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OCR for page 833
Simulation of UUV Recovery Hydrodynamics
S. Huyer, J. Grant (Naval Undersea Warfare Center, USA)
ARSTlRACT:
A novel medhod to compute th 3-D unsteady
hyd odynamics with application to undersea vehicles is
presented This approach solves the vo ticity equation, which
is derived fiom the moment m equation of the No ergot es
equation Most problems of No y interest Evolve
Compressible flow, which c m be described m terms of She
vo ticity alone Velocity is m integ cl quantity of She
mstmtaneous vorticity field Specific geometries me
represented using surface source Ed vo tex panels whose
trength is prescribed to satisfy She no-slip Ed no-flux
bo mdary conditions Vorticity is diffused fiom the vortex
sheets onto She body surface to mci tam c vorticity balance
Vorticity in the flow is specified et pomts Ed the vorticity et
my oth r point m th field is obtained vie Imear i temoktion
Imtemohtion is perfommed by conshucting tetrahed c using
D launcy hiaDguiarizatioa Tetrahed c provide She control
vol me to Meg ate over to obtain th velocity Ed She
cow ctivity of the control pm ts pro ides c basis to conshuct
derivatives A BcldwimLomax eddy iscosity model was
implemented mto the sol non algorithm to model turbulent
flow effects This method was Hen validated for two disparate
flow cases flow past m mmanned undersea vehicle UW
et Rey olds n mbers of one million Ed unsteady flow
development pest c cone Attached flow pest She WV was
compared with mpi iccl turbule t flat pate results Quality
of the flow pest c cone was compared with data obtained wish
e perime hi date Validation of f is method allows for c
subsequent simulation of c UW recovery problem
INTRODUCTION:
Undersea vehicle hyd odynamics pose sigmifc mt
challenges for the computation of complex, f ee-dimemiorul
unsteady flow field. Major examples include mbmarme
maneuvermg problems, low speed mcneuvermg Ed control
Ed mmarmed undersea vehicle UW) recovery Mmy of
These complex flows are characterized by th production of
vo ticity Ed its subsequent mteeaction wish She vehicle This
is generally th case when c vehicle enco mters She wake of
another body For cases where vorticity is dominant, c
Leg mgi m vorticity based method c m be used to compute She
complex m teddy twobody hydrodynamics Here,
Compressible, unsteady fluid flow c m be ohartotenzed by
She mst mt meous vorticity field clone For cases of flow pest
bodies, th vorticity distribution in She bo mdary layer Ed
wake detemmines She characteri tics of She enti e flow field
Vorticity based methods would seem to be c natmal fit to
solving th se t pes of problem This tech ique tilings She
velocityworticity fommohtion of She Navier-Stokes equations
to solve for th se flow variables on c Leg maim mesh
toalonlatiorLtl points adverted by th local flow) The solution
methodology ht. distinct cdv midges over t tditiornl methods
chat rely on fixed g id solutions The Lags mgi m vorticity
method is essentially g id fiee it does not Ply on c g id
( tructmed or mshuctmed) in the r tdirior~tl sense to evolve
':. vo ticity associated with th unsteady flow Vo ticity is
continually generated et th surface Ed is Represented by th
computational pomts chat are also contimmously generated
This m it e. the method natmally adaptive to coherent vortex
shuctmes Ed since the vorticity is tdh eoted by the flow, it is
subject to little m mericcl dfff sion The use of c Leg mgi m
mesh allows for She tIaightforward treatment of mm-i g
surfaces Ed does not requi e incomorction of cdditiorul temms
due to nommerticl reference frames in addition, multiple
bodies are mcluded m c shaighfforward maimer The
r tdirior~tl vo tex method ( horin, 1973) describes She
vorticity field by me ms of isoh opic elements or t lot s', which
have a sthugh Shut depends only on distmce from Heir
center; a fh quently used sh ength dish caution is the G mssi m
Caref I comparison with Theoretical Ed experimental data of
tw -dimensiorul c tlcnlations nsmg isohopic blobs of preform
size has bee repo ted by Sedhi m Ed Ghoniem (2) for a
backward-faci g step High-hsohtion 2-D computations for
flow past a cylinder have been conducted by Ko mom It os
Ed L onard (1995) Ed Snbrtm mitm (1996) Ed Present
th st mdard for using blob methods to compute msteady flow
past surfaces The original medhod has the feature that th
identity Ed location of neighbori g blobs are not needed to
compute She Biot-Savart Meg al (which determines th
pencil 1, so She algorithm for d is computation is simple
Liter works te g, Ko mom It os mdL onard,l995)typically
employ accelerated methods I G eengard Ed Rot hi:, 1987;
Strickland Ed Baty, 1995) to avoid m order No cat nhnon
(wh re N is the mmbe of elements) Even with thee
elaborations, the simplicity of m approach based on th
r tdirior~tl method Mains ah active
There remain sig i f mt difficulties when apply g
vortex blobs to flow past a e f ace, how ver One is the fact
that near She e face the blob vorticity dish9bution (a
G mssi m) actually penehates She surface so a finite value of
vorticity is on She mside of She surface This c m c me
problems when computing th bo meaty conditions Another
is that without s fficient overlap of She blob radii, th
computed velocity Ed associated vo ticih field will be
extremehy noisy In the cnrrem method, which will be
pret nted hter, -. vorticity in She field is piecewit
OCR for page 834
co timmous md is Imearly mterpohted between pomts in 6he
feld No overhp of elements is necessary to mcintain c
moodh velocity md vorticity feld in conbc t, blob medhods
~ely on c superposition of 6he blob f mctiom to determine 6he
feld vo ticity ad subsequent velocity dish~bution if
sufhcient overkp of 6he f mctions is not mci hmed, c noi y
velocity dish lb tion c m re mlt c msmg poor satisfation of 6he
bo mdary conditions ad c diwxgmg solution his overkp
ca be wxy difhcult to mcintain, especiclly near 6he surfae es
6he blobs me cdvected by the flow For 6his recson, mo t blob
medhods (eg Ko moutsakos md Leonard, 1995) employ
periodic interpohtion of 6he blob tre gth onto c ~eg lar g id
to most efhciently mci hm overhp to avoid noisy bo mdary
conditions his ca lecd to artifcicl mmericcl dfff sion,
however, md still does not solve 6he problem of vo ticity
penetrction mto 6he surfa due to 6he blob f mction
A odher difficulty with blob methods is th wide
r mge of sccles that emerge duri g the evohtion of flows pest
c surfae A example is 6he misohopy of the vo ticity
distribution m c 6hm bo mdary kyer For 6hese cases, 6he
g cdie t of vorticity in 6he normcl di~ection is much g ecter
6 m that m 6he t mg nticl di~ection his presents mother
compellmg motivation for developing m clgori6 m not based
on elements of miform size, b t one in which 6he si:D: md
shape of the eleme ts cdapt to th loccl spaticl dishib tion of
6he vorticity feld Blobs, whatever th ir si:D:, h~ve c con t mt
~adius md are the~efme isotropic in m~ture For thm bo mdary
kyers, misohopic elements are desi ed As problems are
e tended to thee-dimensions, 6he ctove probl ms me only
exacerbated For 6his ~ecson, the~e me only minimcl examples
of vortex blob sohtions for flow pest surfaes (e g
Gharaldvmi ad Ghomem, 1996)
We have developed c novel techmique directly
so lvmg the vorticity equation on c Leg mgi m mesh Vo ticity
is pecif ed et points m the field md is Imearly interpohted by
comtructmg tetrahed c This medhod hcs 6he cdvatag that
6he eleme ts me com ted cllowmg 6he vorticity to be locclly
cppro imated es c f mction of position his cpproah was
mtroduced by Russo md St~ain (1994) who examined in iscid
vo ticity felds for two-dimensiom~l flow Huyer md G mt
(2000) extended the method to examine viscous flow pc t
treamlined md bl ff bodies he current cpproah hects
f lly th ee-dimensiom~l flow pest multiple bodies he
voticity is determined on c mmber of pomts m 6he feld
These points me 6hen com~ected to form c set of teDched c vie
c Dehuncy teDched clization clgori6 m Th vorticity et my
odher point m 6he region is thff~ aproximated by linearly
mterpohting 6he vo ticity et the nodal points with given shape
f mctions based on the geometry of the tetmhed ~ Fi st md
second order derivatives me comp ted vie c second order lee t
squares fommohtion from the eleme ts com cted et c given
node ~rshall md G mt, 1995) Th vorticity feld md 6he
locations of the cclcohtion points me updated et eah time
tep Since the formoktion is c Lagrmgim cpproah, 6he
cdvection term is cutomaticclly inchded Viscous dfff sion is
a omplished usmg bodh m effective diffusion velocity (i e
Shicklmd md Bcty, 1995) es w 11 es 6he second order
Lcpkcia ~teg ction of the Biot-Savart i teg cl provides 6he
velocities
A oth r difficulty in t~ectmg high R y olds n mber
flowis aco mting for tmbulence Traditional vortex methods
have relied on mdom walk ad oth r tochc tic medhods to
simohte 6he high R y olds number tubuent flow Thus far,
most deterministic solutions for viscous flow have been
limited to low R y olds number kminar cases In 6he present
method, c deterministic aproah utilizmg 6he full viscous
equ~tions was desi ed Therefore, c tmbulence model has
been i trodu d to acoumt for 6he hck of spaticMesolution
required to properly t~ect fully tubulent flow diectly A
Bcldwm-Lomax tubulence model was implemented mto th
viscous soluion methodology to better model 6he tmbuent
flow charateristics To th athor's k owledge, 6his h~s yet
to be implemented in my vorticity based solution
methodology
This pcper includes um tecdy compubtiorud date
collected for flow pc t c UW et c Rey olds number of one
million md um tecdy flow pest c cone for c R y olds number
of 50,000 Attahed tubuent boumdary hy r flow over th
WV is compared with tubulent velocity profiles obbined
fiom empiriccl ~esults by Spcldmg md Coles (see White,
1974) Time averaged wake velocity dab for flow pc t c cone
is compared with dab prese ted by Calve t (1967) The
dockmg cone md WV are fi ed in spae md the flow field
computed These validation tests are performed to e tablish c
high co fidence level in 6he method This is follow d by c
simohtion of UW ~ecove y md discussion of 6he ~esults
METHODOLOGY:
Surface Defirddon:
A example of m Ummanned Undersec Vehicle
W) su fa mesh is show in Figure I To con truct 6he
mesh, su fae body points md umit nommcls me ~equi~ed
Pseudo points are pkced just ctove the body points A
L hunay tetrahed clization clgorithm Borouchaki md Lo,
1995) is then cpplied This essenticlly conshu ts teDched c
encompcssi g cll points md lea~s c convex sufae Th
su fa then consists of 6he faes of the teDched c, which me
exclusively mcde up of 6he origincl body points The WV
sufae was defmed with 586 points md c totcl of 1150
su fa pcnels These pcnels me 6hen u ed to define 6he
sufa souceadvort:xpcnels
Satisfaetion of the Surface Boundarv Condfition:
Eah pa I on th body surfae carries two velocity
generctors: c sufae vortex dish~bution Iymg in th phne of
th pcnel md c pote tial sou ce The souces me neded,
math maticclly, to ffnu e the no-flu boumduy condition is
met properly Uhlm m md Grmt (1993) show d th~t es 6he
number of pcnels mcrecses to i finity, 6he sou e st~engh
cpproahes :osro since the su fae vo ticity di tribution c m
sati fy 6he no-slip md no-flu boumdary conditions
simultmeously Both distributions me taken to be umffomm
over a individu~l pcnel md lie m m i finitely 6hin sheet on
th su fa Thus 6he vo tex shength parameter charateri i g
c pcnel is the velocity jump aross the pcnel The velocity du
to c potenticl sou e, a, md vortex pcnel tre 3th, 7, on c
sufa Sis:
2
OCR for page 835
A
~_Q:
n = surface normal
S =
... ~ ....
t = tangent to normal and streamwise
Figure 1: Unmanned Undersea Vehicle Geometry
N
u(x) = 2~ l-C7nBn + 1'n x Bn }
n=1
where on and ~ n are the discretized strengths for a panel n
of surface are Sn and:
J1 (x-x')
47rV ~x- x'~3
sn
(2)
The no-slip and no-flux boundary conditions are then
applied to the system of equations to solve for the source and
vortex sheet strengths.
Initial Volume Vorticity Distribution:
After the surface panels are defined, it is required to
initialize the volume vorticity beginning with the surface
vorticity. Nodal vorticity values are located at the surface
body points used to define the geometry. Additional points are
then placed in layers staggered over the body nodes and the
surface panel centroids. These layers are located at a constant
normal distance initially and are separated by a distance
according to a number of boundary layer thicknesses. A single
thickness is defined based on a viscous diffusion length scale
of Where v is the kinematic viscosity and At the time
step. The initial layer is very thin and is of the order of 10%
of a flat plate boundary layer thickness
Linearized Tetrahedral Vorticity Elements:
In the present method, nodal vorticity values are
known and a linear variation of vorticity between nodal points
is assumed. At each time step, a Delaunay triangularization
routine is used to form an unstructured mesh connecting each
nodal point thus forming tetrahedral elements. Delaunay
triangulation effectively optimizes the aspect ratio of all
tetrahedra constructed from a random distribution of points.
For more detail concerning this method, the reader is referred
to Borouchaki and Lo (1995). For a single tetrahedral
element, Zienkiewicz (1977) derives the four shape functions
that are described as a function of the geometric location of the
four vertices. The shape function values are 1.0 at their
respective nodes and 0.0 at each of the other three nodes. The
vorticity over the element can then be expressed as:
(1) = tt)lN1 + 'l)2N2 +(i)3N3 + m4N4 (3)
The velocity (from the Biot-Savart Integral) is:
47t J ~x-x]3
V
In order to solve this integral analytically, vorticity is taken to
be linear as in (3). In addition, the integral is decomposed
according to the divergence theorem and is solved over the
four faces of the tetrahedra. This expression becomes:
(1) 4~ |((x - x') rS
+~k(X )X_ | k (_ _,)t' dS
k=l sk
_ ~ k ( ) X |^V
k=l V
(5)
This expression is used for nearby tetrahedra.
Contributions of tetrahedra at an intermediate distance are
computed by 1-point or 5-point Gaussian quadrature.
Contributions of tetrahedra further away are computed by an
accelerated calculation (Greengaard and Rokhlin, 1985).
Computation of Derivatives:
Since the vorticity is assumed to vary linearly over
the element, the first derivatives will be constant over a single
element and the second derivatives will be zero. A higher
order method to compute the derivatives was desired. A
second order method that expresses first and second order
spatial derivatives across scattered points utilizes a polynomial
fit of the local vorticity using a least squares solution for all
the tetrahedra which intersect a given node. For more detail,
the reader is referred to Marshall and Grant (1995). This is
accomplished by expressing a component of the vorticity
about a desired node as:
= (I)o +ax+by+cz+dx2 +ey2 +fz2
+ gxy + hxz + kyz
(6)
and determining the constants a - k by a least squares fit to the
values of m- mO at points in the neighborhood. x, y and z are
referenced to the local node. For tetrahedral methods, the
elements connected to a given node are known and therefore
the nodal points can easily be determined. Additional points
can be found through a search of neighboring elements. After a
sufficient number of points are found to maintain accuracy, a
second order fit of the vorticity is computed with the values of
the derivatives simply computed at a given nodal point (x = y
=z= 01:
3
OCR for page 836
—=6,—=b,—=c
ax ay az
2 32d 32d 32d
V d =—+—+—= 2 0(d+e+f)
ax2 ay2 az2
C)
Evoludon of the Vorticity Field:
~ i vik id flow, the v locity field trasports vo ticity
m fhe wme way 6s 6 material element T is type of flow is
fhus v ry w 11 suited to L6g mgi m mesh fommulations, wh re
fhe mesh points 6 e trasported by the v locity fleld
How v r, when Vik osity is p~esent, vorticity may be
h m po ted by me ms of her th m 6 dv ction by the v locity fleld
dmely, Vik diffu ion Rcth r thm i trodu ing new 'empty'
pomts mto fhe mesh onto which vorticity may diffuk, w
h m port the exi tmg mesh points wifh th sum of 6 dfffusion
v locity md fhe urokl flow v locity Thus fhe mesh points
tend to mov fiom regions of ki ger vorticity magmitude to
~egions of lesser magnitude, acordmg to fhe dfffusiv
h m po t by viscosity
The concept of dfffusion v locity for k 616rs (ie
voticity mag itude which cm be witten 6s one of fhe
dependent v6 iables) is rk~dily dev loped Begm wifh fhe
vo ticity equation:
a +u Vd =di Vu+VV d
(8)
This equation states fnat fhe matk i61 chmge m
vo ticity is 6 f mction of vortex Ime tretchmg md dfffusion
du tOVik osity Diffusionv locitybasedonthemagmtude of
fhevorticityisdefmed65:
v = VV(k Q)
(9)
wh re Q is defmed 6s the rcaki magmitude of th vo ticity
v tor This expression is mse ted i to equation (8) md after
some algeb~a, the ~esult is:
—+ (u +v) Vd = d Vu
at (l O)
+Qpd V (k Q)+vV d v Vd ~
Here, d is th mt v ctor tmge t to fhe vorticity v ctor
deflnedas: d =—Q
The mesh pomts 6 e now h m ported a ordi g to:
d _ _
—=u+v
dt
md fhe vo ticity is ev iv d acording to (10)
TheL6g mgimpointsw readv ctedacordingto m
Adams-B6shfo th method to mamtam second order acu ay
m time Since time tep r mainr comtat:
x~eW = x old + I S(u + v)At 0 S(u + V)ol6 ~t (12)
BaldwimLomax Turbudenee Model:
Since spati61 resolution conshamts du to th
computatiomd co t of the c61cuation prohibit di ect m meric61
simulation of high Rey olds number flow past the bodies of
i te~est, 6 B6 Idwm-Lom6 tu bUIk e model w6 s implemented
i to th code This tubuik e model is summ6 ik d by
Wilcox (1993) it is m 61gebraic eddy iscosity model that
was dev loped for Uk in computations wh re boumduy kiyer
prope ties mch 6s boumd6 y kiyer 6hickness 6md edge v locity
6 e dffficult to determine T is is 6he case for the p~esent
umteady flow cases Smce 6he flow is umsteady ad th
computatiomd points 6re effectiv Iy mdomly k 6ttered. thek
qumtities would be ne6 Iy impossible to detemmine Th
B6 Idwm-Lom6 model contkim 6m i mer 16 y r 6md outer ki yer
eddy viscosity:
Im~er L6y r:
VT = 12 t |d]
mt = ky[l—e (i ° )]
uter L6yer:
VTo ctCc~Fw keFkleb(y;ymax/ckieb)
Fk leb = [I + 5 5( 3 ) ]
Fw ke = mm| ma::Fmax; Cwk YmaxU2ir /FmaX ] (1 4)
k; [ y ]
The ciokure coefficients 6re:
k;= 0 4, ct= 0 0168, Ao =26
Ccp = 1 6, Ck leb = 0 3. Cwk I
(I 3)
(I 5)
In order to detemm me y+, the fi iction v locity must be
detk mined in the present computations, 6he fiiction v locity
is 6ssumed to be related to th su fae vo ticity v61u so that:
UT ~ 7;; [i~i
y = UTy / V
(I 6)
(I 7)
In the 6bov formokition, y is the norm6 I di tance to
th w6 11. Ymw is th valu of y where FmW is foumd 6md Ud ~ is
th m6 imum v61u of U for boumd6 y 16y rs For fiee she6
16yers, Udr is the dffference betw k the madimum v locity m
th 16yramddhev61u of U6ty=ymw
In detemminmg wh ther to use 6he imer kiyer or the
outer 16y r eddy iscosity, FmW 6md Ymw mut be prope Iy
detk mined To properly determme 6hese v61u s, the
computational points must be directly 6bov 6he body su fae
For tructu cd g ids, thek poi ts 6 e k~sily determmed but for
ramdomly spaed poi ts, it becomes mme dffhcult A
4
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subroutine was written to organize the volume points
according to their nearest body point. In this manner, all
points in the volume are associated with a single body point
that it is most nearly above. These sets of points are then
sorted to increase in the normal direction and then FmaX and
Ymax determined. If y is greater than Ymax, the outer layer eddy
viscosity equation is used and if y is less than Ymax, the inner
formulation is used. Equations 14 and 15 are then modified
using the total viscosity defined as:
Total = v + v eddy
(18)
Vorticity Boundary Condition:
After the surface source and vortex sheet strengths
are computed it is necessary to transfer the vorticity in the
infinitely thin sheets into the volume. This is accomplished by
adjusting the surface nodal vorticity values to satisfy the no-
slip boundary condition exclusively. To do this, surface
vortex sheet values must be minimized thus requiring:
(omVm = 1'mAm
(24)
The left-hand side is the volume integral of the first vorticity
layer at node m that must balance the vorticity in the infinitely
thin layer expressed in the right hand side term. The vorticity
is desired at body point m that can satisfy the no-slip boundary
condition. An iterative scheme is then used to set the
vorticity. The surface velocity boundary conditions are
computed and the vorticity sheet strengths determined. The
vorticity is then determined by equation 24. The surface
velocity boundary conditions are then re-computed and
vorticity sheet strengths computed again. This iterative
procedure continues until the magnitude of the vortex sheet
strength is below 0.01. This typically only requires 2-3
iterations and converges quite rapidly. The reason for this is
that the layer of vorticity is very thin so there is little
difference in the velocity generated by the infinitely thin sheet
and the thin vorticity elements connected to the surface.
Initial Volume Vorticity Distribution, Euler Layer and
Point Creation:
After the surface panels are defined, it is required to
initialize the volume vorticity on a set of points. Nodal
vorticity values located at the surface body points are used to
define the geometry. Additional points are then placed in
layers staggered over the body nodes and the surface panel
centroids. To ensure that sufficient resolution of the boundary
layer vorticity is maintained close to the surface, a thin layer of
"Euler layer" of fixed points is used. Typically, the field
points are located normal to the body nodal points in
successive layers. This also allows for the computation of
derivatives using standard finite difference formulas as an
alternative to the least squares approach summarized in
equations 6 and 7. Vorticity evolution for points in the Euler
layer is performed using the Lagrangian form of 8-10 then
interpolating the values back on to the original point positions.
On the first time step, the Euler Layer consists of seven sub-
—=~
a.)
b.)
Figure 2: a) initial point distribution and b) point distribution
at t = 3.0.
layers and ten additional sub-layers of points are Lagrangian
(adverted by the local flow). Figure 2a shows a close-up view
of the typical initial distribution of points and subsequent
cross-section of the tetrahedral mesh for a cone. Figure 2-b
shows the developed point distribution and mesh cross-section
at a later time.
On the first time step, an initial boundary layer
thickness is assumed, which is 10% of a fully developed,
turbulent boundary layer on a flat plate. The boundary layer is
assumed to be attached and of uniform thickness modeling an
impulsive start. The purpose of this is to more easily initialize
the vorticity field with the vorticity remaining close to the
wall. The vorticity on the surface is determined from the
boundary conditions and the vorticity in the volume is
assumed to exponentially decay. The vorticity values on the
surface and in the boundary layer are then adjusted using an
iterative scheme to satisfy the no-slip and no-flux boundary
conditions.
New Lagrangian points are continually created above
the Euler layer. Extended panels are formed that are
effectively the outer faces of the tetrahedra formed in the Euler
layer. Since the points in the Euler layer are not adverted, the
tetrahedra in this layer can be explicitly set for all times. These
extended panels on the outer surface of the Euler layer have
the same characteristics as the surface panel directly
underneath. New Lagrangian points are formed by first
determining the closest point to the extended panel that is
above the panel. To determine if the point is above the panel,
the centroid of the surface panel is determined and a radius
constructed to include the three panel vertices. If a point lies
within the radius, its normal component is computed. The
closest normal point within the radius is then found. If this
point is greater than a prescribed value, a new Lagrangian
computational point is created. The new point is located in the
tetrahedron formed by the closest point and the three points
connected to the nodes of the extended panel. The vorticity is
5
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Representative terms from entire chapter:
flow past
103 104 105
10° 1 1 ,,, 1 ,. 10°
. . . -
a,
Is
. -
~ 1 o-1
._
o
._
o
an
in:
W: ~ ~X_ '
I, `~ at- 3`Vortex Blob
\\ ~ £ - 1U
_ ~ \ ' ' ' A
~,~=10
direct
_ 10-4
10-~
~ 10-3
0-3 ~ ~ ~ ~ ~ ~ ~ ~1 ~ ~ ~
103 104 105
Number of Points
Figure 3: rms error in the velocity calculation for various
accelerated calculation error bounds.
t=O.O
Axial Vorticity
U.UU' 7 _
1 2 3
Radius, r ~ Azimuthal Vorticity
t=2.0
~0
t=1 in
' ~ 1,,
1 2 3
Radius, r
>~ 0.125 _
(a) ~ ~
0.100' ,
0.075
0 050
t = 2.0 ~~—
1 .0
0.9
t~ 0.8
0.7
0.6
0 5
Vorticity `o 0.4
:~ Magnitude ~ 0 3
z 0.2
0.1
0.0
Figure 6: Instantaneous tetrahedral grid cross-section in the x-
y plan
-—————en
i,.._—
me... - _
o
0.10
0.09
t~ 0.08
8 0.07
0.06
~ 0.05
am- ~o 0 . 0 4
~ 0.03
Z 0.02
0.01
0.00
Figure 7: Instantaneous vector plots in the x-y plane of the
turbulent flow past a UUV. The vectors are colored based on
the vorticity magnitude.
the vorticity is diffused onto the initial layers of points forming
a fully attached boundary layer over the cone as well as the
UUV as was described earlier. As time progresses and the
flow develops, a separated region forms behind the cone
characterized by a cohesive vortex ring structure. This vortex
forms close to the surface of the cone and the resulting wake
structure impacts the downstream UUV.
Test runs were conducted at both the Waterways
Experimental Station Cray C-90 and at the NAVOCEANO
Cray C-90. Average run times required approximately 36
CPU hours.
RESULTS:
A cross-section of the tetrahedral mesh depicting the
instantaneous flow past a UUV at zero angle of attack for a
Reynolds number of 1,000,000 is shown in Figure 6. This
figure illustrates the resolution of the grid points used to
compute the unsteady turbulent flow. Figure 7 shows snap-
7
BOUNDARY LAYER PROFILES
Re = 1,000,000, x/LUUV = 0.7
—VORTEL Al
—Empirical TBL Profile (Cole)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0
Streamwise Velocity
Figure 8: turbulent boundary layer profiles comparing the
current computations with empirical flat plate results from
Spalding and Cole.
BOUNDARY LAYER PROFILES
Re = 1qOOOqOOOq x/LUUV = 0.7
—VORTEL
—Empirical TBL Profile (Cole) /
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Streamwise Velocity
Figure 9: Same as Figure 8 except scales adjusted to highlight
near wall region.
0.12
0.10
._
In
it, 0.08
-
~ 0.06
Q 0.04
0.02
0.00
TURBULENT INTENSITIES
Re = 1,000,000, x/LUUV = 0.7
.
,
[
r ~
it__
——U'
~V'
~~W'
u'(Klebanoff)
~ v'(Klebanoff)
0 w'(Klebanoff)
0.0 0.2 0.4 0.6 0.8 1.0
Normal Distance
Figure lo: Turbulent intensities in velocity components (u', v'
and w') compared with flat plate empirical results from
Klebanoff.
shots of the vector plots taken at x/1 = 0.7 for t = 2.0, 2.1 and
3.0. The vectors are colored based on the vorticity magnitude
with the scale shown to the right. Maximum vorticity
magnitudes of 100.0 were chosen to illustrate the boundary
layer flow. Maximum surface vorticity values exceeded 2,000.
These plots appear very similar. Upon closer inspection,
however, there are qualitatively small changes in the overall
flow as the unsteady vorticity field is advected over the UUV.
These plots illustrate the unsteady attached flow over the UUV
for high Reynolds numbers.
Boundary layer axial velocities over the UUV were
time averaged between t = 2.0 and t = 4.0 to obtain mean
velocity profiles and the turbulent intensities in the velocity
components (u', v' and w'). Mean axial velocity profiles for
Re = 1,000,000 are plotted in Figure 8 for the current
computations and for empirical results obtained from flat plate
boundary layers for Spalding and Law of the wake corrections
of Coles (see White, 104. As can be seen, the computed
velocity profile agrees very well with empirical solutions from
Spalding and Coles. Even in the inner layer, the agreement is
good except for slightly lower velocities at y/d between 0.01
and 0.04.
Figure 9 shows the same mean velocity profile with
the scales compressed to highlight the near wall behavior. At
the wall, the computations agree very well. From y/d = 0.01 -
0.04, it appears that the computational solution bows out
somewhat. Good agreement is seen thereafter from y/d = 0.04
-0.1.
Figure 10 shows plots of the turbulent intensities
(u',v' and w') as a function of distance with comparisons with
these statistics obtained by Klebanoff (see Hinze, 19754.
Since a turbulence model is used, the turbulent intensity
statistics were expected to be minimal. The computations,
however, are showing fluctuations in all three velocity
components due to the inherent unsteady computations of the
present method. u' appears to show the best agreement
although it does not reach the same maximum near the wall as
Klebanoff's data. v' and w' show poorer agreement. This is
likely due to the lack of spatial resolution of the surface. The
fact that the turbulent intensities are non-zero and the fact that
u' follows the same trends as the experimental results
demonstrates the potential of the current code to represent
turbulent flow without a turbulence model. Of course, the
spatial resolution would need to be greatly increased to
properly accomplish this.
Unsteady flow past a 45° cone was also computed
and results of the unsteady flow field can be seen in Figure 11.
Here, vector plots, representing the instantaneous velocity
field, are colored based on the z-vorticity component with the
x-y plane displayed. The surface is shaded based on the
vorticity magnitude. This plot compares the instantaneous
unsteady flow at t = 2.0, 3.0 and 4.0 for Re = 1,000 and Re =
50,000. For Re = 1,000, the flow was assumed laminar and
for Re = 50,000, the turbulence model was used. In both
cases, a ring vortex is produced in the wake of the cone and
can be seen to grow in size as time is increased. The laminar
flow case shows a much more coherent, well-defined vortex
compared with the turbulent flow case.
Quantitative comparisons of this flow condition are
shown in Figure 13 for laminar and turbulent flow
.,......
| Z-vort city
·-~.-..Surface:
Vorticity
Magnitude
—x
~..·~-~:
Re = 1,000 (1aminar) Re = 5O,000 (turbulent)
Figure 11: Unsteady flow development past a cone for the
laminar flow case (Re = 1,000) and the turbulent flow case (Re
= 50,0004. Velocity vectors are colored based on the wake z-
vorticity and the surfaces are colored based on the vorticity
magnitude.
Z-VORTICITY DISTRIBUTION
. - ~~
~ -0.75
o
N -1 0
_. ~
-0 5 -0.25 0
-20
Re = 50,000
Re = 1 ,000 30 ~
y/D
Figure 12: Z-Vorticity distribution at t = 4.0 in the wake of
the cone at x = 2.0 (relative to the cone apex) for Re = 1,000
(laminar flow) and Re = 50,000 (turbulent flow).
computations. The z-vorticity component in the wake was
computed at one cone diameter downstream of the stern of the
cone (corresponding to the location of the vortex core). The
computational data shows the instantaneous velocity field at t
= 4.0. As can be seen, data for Re = 1,000 show significantly
higher vorticity with peak values on the order of 20.
Turbulent flow vorticity values for Re = 50,000 are much
more modest by comparison with values approaching 10. In
addition, the vorticity for the laminar flow case appears to be
STREAMWISE VORTEX VELOCITIES
Vortex Wake of 45 degree Cone
i.,.,'. ~ ~
~ ~ I~
_L 1 1~
_ \~x
Re=50,000
Re = 1,000
- ~ Calvert (Mean u)
Figure 13: Wake velocity distributions for a 45° cone 0.88
diameters downstream of the cone stern for Re = 50,000 and
Re = 1,000 at t = 5.0. The mean wake profile from Calvert is
shown as a reference.
0.5 0.75
more concentrated within a well-defined core. The turbulent
flow case appears to be more spread out.
Streamwise velocity component data are shown in
Figure 12 for laminar and turbulent flow computations. The
streamwise velocities in the wake were computed at 0.88 cone
diameters downstream of the stern of the cone. This
corresponds to the same location as Calvert (1967) for his
experimental wake velocity data for flow past a cone. The
computational data shows the instantaneous velocity field at t
= 5.0. Data from Calvert are time averaged. As can be seen,
data for Re = 1,000 show significantly stronger reverse flow
velocities along the centerline of the cone with reverse flows
approaching -1.5. Also, maximum flow velocities are on the
order of 1.5 near the shear layer where the freestream meets
the vortex wake. Turbulent flow computations produced
milder flow velocities with maximum streamwise velocities of
1.1 seen in the shear layer and reverse flow velocities reaching
-1.0. These data also compare well with experimental results.
The initial decrease in streamwise velocity (y/d = 0.3 - 0.6)
compares quite favorably as do maximum streamwise
velocities. Experimental results show reverse flow velocities
on the order of-0.5. Since we are comparing instantaneous
computational results with time averaged, however, the
computational results appear highly encouraging.
UUV Docking Simulation:
The docking cone was defined using 442 points
resulting in the formation of 880 panels (see Figure 144. A 45°
cone angle was used with 0.0954 diameter and the stern face
of the cone was flat. The nose of the cone was placed at the
origin of the global Cartesian coordinate system and the nose
of the docking UUV was placed at distances ~ Xuuv ~ of 0.5,
0.25 and 0.125 relative to the origin.
Figure 15 shows the initial flow development for the
test case of the UUV at a position 0.125 downstream of the
origin (slightly less than one cone diameter downstream of the
cone stern). The surface displays the source/vortex panels with
the surface colored based on the y component of surface
UUV/Cone
- xuUv~
Figure 14: UUV, Cone and tandem UUV/Cone geometry.
Figure 15: Velocity vector plots at the tetrahedra centroids
depicting the unsteady flow development past a cone/docking
UUV from t = 0.0 - 0.5. Re = 10,000, Xuuv = 0.125. Vectors
are colored based on the streamwise velocity component and
the surface is colored based on the y vorticity components.
vorticity (from +5004. The vectors in the field represent the
velocity at the tetrahedra centroids and are colored based on
the streamwise component of the velocity with red vectors
displaying maximum velocities over two times freestream
values and blue vectors displaying reverse flow velocities. At
t = 0.0, the effective attached boundary layer flow can be seen.
Notice the high velocities at the top and bottom of the cone.
Immediately aft of the cone, the velocity vectors point toward
the axis of the cone with zero streamwise velocity component.
The initially structured mesh points over the UUV display a
well defined boundary layer produced by diffusing the initial
vorticity onto the points over the surface. After 10 time steps
(t = 0.05), the separated vortex ring structure can clearly be
seen aft of the cone. By t = 0.1, this vortex impacts the
leading edge of the UUV. Notice a slight reduction in surface
vorticity by the decrease in redness near the leading edge. By
t = 0.15, the vortex in the wake becomes larger and the surface
vorticity becomes redder indicating increased vorticity. By t =
0.2, impact of the vortex ring structure on the nose of the UUV
9
~ -,''$',~.j,., , f~"',,,,,,.,,. ,, ,.,., ~ .~ .2::"2j" ~~ " it' ~ '
~~"'~ ~~ ~ ""I,,., ' ""~ ,".'',.,''~' :.:2.2'
~ ....~:~ ::~:: — , " '~ ,,''"':"" ~'~'~'~'~'~'''~"''"'. ~~'~'''~''' ~~ ~'~'"."~'~'"~ :.'' ,' ~ Y-c'o,
Figure 16: Velocity vector plot comparison of flow along the
centerline in the x-y plane for Xuuv = 0.125, 0.25 and 0.5 at t
= 1.0.
Figure 17: Velocity vector plot comparison of flow along the
centerline in the x-y plane for Xuuv = 0.2 and the UUV center
vertically displaced - 0.05. The surfaces are colored based on
the surface pressure and the velocity vectors are colored based
on the z-vorticity component.
creates an asymmetry. In addition, notice the blue surface
coloration indicating opposite signed vorticity production at
the nose of the UUV. Beginning at t = 0.25, periodic
asymmetry can be seen in the vortex structure between the
cone and the UUV. Finally, notice the increase in boundary
layer thickness and the formation of smaller vortex structures
as the vorticity from the cohesive vortex ring is ingested into
the UUV boundary layer.
Figures 16 shows comparisons of the flow for Xuuv =
0.125, 0.25 and 0.5 at time 1.0. For Xuuv = 0.125, the flow
from top to bottom is mostly symmetric although the alternate
variation in vortex size does appear to result in increased flow
over the upper surface by t = 1.5. For Xuuv = 0.25 and 0.5,
the UUV is sufficiently downstream of the cone so a definite
oscillation due to changes in wake vorticity shedding occurs.
This wake oscillation results in different impacts on the
trailing UUV. This results in an overall thickening of the
UUV boundary layer with areas of vorticity accumulations,
which include the vorticity produced by the cone.
In Figure 17, the UUV is displaced a vertical distance
of 0.05 below the axial centerline of the cone. The cone is at
0° pitch angle and the Xuuv = 0.2. Again, the surface is
colored based on the surface pressure and the velocity vectors
are colored on the z-vorticity component. In this case, the
vortex structure on the upper portion of the cone is allowed to
freely advect over the UUV while the vortex on the lower
portion of the cone impacts the UUV. At t = 0.1, the cone
PRESSURE COEFFICIENT vs. TIME
Point Calculations, Xuuv = 0.125
n
1 _
-0.8
-1
Figure 18: Surface pressure on the UUV as a function of time
for z = 0.0, upper surface and x = 0.0, 0.06 and 0.5
corresponding to nose stagnation point, point of maximum
UUV radius and UUV body center respectively. Xuuv =
PRESSURE COEFFICIENT vs. TIME
Point Calculations, Xuuv = 0.5
-n4
-nn
-o.S
-1 -
~ .75
Ill—X=0.0
—x=0.06
Non-Dimensionai Time —x=0.5
Figure 19: Same as Figure 8 except Xuuv = 0~5.
vortex wake appears fairly symmetric. By t = 0.2, the initial
portion of the wake impacts the UUV. This results in the
formation of a vortex structure near the leading edge. By t =
0.3, this wake induced vortex begins to advect over the UUV.
A low-pressure region is indicated immediately downstream of
this vortex. By t = 0.4 and 0.5, it appears that the entire wake
produced by the cone advects over the upper surface of the
UUV and combines with the boundary layer vorticity of the
UUV. There did not appear to be any portion of the cone
wake that advected over the lower portion of the UUV.
Figure 18 shows the point pressure distributions as a
function of time for the Xuuv = 0.125 test case. The point
locations correspond to UUV nose stagnation point (x = y = z
= 0.0), point of maximum radius on the UUV (x = 0.06, y =
0.045, z = 0.0) and the center of the UUV (x = 0.5, y = 0.045,
z = 0.04. This plot demonstrates the transient nature of the
surface pressure and highlights the impact of the cone wake on
the UUV. The pressure at the nose shows an initial stagnation
pressure of 1.0. By t = 0.1, the pressure decreases to 0.0. By
this time, the vortex ring structure is well formed in the wake.
At t = 0.25, there is an increase in pressure which correlates
with full impact of the vortex on the UUV nose. As time
progresses, there are fluctuations in the pressure.
Pressure at maximum radius (x = 0.06) shows a
different behavior. Initially, Cp values of -0.3 are seen before
impact of the cone wake. There appears to be an increase in
10
UNSTEADY UUV FORCES
03' Xuuv=0.125
0 2 - .
0.1 - .
-0 1 - .
-02-
-0 3 -
Non-Dimensional Time
UNSTEADY UUV MOMENTS
oo4- Xuuv=0.125
oo3
non
n n1
-O.0
-0 02
-0 03
-o .04—
. .—Cmx
—Cmy
—Cmz
Non-Dimensional Time
Figure 20: Unsteady force and moment coefficients for the
UUV with the origin at 0.125
UNSTEADY UUV FORCES
o.3 T XU UV = 0.5
_
0.1 -
o- ,
-0.1 - .
-0.2-
-0.3 -
0 04 T
0 03
002
0 01
n
-0 01
-0 02
-0 03
-0 04-
Non-Dimensional Time
UNSTEADY UUV MOMENTS
Xuuv = 0.5
Non-Dimensional Time
Figure 21: Unsteady force and moment coefficients for the
UUV with the origin at 0.5
Cp to 0.0 at t = 0.35 which correlates with boundary layer
ingestion of the wake vorticity. Additional fluctuations
correlate with the ingestion of accumulated vorticity in the
boundary layer. At the center of the UUV (x = 0.5), Cp values
remain at approximately 0.0 until t = 1.125 where a sharp
decrease in pressure is observed. This Cp spike is quite
transient and correlates with a locally strong vortex in the
boundary layer.
Figure 19 shows the point pressure distributions as a
function of time for the Xuuv = 0.5 test case. Nose pressure
shows stagnation Cp values of 1.0 to t = 0.45. This is
followed by a sharp decrease in Cp to 0.0 by t = 0.5 followed
by an immediate increase back to Cp = 0.8. Cp values slowly
decrease with fluctuations in pressure seen. Pressure at the
maximum radius shows steady Cp values of -0.5 up to the
point of wake impact on the UUV at t = 0.5. As can be seen,
at t = 0.55 a sharp increase in Cp to 0.2 is seen followed by a
decrease in Cp back to -0.5 by t = 0.75. Pressure then remains
approximately constant until just after t = 1.0 where additional
fluctuations are seen as the wake becomes ingested in the
UUV boundary layer. At the UUV center, Cp values of 0.0
persist out to t = 0.7 where a decrease to -0.3 is seen at t =
0.75. This correlates with the passage of a vortex. As time is
increased and the vorticity is advected downstream, Cp values
recover back to 0.0 with fluctuations in Cp seen thereafter.
Figures 20 and 21 show the UUV forces and
moments as a function of time for UUV positions of 0.125 and
0.5 respectively. In these cases, the forces are non-
dimensionalized by the product of the freestream dynamic
pressure and wetted surface area; the moments are non-
dimensionalized by the product of the freestream dynamic
pressure, wetted surface area and vehicle length with the
moment taken about the vehicle center (x = 0.5, y = 0.0 and z
= 0.0. For all three cases, the z component of the force is
negligible due to the assumption of symmetry about the y-axis.
As a consequence, the y-moment component is negligible as
well as the rolling moment about the x- axis.
For Xuuv = 0.125, drag coefficient appears to remain
level at 0.025 out to t = 1.3. After that, there is a rise and
fluctuation to approximately 0.04 followed by another
decrease. Normal force (Cfy) fluctuates about the zero value
until just before t = 1.0. Afterward, there are several
fluctuations with a large transient peak in Cfy of 0.3 at t =
1.35. This is followed by another peak of 0.1 at t = 1.5 before
the forces diminish again. The pitching moment (Cmz) shows
small values until t = 0.75 where a decrease in moment to -
0.01 is seen. Afterward, there appears a definite fluctuation in
the UUV moment with peak values on the order of 0.02 but
are very transient in nature. It is interesting to note that even
though the wake impacts the UUV by t = 0.15, force
fluctuations are not seen until t = 0.75 at the earliest.
At the furthest downstream location of 0.5, it appears
that all fluctuations in the UUV forces and moments are
diminished. The only fluctuation appears at t = 0.5 just as the
cone wake impacts the UUV. After the cone vorticity is
ingested in the boundary layer, vehicle normal force and
pitching moment return to effectively zero values and the drag
coefficient remains constant at 0.025.
Figure 22 shows the unsteady forces and moments on
the UUV for the cases for Xuuv = 0.2 and YUUV = -0.1. Drag
force remains fairly constant at 0.02 and side force remains
negligible. Notice that the normal force becomes negative by t
11
UNSTEADY UUV FORCES
0.3 T Xuuv = 0~25, Yuuv = -0.10
0.2
0.1
n
-0.1
-0.2
-0.3
.~
-
~=—- =
Non-Dimensional Time
UNSTEADY UUV MOMENTS
o.o4 XUUV = 0-25- Yuuv = -0.10
0.03
0.02
0.01
n
-
-0.01
-0.02
-0.03
-0.04
—Cmx
. Amy
—Cmz
_ _~
0.25 ~Y5 0.75
Non-Dimensional Time
Figure 22: Unsteady force and moment coefficients for the
UUV with the origin at (0.25, -0.10, 0.04.
= 0.1 with values remaining fairly constant up to t = 1.2 at Cfz
= 0.01. After that time a decrease in normal force to Cfy = -
0.1 is seen out to the end of the run. Pitching moment shows
the effect of the displaced wake as well. Initially, pitch
moment becomes positive by t = 0.4 with Cmz values on the
order of 0.01. The increase in normal force correlates with an
increase in pitch moment suggesting a nose down pitch
moment with downward forces proximal to the UUV nose.
DISCUSSION:
Unsteady Wake Development:
The velocity vector plots displayed the initial
unsteady flow development aft of the docking cone and the
resultant impact of the wake on the docking UUV. After the
flow travels approximately one cone length, the initial vortex
ring structure aft of the cone is formed. This ring structure
then becomes elongated and appears to form an elliptically
shaped vortex ring by t = 0.2. As long as the UUV is not in
close proximity (e.g. Xuuv = 0.5), the major axis of the vortex
ring appears to reach a maximum size about twice that of the
minor axis. For the UUV in close proximity, the cross-section
of the vortex ring remains circular with fluctuations in the
local diameter of the ring vortex as the flow develops.
The presence of the UUV appeared to alter the
frequency of the force fluctuations on the UUV. For Xuuv =
0.125 and 0.25, high frequency fluctuations in drag and cone
pitching moment were seen. For the far downstream UUV
case, however, a relatively well-defined Strouhal frequency of
0.225 was observed. This suggests that the UUV alters the
Bode Plot
Vertical Position Gain due to Pitch Moment
nn F ~ n 1
1 .nn E-n 1
1 .00 E-O 1
,C
1 .00 E-02
1 OO E-0 2 - \
1 .00 E-03 -
1.00E-04 - . . ~
1 .00 E-03 1 .00 E-02 1 .00 E-O 1 1 .00 E+OO
Frequency (Hz)
Bode Plot
Vertical Position Gain due to Vertical Force
1 .00 E-02 1 .00 E-O 1 1 .00 E+OO
Frequency (Hz)
Figure 23: Bode plots displaying the gains in vertical position
due to pitch moment and vertical force.
vorticity shedding process of the cone as it comes in close
proximity. As the flow continues to progress, asymmetry in
the vortex ring structure results in the shedding of 'chunks' of
vorticity. These vorticity 'chunks' do not appear as well
formed vortex rings, rather they may more appropriately be
described as convoluted hairpin vortices. It is this vorticity
that impacts the UUV and alters the local pressure distribution.
For the far downstream cases, there appears to be little
influence of the wake on the integrated forces and moments
although significant variations in surface pressure were
observed. As the UUV gets closer to the docking cone,
however, significant unsteady pressure and loads are
produced. There is no longer a stagnation point at the nose of
the UUV and the low pressure at maximum UUV radius
normally seen is diminished. The impacting vorticity leaves a
definite signature in the pressure distribution and generates
significant normal force and pitching moment.
Full-scale UUV Recovery:
A full-scale UUV is on the order of 6.096 m (240")
long with 53.34 cm (21") diameter. The cone diameter is of
the order 55.88 cm (22"~. It is envisioned that the UUV will
dock with the cone for a submarine forward velocity of 2.572
m/see (5 knots). Using these numbers, a force coefficient of
0.01 corresponds to a force of 280 N (63 lbs.) and a moment
coefficient of 0.01 corresponds to 1625 N-m (1200 ft-lbs.~. A
non-dimensional time unit of 1.0 corresponds to a full-scale
time of 2.11 seconds. Fluctuations with a non-dimensional
period of 1.0 therefore correspond to a full-scale frequency of
0.474 Hz.
Figure 23 shows Bode plots for the 21 " UUV
displaying the gain in vertical position for inputs of pitching
moment and vertical force. These plots effectively
demonstrate the response of the vehicle to unsteady loads. It
12
css mes 6~t c smusoidal forcmg f mction is imput with given
smplit de Actucl dispkcement (smusoidal) c m be ~ehted to
6he gam by 6he equation:
y(meters) = G. F
25)
wh re y is 6he di plac ment, G is the gain md F is 6he forcmg
f mction (either force or pitchmg moment) 7hese plots show
some ~evecling characteristics First of cll, the~e is c pesk
fiequency et 0 02 H 7bis sugg sts 6~t 6he vehicle will be
unst~ole to fmcing f mctions with c period of 50 seconds As
fiequency is mmecsed, th gains decrecse sigmflcmtly so th~t
by I Hz, the gam is only on th order of 10 for the pitchmg
moment At even higher fiequencies, 6he gain d ops off
sharply so thst, in effect, 6he vehicle doesn't even respond to
6he fmces
As could be seen from Figmes 9-11, 6he high
fiequency vortex sheddmg res 5ted m force fluctuation periods
on the order of 0 I seconds for remlti gfiequencies of 10 H
7he corresponding gam is on th order of 10 res 5ting m
negligible chmge m WV hajectory Stro~l sheddmg
fiequencies on 6he order of 0 75 H w ~e observed fiom 6he
moment fluctuations m Figme I 1 7he conespondmg gain m
pitchmg moment is 1 356xl0 md is 1 334xl0 3 in vertical
force Assuming c forcing fi:mtion 6~t produces 1625 N-m
(1200 f-lbs) of pitchmg moment md 280 N (63 ks) of
wxticcl force, the re mlting chmg m h cjectory is on th order
of 9 I mm (0 36 inches) 7his chmge should be corrected
during c dockmg procedure md is ecsily wi6hin 6he wmdow of
en or
Alhough th vortex sheddmg off of 6he cone will
likely have mmimcl impact on th tmjectory of the WV,
motion of 6he submarme or 6he p~esence of oce m cmrents will
cmse the cone wake to vary es w 11 A simmsoidal wave
pattem could be e tablish d on 6he wake of 6he cone For
modemte ocem cmrents, md tempmal variction with 10
second period s perimposed on 6he cone wske would not be
umecsoruible 7his wave pattern co 5d ~esult m pitch md
moment fluctuatiom on the order of 0 I H 7he bode plots
shows c gam in pitch moment of 3 8xl 03 md gain m vertical
force of 858xl02 For c 1625 Nm (1200 f-lbs) pitch
moment smplitude md 280 N (63 lb) verticcl force smplit de,
6his would res 5t in chmges m vehicle hajecto y on 6he order
of 1 37 meters (4 5 feet) 7his cltered trajecto y could be
sigmfficmt m terms of dockmg th UW Futme flow
cclcoktiom sho 5d focus on th se types of phenomemr
CONCLUSIONS:
A novel Lag mgim vorticity method hcs been
p~esented to compute 6he mstecdy hyd odyr~smics cssocisted
with r~l mmarmed mdersec vehicles Since 6he flow is
mcompressible, 6he pressure term is not ~equi~ed for sohtion
leavmg c velocityworticity form 5ction ~ fact, since 6he
velocity is c di~ect integ al qDmtity of th vo ticity, this
medhod demonshates th~t 6he m tecdy flow cm be descobed
by the vorticity clone 7he Lag mgim m~tme of 6he
calcoktion ~equi~es only th~t the diffusion temm be solved
e plicitly The cdvection temm is cutomaticclly mchded since
6he points me moved wi6h the loccl flow The dfff sion
velocity concept is clso used to move 6he points mto regions of
13
:D:ro vorticity This avoids 6he dffhc 5ty of e tablishmg empty
points to dfff se 6he vo ticity onto The diffusion eqDrtion
was modffied accordingly md solved st ecch time step
fffects of g id resohtion md validation of 6he velocity
cclcohtion md dfff sion clgorithm w ~e conducted by
comparmg comp htiork~l results for crurlyticcl sohtions for
Hill's sphericcl vort:x md c col mnar vortex
The mstecdy tmbulent flow pc t c WV et Re =
1,000,000 md the unstesdy flow development m th wske of
th cone w re bodh inve tigated to d momtmte 6he
effectiveness of 6he curre t medhod Mem tmbulent bo mdary
Izyer velocity proflles cg eed quite w 11 with empiriccl res 5ts
fiom Spcldmg md Cole Turbule t fluctuations cg ed
s mrisingdy w 11 for o'but not es w 11 for v' md w' This was
expected due to lak of surface ~esol tioa The fact 6~t
turbulent qDmtities are comp ted et cll demon trstes the
pote tial of 6his method to compute turbulence di~ectly
Unstecdy flow past c cone show d 6he vortex ri g
shuctme produced Initiclly, th vortex ring was symmehic
but developed csymmehies as 6he flow developed Th
Isminar flow case show d c much mme coherent vortex
compared wi6h the turb 5ent vortex This is exactly as
i tended md was the recson for impleme ti g c turbulence
model Consequently, 6he tmbulent flow res 5ts for 6he wake
velocity proflles w re in much better cg ement wi6h
experimental test cases
Simoktion of UW ~ecove y hyd ody smics
di plcy d some intere tmg flow fleld phenomena Th
m tecdy flow development in 6he wske of 6he cone show d
th vort:x ring tructme produced Imticlly, 6he vo tex ring
was mmetric but developed c mmetries es th flow
developed As 6he csymmet y was mmifested, sheddmg of
vorticity was observed The shed vorticity did not cppear as
w 11-formed vortex rmg but es convoluted hai pin vort:x
shuctmes Proximity of th WW sffected 6he way m which
th vorticity was shed For cases whe~e 6he UW was close to
th WV, th separcted vorticity region betw en th WV md
th cone mcintained cohe~ent vortex ring shuctmes whose
csymmetry varied with time As the WV was plaed
dow tresm, the vo tex rmg shuctme becime elongated md
ellipticclly shaped Here, defimte 'ch mks' of vorticity w re
shed which impacted 6he WV
As th wske of the cone impacted th WV,
sigmfficmt trmsients in loccl pressme were observed The~e
was c reduction m stagnation p~essme et 6he nose of 6he WV
md mmecse m p~essme et the point of maxim m ~adius of 6he
WV Pressure fluctuations w re observed es the vorticity
domirurted wske impacted the nose of the WV Cclculations
et midbody show d fluctDrtiom m pressure es w 11 Ew~n
though 6here was k ge pressure fluctuations, i teg cted force
computations show d little fluctuations during mitial wake
impact Only es the flow becime developed w re sigmflcmt
fmce fluctuations seen Also, the mcgmit de of the force
fluctuations rapidly diminished, es the WV was pkced
dow tresm
Verticcl pkcement of th UW produced cltered
wske structure md mstecdy loadmg on 6he WW Placing the
WV below 6he cone clte~ed 6he vort :x wake mte~action wi6h
th UW md clso produced negative normcl forces md nose
dow pitch moment For this case, sim~soidal fluct stiom in
pitch moment on 6he order of 1200 ft-k s md verticcl forces of
60 ks w re observed This was likely dp to 6he overcil
moment m defcit m 6he wake remiti g inhigher pressure on
6he WV ppper sp fae
A crurlysis was perfommed to determip the lespit mt
chmges WWtr je toryfor full- leUW hw sshown
6~t fne flpctpati ns dp t vortex sheddmg were suffimently
high to h~ve mmimcl impat on th WV hajecto y The
plesep e of th wake however md modffications of the wake
dp to submarip motion or ocea cp re ts may cdversely
impat 6he WV tmjectory A wave-like wake shuptme wi6h
10-second period was show to citer th hajecto y of 6he WV
by 1 37 meters (4 5 feet) Fptme p~lcpktions shopid focps on
6he citerction of 6he docking cop wake dp to mbmarme
motion md its lespitmg impat on the UW
Now that c tmbplep e model hcs been mtroduped
mto th code, it is possible to hect c wide ra~ of e giperm
problems of pa~l mtelest Cm ent work mcl des examipatio
of m tecdy flows dp to bow 6 p ters for low speed control,
mbmarip maneuvermg problems md UW recove y
problems O going work will be plesented in f tpre pcpers
ACKNOWLEDGMENTS:
This work was ponsored by the United Kmgdom
D fence Evalpation md Research Agep y mder FMS case
mmbers UK-P-GVK md UK-P-GUW, Rchard Breward,
DEPA Prog cm Marug r, Simon Corfield, Techmical L cder
md M k H big Pr ject E g p er md by the O fice of Ncval
Rese~ch mder o tra n mb 9 WX20012 Dr Spiro
L koudis Prog cm Maruger
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DISCUSSION
U. Bu garel i
Instituto Naziona e per Studi ed
Espenenze di Architettu a Nava e, Ita y
In the equation of vor icity, how do you
compute the coefficient u at t=0 for
sta ting the computation?
AUTHOR'S REPLY
At t = 0, au impu sive start is assumed.
In this case, the freestream velocity is
accelerated to 1.0 in one hme step. The
vor icity is dist ibuted on au iritia set of
points in the boundary layer with the
point dist ibubon nomsal to the surface.
The thickness of this initia bounds y
layer is assumed to be 10% of the f Ha
thickness of a flat plate bounds y layer.
(: n the f rst t me step, the no s ip, no-
flux bounds y conditions are satisfied
with the su face vortex sheet. The
surface v or icity is then set so that
cDdV=7dA where to is the surface
vor icity, 7 is the surface vortex sheet
strength, dV is the volume of the
elements connected to the surface node
and dA is the area of the panels
connected to the su face node. The
vor icity is then assumed to
exponenha Iy decay through the i iha
layer with a maximum v or icity at the
surface. Since the dist ibution of
v or icity will affect the velocity at the
surface, au iterative method was used to
re-ca cu ate the surface vortex sheet,
surface v or icity and layer v or icity. By
the end of this iterative process, the
v or icity and velocity of each point are
known.
