces for Afterbodies 1, 2 and 5, respectively. The body
profiles are included in the figures in order to show the
relationship between the pressure gradients and the stern
shapes. At the region near the end of the body where the
stern and the hub meet, the computed pressure shows a
sharp decrease followed by a equally sharp increase. Based
on some different numerical schemes, this peculiar feature
has also been encounted by Chen and Patel (18) and Lee et
al (20) in their computations. Chen and Patel attributed
such phenomenon to the rapid change of geometry near the
hull-hub juncture as well as the upstream influence of the
complex pressure interactions in the tail region. A solution
we obtained with a panel method exhibited the same fea-
ture, as long as enough panel resolution around the hull-hub
juncture region was provided. Unfortunately, the experme-
ntal data lack the resolution needed to verify this feature.
Figures 6, 7 and 8 show the comparisons between the
computed and the measured velocity components at sev-
eral streamwise locations for afterbodies 1, 2 and 5, respec-
tively, where rO denotes the radius of body surface . For
Afterbody 1, the agreement is very good. The computa-
tion predicted the development of the boundary layer very
well. For Afterbodies 2 and 5, the agreement in general is
good, except the radial velocity profiles which are less sat-
isfactory. The agreement deteriorates as the measurement
location moves further downstream. The radial velocity
component is relatively small and is more difficult to mea-
sure accurately. For both Afterbodies 2 and S. the maxi-
mum difference between computed and measured values is
about At percent of free-stream velocity. The error bound of
64
the measurement is 2 percent. The measured axial veloc-
ities for Afterbodies 2 and 5 are progressively slower than
the computed velocities as the end of stern is approached,
although the difference is small. Near the end of the stern,
the measured axial velocities for both Afterbodies 2 and 5
show an inflection point which is not captured by the com-
putations. The pressure gradient of the stern boundary
layer is directly affected by the fullness of the stern shape.
The stern of Afterbody 1 is the least full among all three
bodies studied here. It caused only a mild adverse pres-
sure gradient in the boundary layer surrounding the stern
region. This may explain the reason why the simulation for
Afterbody 1 is the most successful.
In each computation presented above, 220 iteration
steps were taken. The L2 norm of the residual dropped
two orders of magnitude from 10-3 to 10-5 . 220 iteration
steps were chosen since further iterations produced only in-
significant variations. Each computation requires 17 CPU
minutes on a CRAY-YMP machine.
Flow over an Axisymmetric Body with a Propeller
Flow experiment for an axisyn~netric body with a pro-
peller were conducted at DTRC by Huang et al(4,G,7~. A
propeller was mounted on Afterbody 1 at x/L = 0.983 .
The geometrical and hydrodynamic characteristics of the
propeller are given in Huang et al (4) and Huang and Groves
(7~. The experiment was performed at a Reynolds number
of 6.6 x 106 (based on body length).
Numerically, the propeller effect is simulated by imbed-
ding body forces in a disl; of finite thickness located in the
propeller region. The details of this type of formulation
can be found in Stern et al (8~. Distribution of body forces
depends on the propeller's characteristics such as, thrust
coefficient CT, torque coefficient Icy, advance coefficient
J and radial circulation distribution G(r). The axial and
circumferential body force per unit volume are obtained
from the following equations:
CTR9pG(r)
MY IR P G(, )? d
4It R3G(r)
1rrJ2 /YX .fR P G(r)' do
where f bx and f be are the body forces per unit volume in
the axial and circumferential directions, respectively, Rh
and RP are the radii of propeller hub and blade, respec-
tively, and AX is the thickness of Else disk. The following
propeller data were used ill our computation:
J = 0.370, It T = 0.227, It q = 0.0453, CT = 0.370
The computed body forces are then incorporated into the
right Lund side of Eq. 16 and forms a part of the residual.
Based on the identical grid size and grid distribution used
previously, a computation was carried out for 220 iteration
steps. The L2 Noreen of tl~e residual dropped two orders of
magnitude from 10-3 to 10-5 without suffering frown any
numerical instability.
From the results of the computation, the influence of
propeller action can be detected up to about two propeller
diameters upstream. The differences between axial veloc-
ity components /\u=/VOO Title and without the propeller
in operation at two streamwise locations are shown in Fig.
9 . The measurement locations are at X/L = 0.954 and
X/L = 0.977 . The agreement between the computed and
the measured values is very good. The differences in pres-
sure distribution on the afterbody surface upstream of the
propeller plane are shown in Fig. 10. The results of the
computation are to the same degree of accuracy as those
reported by Huang and Groves (7~. The swirl velocity was
also computed but due to lack of experimental data to com-
pare with, it is not presented here.
Flow over Tangent-Ogive Forebody
NVith the same numerical scheme discussed above and a
modified turbulent model the flow over a 3.5 caliber tangent-
ogive forebody was studied (21) at angles of attack of 20°
and 30° and at Reynolds Lumbers in the range 0.2 -
3.0 x 106 . The purpose of the study was to investigate the
Reynolds number effect on low speed vertical flow and to
validate the numerical scheme. A C - O type grid was
generated for the purpose of computations. The grid size
was 97 x 40 x 91 in ~ (streamwise), ~ (normal) and
~ (circumferential) direction. Fig. 11 shows a comparison
of the computed and the measured surface flow patterns
at a Reynolds number of 0.S x 106 (based on diameter)
and an angle of attack of 20° . The lines indicate that
the primary separations are in good agreement. Top views
show two distinct regimes in which the surface streamlines
appear to collocate. Figure 12 shows a comparison of the
computed and the measured circumferential surface pres-
sure distributions. The Reynolds numbers are 0.S x 106
and 0.3 x 106 and the angle of attack is 30°. The Reynolds
number effect is snore pronounced on the leeward side, near
the nose. The agreement is good.
VI. Conclusion
The 3—D incompressible Navier-Stokes equations was
discretized by the flux-(lifference splitting and the implicit
high resolution schemes. A discretized system of equa-
tions was solved by an implicit hybrid algorithm, where
a symmetric planar Gauss-Seidel relaxation was used in
the streetwise direction in combination with approxi~na-
tion factorization in the two remaining directions. The al-
gorithn~ is highly vectorizable and suitable for computation
on a modern superco~nputer.
For the simulation of afterbody boundary layer flows,
this method is proven to be effective. Inclusion of the body
force propeller model poses no additional problem. To ob-
tain a converged solution with the propeller model included,
it does not require snore iteration steps in comparison with
the case without including a propeller model. The method
was also shown to simulate low speed vertical flow with
good results.
Acknowledgement
This study was supported by The Office of Naval Tech-
nology. The computing time of CRAY-YMP was provided
generously by NASA Antes Numerical Aerodynamic Sin~u-
lation (NAS) Program.
65
References
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ppl57-163
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Three-Dimensional Equations of Motion for Rectangu-
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"Propeller/Stern/Boundary Layer interaction on Ax-
isymmetric Bodies, Theory and Experiment", DTRC
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No. 4, Dec. 1988, pp 246-262.
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wind Schemes for the Three-Di~nensional, Incompress-
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No. 11, Nov. 1988, pp 1321-1328.
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13. Roe, P.L.,"Characteristic-Based Schemes for the Euler
Equations", Ann. Rev. Fluid Mech., Vol. 18, 1986, pp
337-365.
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Schemes and Their Applications", Journal of Compu-
tational Physics, Vol. 68, 1987, pp 151-179.
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AIAA Paper 78-257 Jan. 1978.
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Mesh Generation Using Transfinite Interpolation",VKI
Lecture Series 1983-04 Vol. 1, Mar. 1983.
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66
18
20
BL
8.6
Chen, H.C., Patel, V.C.,"Calculation of Stern Flow by
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AIAA Paper No. 89-0337, Jan. 1989
2 _
e i_ rrterxrg ~ ~ =
-
Att~~b~g 2
8.65 8.7 a.7s
z/L
I 1 1
8.8 e.8s 8.9
Fig. 1 Axisymmetric afterbodies
8.95 1
Fig. 2 Partial View of a C—O Grid around a Body
84sr
8.375 _
8.3 _
8.225 _
8.15 _
x x x Experiment
- N-S Solution
C'' -I
x/L
Fig. 3 Pressure Distribution on Afterbody 1
B.375
8.3
8.225
8.15
8.875
-. .875
-~.15
-8.225
_a ~
_ ._
-a.375
8.45 9,45
- x x ~c Experiment
N-S Solution
xT
;;C~8 _ C ~ ~
_ ~1
-8.45 1 1 1 1 ~ 1 1 1 1 1 1 _
-8 ~ 8 8.1 8.2 8.3 8.4 8.5 ~ 8.6 B.7 8.8 8.9 ~ 1 ~ 9.45
x/L
Fig. 4 Pressure Distribution on Afterbody 2
O.1 s 0 0 0 Experiment
N-S Solution
B.375
S _
0.3 _
B.22s
8.15
e.07s
;338 _
-C.875 _
-~.15 _
-3 .225
_D ~
_._ _
-a.37s _
x x x Experiment
N-S Solution
~x.
, , ,
8 8.t 8.2
8.3 8.4
8.5 8.6 8.7 8.8
1 1
8.9 1 1.1
z/L
Fig. 5 Pressure Distribution on Afterbody 5
(6a) Axial Velocity Profiles
x/L = 0.846 ~/L = 0.914 ~/L = 0.934 x/L = 0.964 x/L = 0.977
_
~ ,
~,1 0
0.os
o.oo
5 ,
1 ~'t 1
1
~7
0 1 0 1
0 1 0 1 0 1
ulV~
0 0 0 Experiment (6b) Radial Velocity Profiles
O. l 5 N-S Solution
r/r = n 8~46 z/L = 0.914 =/L = 0.934
0.10
0.05
0.00
x/L = 0.964 z/L = 0.977
~, ~
~___ ~ ~ ~ ~
O.O 0.1 O.O 0.1 O.O 0.1 O.O 0.1 O.O . 0.1
~Ur /V-go
Fig. 6 Velocity Profiles on Afterbody 1
67
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7/(°~—~)
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¢ 11' ~ ~ U- VV--: ,
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11 0 0 - 0 0 ~ ~ - ~8
co
1°1 - o 0 0 0 0 0 0 - ,
:: , _ ~
~ cn _
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~2
0 0 0 0 O~ '
.
o
o
T/(°.1 - .l)
co
G)
o
r~
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11 ~ [
° ° ° ° °°od
o
t—
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,~N
/~^
/oOO
- o
11
.
o
n
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~: °
_
n ~ 0
U)
~ cn
X
o
o
o
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o
o
O U~,
68
:0
O
O
'
o
oo
° o°~
11 o
1 ~H 1 o ,~~
~ 0 Ln 0
~ ~ 0 0
. . . .
O 0 0 0
7/(°d—.1)
7/(°~—~)
c~
0 0
f~
0 ~
-
° c
o
0 cn
~o ~
~ ~ z
0 ~ 0
~ ILl
~ P~,
0 .~
0 0
-
0
~o
· -
o
o
0
o
o
c~
o
o
o
o
o
O ;:s
i
o
o
u:
o
X
o
u)
~:
o
:^
._
c;
-
0 tl4
o
o
o
o
o
x/L = 0.954
a.06 L 0.06
.05
.04
D...01 _
=/L = 0.977
.05
.04
.03
0.01 _
x x x Experiment
N-S Solution
1 x
_ _ ~
_~
a l l l l
1 3 0.05 0.1 D.15 ~ 8.05 8.1 0.1S D.2
Au~ /VOO
Fig. 9 Measured and Computed Axial Velocity Increase on
Afterbody 1 Due to Propeller Suction
~ 4
a) top vlow
~//.~
y
compahtlon
//~
b) side view
O.2
5
_ . _ _
t<)~.1 _
0.05
x x Experiment
N-S Solution
. X 1 1
7 D.8 D.S 1 1.1
x/L
Fig. 10 Measure and Computed.Surface Pressure Decrease
on Afterbody 1 Due to Propeller Suction
1.C
x/D = 6.0
.5
.,[ ~ 1 1 1 1 1 1 1
0 20 40 60 80 100 1 = 140 160 180
~_
1.C,
.s .
n
1
x/D = 3.5
— _,ia'
-.5- ' 1 1 1 1 1 1 1 1 1
0 20 40 60 ~0 1 00 1 20 1 40. 1 60 1 80
'~:x~=2.0 ~
_~
-.751 1 1 1 1 1 1 1 1 1
0 20 40 60 ~ 100 120 1" 160 180
·7sr
.25
y
Fig. 11 Surface Flow Paterns - 25
__
_.7~
69
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~ a-~
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