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OCR for page 848
Reynolds-Averaged Modeling of
High-Froude-Number Free-Surface Jets
D Walker (ERIM International, Inc. USA)
Abstract
For turbulent free surface flow, the main effects to be
considered are modification of the near surface tub
bulent structure -- the free surface boundary when
the local Froude number is low, and the effect of the
fluctuating interfacial forces when the local F oudr
number is high This study focuses on the develop
ment of n consistent approach to calculating turbo
lent free surface flow at arbitrary F oude number
A frnmew rk, centered on the PeJnoids aver bed
N. A''' Stokes equations and the code CFDSHIP
IOWA, is described where the standard k e tub
bulence model has been augmented with an nigh
brnic P y old stress model to capture near surtacr
stress nnisotropJ in addition, turbulence generated
w wes have been described using n wa:v~nction model
with n source term for the energy transfer from the
sub su face turbulence to unsteady free surface dim
turbancm The effect of the su face fluctuations on
the flow are modeled -- an npprcximate shear stems
boundary condition at the free surface Pmults are
presented for predictions of the mean fl w and turbo
lence quantities, as well as fre~surtace fluctuations
with comparison to experimental data
1- Introduction
Engineering predictions for turbulent flow near n free
surface are of inter t in applications ranging from
ship hJdrodJnamim to manufacturing processes The
nonlinear nature of free su face boundary conditions
along with the nonlinearity of the underlying Na:vier
Stokes equations, make this problem anal tically in
ton table, an d computationally challenging as w 11 A
major impediment to the accurate modeling of tur
bulent free surface flow, i e fl ws with n gas liquid
inte face, has been the lack of an appropriate form for
the P Jnoids weraged equations Unsteady surface
elevation fluctuations cause the boundary to be in
di tinct, or'fuzzJ, in the conte t of the weraged set
of equations PecentiJ this issue was nddrersed in
Hong & Walker (2000) where, starting from the gen
ernl, variable property N. A''' Stokes equations, the
authors derive n set of P Jnold~ weraged equations
which apply nCrOSS the entire gas liquid domain The
resulting governing equations, most conveniently em
pressed in term of density w ighted weragm, are of
the level set form (see e g Chant et cl 1996), where
the interfacial boundary conditions are eflectiveiJ em
bedded in the field equations
In most approaches where the P ynold aver bed
Nwier Stokes equations are applied to problems of
ship hydrodynamics, the form of the equations used
assume constant fluid properties (see e g TnEnrn &
Stern et cl 1996) They also assume that the nix
water interface is idealized to water fl wing bel w n
void with zero pressure, density and vi cosity The
mean free surface is deformable, but is assumed to
be steady, and therefore n well dean d, sharp inter
face, at which appropriate boundary conditions are
applied While assuming that there is n void above
the water is n reasonable approximation for this prob
lem, ignoring the free surface flu tuntions can lend to
serious errors in the predicted behavior of the fl w
For n high F oude number free surface jet, Hong &
Walker (2000) show d that surtac~elevation fluctu
ntions help drive secondary fl ws which can change
both the qualitative and quantitative features of the
fl w Hence, inclusion of free su face effects is ret
quired to obtain accurate predictions
The obje tive of this study was to develop an up
preach to predi ting turbulent free - :- ace flows in
the conte t of the conventional P y old averaged
N. A' '' Stokes equntions in the approach developed,
the exact equntions derived by Hong & Walker (2000)
are reduced to an npprcximate form for small surface
fluctuations This all ws high Froud~number flow
with un tends surface flu tuntions to be treated with
n conventional PANS code such as CFDSHIP IOWA
(Stern et cl 1996, Sahara & Stern 1996), rather than
using a level set approach Surface fluctuations are
—r -e e t via coupled solution of a w we action spew
trum model and their sflsct on the mean fl w are
OCR for page 849
captured using an zpprcximate tress condition at
the free surface The effect of the free surface on
the sub surface turbulence at low F oude number is
captured using z near surface turbulence anizotropy
model ..
The zb he developments are first described, fol
I wed -- an application of the resulting model to
turbulent free su face jet flows, and comparison of
the results to experimental data First z I w Froud~
number jet is calculated, where there is little wwe
generation and near surface znisotropJ dominates the
bed wtor of the fl w; here the approach perform
quite w 11 when compared to I w Froude number
jet data Next z one w J coupled prediction of z
high Froud~number jet is presented, where the sun
face flu tuations are predi ted using the w we action
model, but the surface fluctuations are not used in
the boundary conditions for the sub su face flow (i e
the subsurface fl w is treated as z I w Froud~number
fl w) The predicted surface fluctuation distribution
is in good agreement with the observations Finally, a
two w J coupled calculation is performed, where the
surface fluctuations are used in the boundary condi
tions on the subsurface fl w This agrees Ims well
with the experiment, due mainly to z lack of an ef
fective, rational method for switching between the
low F oude number and high F oude number treat
ment of the free su face
2 Governing Equations
2.1 Reynolds Averaged Navier
Stokes Equations
In this section, the Reynolds weraged N. A--- Stoker
equations for application to the problem of turbulent
free surface flow are presented They are then ape
cialized to the case of small surface fluctuations, and
an zpproaxi mate set of free surface bound ary condi
tions are derived
The Exact Averaged Equations
The beEwtor of the mean fl w is described by the
P y old veraged form of the N. A' '' Stokes equal
tions The appropriate exact form of these equations
for application to general two fluid problems was pro
seated -- Hong & Walker (2000) For these purposes
the time werage of z quantity is defined as
tions, but much shorter than the time scale for the
v riation of the mean fl w (zfler Hinge 1975 pp 6
and 20)
The time weraged momentum equation for turbo
lent two fluid flow is
FLU\ ^ ~u\1 SP Sp \U)
P L t +U>~] = 5~\ 5~>
~ SU\ SU
+ S~) ~ S~) + ha\
+ S~) ~ S~) + an\ +—(ESPY)
+ [(Pt po)92 + ( 7)] 35Gf(G) (2)
This equ Lion is written in term of density w ighted
werages C: = p:/p; (USE) is viscous stress related
term (see Hong & Walker 2000) Here, the mean free
surface position is given -- G(x\,t) = 0 and f(G) is
the probability density fin tion (p d f) of the free flue
tuating free su face position This equation, together
with the continuity equation
+ (p A\) O.
St An\
and the evolution equation for the mean free surface
( t 5~\) (Po Pt) 5~\
constitute z complete set of g Kerning equations for
the flow
If it is assumed the two fluid problem is water flow
ing bel w z void with zero density, Viscosity and =: ~
sure (i e pa = q~ = P = 0), then thme equations
reduce to
_ I, + U SU\~
(3)
:(~\,t) = T; Q(~\,t)dt (1) in Edition,
It is assumed that the averaging time T will be much
longer than the time scale for the turbulent fluctuate
P SP \u)
go\ S~)
~ L/SU\ SU>\1
+ 5~>LL~+~\J]
~ Yule S~
+ S~) ~ S~) + An\
+ pope y f(G), (5)
surfs cs reduces to
( St + U,~5 ) f (G) p 5~ = 0 (6)
OCR for page 850
The resulting equi tions (3), (5) i nd (6), hi e sev~
eri I desiri ble chi ri cteristics In the limit of vanishing
surfi c~elevi tion fluctui tions, the mei n fluid prope~
ties pi; nd T become consti nt, the densitJ w ighteG
iweri ges reduce to simple time iweri g i, i nd the
terms on the third line of (5) vanish The p d f f(G)
becomes i delti function indiciting the position of
the free surfi ce Hence, the equi tions reduce to thr
conventioni I F NS equi tions, but of i level set form
simili r to thi t of Chi ng et cl (1996) In the limit of
limini r fl w, theJ reduce to the exi et level set fo~
mull tion of Chi ng et cl
It should be noted thit the set of equitions (3),
(5) i nd (6) i re mithemitici IIJ exi et, in the sense
thi t no modeling hi i been introduced; h wever, theJ
do not repr lent i closed sJstem of equi tions As is
the cl ie with the usui I PeJnoids iweri ging process,
i dditioni I unknowns i re introduced due to the loss of
informition For the present set of equitions, th ie
i dditioni I unknowns i re the PeJnoids stresses ~ ;,
the fluctui ting stri in ri te terms ^, i nd the p d f of
the su fi ce elevi tion f(G) For the unkn wns beJond
the PeJnoids stresses, i pproprti te turbulence models
must be developed to i 11 w the set of equi tions to be
closed
Approximate Equationz for Small Surface
Fluctuationz
One would like to hi e i n i pprcximi te form of th x
equi tions which cl n be solved using i conventioni.
PANS solver of the tJpe empioJed in ship hJdrodJ
ni mics To obti in this, the derivitiv i conti ining
the vi rti ble mei n fluid prope ties in the momentum
equi tions i re expi nded, i nd then the terms involv
ing gri dients of the fluid properties i re collected to
gether This Jioids i momentum equi tion of the form
:SU\ ~ SU\1 SP _ 3u;
PL e+7i5~i]= 5~\ P5~;
+ T ~
+ {P i~]
/57\+5U\+~\q
~ S~i S~\ J ~ S~i
~ iNl
J]
( i \ )
(~i S~\) } S~\ ) ( '
If G(x\,{) is deflned, for i single vilued free su~
fi ce, i i the normi lized disti nce from the mei n r-^-
surfi ce y(~,y,{) i i
where ~ is the r m s surfi ce elevi tion, then in the
limit of :' ~ 0, f(G) becom i i delti function i nd
the terms multiplied bJ f(G) decouple from the fleld
equi tions These terms then become boundi rJ con
ditions, i pplied i t the loci tion indici ted bJ G = 0
Under these circumsti nc i, the the fluid properties
in the equi tions become consti nt
For smi 11 but nonzero 4, it will be i isumed thi t
the termi multiplied bJ f(G) will still decouple in this
fi ihion, i nd for the momentum equi tion, the result
P ( ~e i 5~; )
5; 52 1\ ~\~;
5~\ + ~ 5~2 P 5~ ,
where the overbi indicites i simple time i eri ge
(equi I to the densitJ weighted iweri ge when the den
sitJ is consti nt) The boundi rJ conditions for the
momentum equi tion i rise from the termi in (~N i
volving f(G),
{ ~ P i ~ (~i S~\)
(~i 5~\) } 5~\f( ) (10)
Evi lui ting (10) for ~ ~ 0 requir i integri ting the
expression over the nei r surfi ce region deflned bJ
f(G) 7; 0 This integri tion Jioids the i pproximi te
boundi rJ conditions for the three components of the
momentum equi tion, to be i pplied i t z = ~ For the
si ke of simplicitJ, it will be i isumed thi t the mei n
free surfi ce is essentti IIJ flit, y(x,y,{) = 0, but thit
the su fi ce fluctui tion level vari i in spi ce (y' 7; 0)
Under these conditions, integri ting (10) Jioids
Sz 2q ( + P )
~1 I pg 542
Sz 2 ~ S~
~7 I pg 542
Sz 2 ~ Sy
Here, the boundi rJ condition for the z momentum
equition (11) is unchinged from thit for the con
ventioni I PANS equi tions H wever, for the x i nd
y momentum equi tions, instei d of zero shei r stress,
there is n w i n i ppi rent shei r str is i cting on the
mei n free surfi ce The ml gnitude of the i pplied
shei r str is is proportioni I to the loci I gri dient of
the surfi ce elevi tion varti nce d/2 This result is con
sistent with the i ni I tici I results of Hong & Wi Iker
(8) (2000) for high F oude number jet sprei ding
(9)
<11)
412)
413)
OCR for page 851
Near Surface Reynolds Srresz Modeling equal I
TLO .3
The approximate governing equations are solved up
ing the code CFDSHIP I OWA it incorporates a Stan
Lard k e turbulence model with the P Jnoids
stresses specified in terms of an eddy viscosity as
_ (SU\ SU
2U\29 u ~ = V2 <~ + S~
~ ) 3k5\>, (14)
where ~2 = Crk2/s, k is the turbulence kinetic en
ergo, e is the dissipation rate for ., and Cry = 0 08
Since the products of ~2 and the normal strains are
typically small compared to ., the k e model will
result in essentially isotropic turbulence (near equip
PeJnoids normal stresses)
For zero, or low, F oude number where the free
surface is msentiallJ flat (i e where there are no un
steady turbulenc~generated wwes) the turbluence
becomes anisotropic The transfer f energy from
the ve tical Angelo itJ flu tuations (w2) to the other
two components (u2 and v2)is captured -- the near
surface redistribution model of Daly & Harl w (1970)
and Shir (1973) This model, originally intended for
implementation as a near surface (a tuallJ near wall)
correction to the pressure strain correlation (by) SN
model for the P y old stress transport equations is ~ x
given by t
T\Z = k ( 235.i 3537 U3U75\3) f (2)
, (15)
where f (2) = cell(? —:), and id = k3/2/z is the
local turbulent length scale introducing this into
the algebraic stress model formalism of Podi (1984'
yields the toll wing expression for the 'corrected' w
P y old stems in the near su face region:
2 w
w =
Here, w on the right hand side of the expression E
the value obtained while neglecting the near su face
effects Far from the free surface f (2 ) = 0 and, hence
SO w2~orr = 0,
are given by
(16)
f (2) ~ co and 9
and v2 stresses a-
2~2~orr = it + 2 + 1/f (2)
V2 0,, = V2 + W (18)
is seen that as the free surface is approached,
are each increased bv and amount
to w /2, conserving turbulence kinetic energy
-..ear stresses are given by
w .0.. = w
Here, z~v is un fact d, but the us and vw shear
stresses are reduced to zero at the free surface, again
consistent with the boundary conditions
(19)
, = 2 (20)
VW
1 + Of (2)/2
(21)
2.2 Surface Fluctuation Model
Wav - Action Spectrum Model
The unsteady free surface fluctuations are modeld up
ing a w we action spectrum model like that described
-- Komen Be cl (1994) This type of model can
predict the evolution of the surface w e directional
spectrum, including wave generation, and the inter
action of the wwes witht the nea~surface velocity
field The wwe a tion spectrum is governed by .
advection equation with source term:
+ ~ (CON) + y (CyN)
+~ (CON) + 59 (CON) = ( ' ), (22)
where x and y are spatial position variables on the
free surface, and ~ and 9 are wwenumber and di
re tion variables for the action spectrum (a = 27r/N,
where ~ is the wavelength) The vectors x = (x,y)
and z = (a, 9) will be used to represent spatial and
spectral position, respectiveiJ N(x, e, t) is the action
spectral density defined as
N(x, e, t) = E(x, 8, t)/o
(23)
the energy spectral density The frequency
to the wwenumber ~ by the dispersion
0(~) = >/~ (24)
The x and y direction component of the wave
propagation velocities are given by
C0 = U + Cg con 9, Cy = V + Cg sin 9, (25)
where U(x,t) and V(x,t) are the x and y direction
mean velocities at the free su face, and the wwe
group vet ocitJ is
3( ) 2.~
(26)
OCR for page 852
The other two velocities Cal and C3 are energy grated to obtain ntwo dimensional spectrum by inter
prop fiction velocities in the spectral domain caused grating over one Cart inn coordinate direction This
by spatial variations in the surface velocity field yields
They are given by
Cm = ~ [COS
S~ + sin2 9
4SV SU 1 AN tion of ~ only it is independent of dire tion 9 due
+ sin 9 cos 9 ~ ~ + ~ ~ :~ (27) to the isotropic nature of the original pressure spew
trum Combining this with the dynamic free surface
boundary condition (in inviscid form) yield
So) = ~ i k (4 + ~2Ft22/We2)2 ( i2) ~ (a4)
which is applied for ~ > 27/e2; 5(~) = 0 for ~ <
27 /e2 Here, Fr2 = (k/gi2)~/2 and WE = 7/ke2 are
the local turblent Froude and Weber numbers based
on k and the turbulent length scale If for the given
free surface location, and the constant was deters
mined by matching the pe k of the predicted surface
elevation variance to experiment
and
C3 = [cos2 9
Em) ~ ~ '0/3
for the two dimensional sDe trum which is n funk
(30)
sin2 9 TV
+ i 9 9 ( V :7)] N (28)
The source term S(~, 9) on the right hand side of (22)
is discussed in the next section
This set of equations can be solved for the action
spectrum for spatial region 7? subject to appropriate
initial conditions N(x, z,t = 0) on 7? and boundary
conditions on 57? For potions of the wave spew
trum with prop cation velocities which carry energy
into 7Z, the 'incident' wave spectrum N(x,z,t) on
the boundary 57? is set to zero; for the portions of
the wave spectrum for which the prop cation velour
ities carry the energy outward from 7Z, an effective
'outflow' condition is used (continuity of the action
flux at 57~) in the spectral domain, similar condi
tions, zero for ink w and continuity of flux for outflow
are applied at the upper and low r w enumber (a'
boundaries The boundary conditions in 9 are that
the spectrum is periodic The velocity field U(x,t),
V(x,t) is an input to the wave spectrum model and
is obtained from the PANS solution
Turbulent Source Term
The unsteady pressure fluctuations in the subsur
face flow cause the free su face to move and, as
result, generate waves in this procms energy is
tran ferred from the subsu face turbulence to the
wave field The two dimensional w e generation
spectrum S(~, 9), which appears on the right hand
side of the wave n tion balance equation represent
this transfer process This term is derived from
the three dimensional energy ape trum for isotropic
turbulence, combined with the dynamic free su face
boundary condition The isotropic pressure spec
trum, for the inertial range is given by
E~(k) ~ k 7/3 (29)
where k is the three dimensional w e vector, and
k = k This three dimensional spectrum is inter
3 Numerical Implementation
The computations for this study were carried out up
ing n modified version of CFDSHIP IOWA Changes
were made to incorporate the nen~surtace stress
nnisotropJ, coupled solution of the w we n tion spec
trum model, and the npprcximate dynamic free
surface boundary condition which relates the Spar
ent stress on the su face to gradient in the surface
fluctuation variance
Modeling of the near surface stress nnisotropJ was
accompli Led in the context of the basic k e model
which is included in CFDSHIP IOWA The ability to
treat non isotropic turbulence was added by includ
ing terms in the momentum equations which repro
sent the deviations of the P y old stress gradients
from those predi ted -- the standard k e model
The form of the PANS equations then look like
(SU\ _ SU\N ~ (_ 2 ~
phi— + U,y J = ~ ~P+3PkJ
+ P yea [(a + 2) ( S~> S~\ )]
p ~ I \7 >,
where Zx7 \7 ~ = 7 \7 ~ ,0,, 7 \29 u
tional source term in the momentum
implemented using second order finite
prcximations
(32)
OCR for page 853
The solution for the wave z tion spectrum was 4 Results
obtained using z new subroutine in the CFDSHIP
IOWA code The new subroutine implements z dim
crete approximation of the the zdvection equation
governing the wave action spectrum using z fully
implicit scheme based on first order upwind deferent
ing for the zdvective terms (A third order upwinc
scheme is currently being implemented ) The wave
spectrum solution requires the surface valum of the
mean velocities, as well as k and e and is zdvancec
in time in z coupled fashion with the PANS solution
for the sub surface fl w
The w we spectrum is integrated to determine the
surtac~elevation variance r/2 These data are ret
turned to the PANS solver for use in specifying the
apparent tress acting on the free surface
Czlculations were carried out for z re tangular vol
ume 48d x 20d x 26d in the x, y and z directions rig
spectivelJ, where d is the jet exit diameter The free
surface was located at z = 0, y = 0 corresponded to
the vertical symmetry plane of the jet, and the jet
exit was located at z = 2d and y = 0 on the plane
x = 0 The CFDSHIP IOWA code was implemented
on z non uniform Cz tesian grid with 33 x 32 x 47
grid points in the x, y and z directions respectively
sh wn in figure I Zero velocity w prescribed on the
x = 0 boundary, except in the region of the jet ori
fice (a circle of diameter d centered at x = 0, y = 0
and z = 2d) The flow w assumed symmetric
at :ut the plane y = 0 Outfi w boundary conditions
were used for the plane z = 48d The free surtacr
was treated as z fiat. shear free boundary (7 = 0)
The remaining boundaries, the bottom at z = 26d
and outboard boundary at y = 20d are zctuallJ no
slip boundaries, but are treated as symmetry planes
to minimize grid density requirements (the velocities
are near zero at the boundaries, an way)
The spatial grid for the wave action conservation
equation corresponded to the PANS grid on the free
surface and m--i ted of 33x32 grid point Cheapen
tral grid consisted of 32 directions equally spaced over
O < 9 < Or and 40 logarithmically spaced wwenum
hers The range of w wenumbers was covered physical
w welengths from d/30 to 30d The w we action equal
tion was advanced in time using local time stepping,
where the tim~step was based on the local values
of the spatial and spectral zdvection velocities The
la he range of w enumbers required results in z large
range of w we group velocities, an d the I wer vet ocitJ
wwes converge very slowly without local time step
ping
In the computations, and in the results to be pre
seated, all quantit is are normalized twiththe jet exit
diameter d, and the jet exit velocity He
In this section, result will be presented for three
casm The first case will be z I w Froud~number
jet, Fr = 0 and Us = 12 700 For this case, the
main effect of the free su face is the anisotropJ of
the near su face PeJnoids stresses These computat
tional result will be compared to experimental data
for Fr = I O an d file = 12 700 (Hong & Walker 2000),
where there is little wwe generation The second
case examined will be for Fr = Uc/(gd)~/2 = 8 and
Its = U d/~ = 12 700, where the w e action equal
tion is solved for the unsteady turbulence generated
wwes The surtacefiu tuation model in this case
is 'one way' coupled, in the sense that the surfacer
fluctuation variance is not 'fed back' into the free
surface boundary conditions This sh ws that the
w we z tion model produces reasonable result, when
compared to su face fiuctutation measurements The
final case is z two way coupled high Froud~number
prediction, with Fr = 8 and file = 12 700 Here the
near surface correction which produces the turbulent
stress znisotropJ is disabled, but the gradient of
the su face fluctuation variance is used for the free
surface boundary conditions These two way coupled
results are compared to experimental data for both
the velocity field and the free surface
In the sections that toll w, the discussion of ret
suits will center on comparisons of transverse planes
of experimental and computational results Compaq
isons of computed and experimental results for sup
face fluctuations in on the frss surface plans will also
be presented For rsfsrsncs and to orient the reader,
figure 2 show the predicted mean streamviss veloc
ity U/U for the entire computational volume The
vertical symmetry plans plans of the jet is shown, as
are the far d wnstream exit plans for the volume, and
the frs~surtacs plans Ths origin of the jet at the up
stream inlet plans of the volume at x = 0 is clearly
visible, as is the decay in velocity with streamvis dis
tancs and intern tion with the f es surtacs 'Float
ing'zbovs the volume in figure 2 is z plans sh wing
the computed energy distribution :'2/dfor the from
surface fluctuations
4.1 Modeling of a Low-Froude-
Number Jet
The first case to be examined is z I w Froud~numbsr
jet, e: Fr = 0 and file = 12 700 The f es sup
face will he tzEsn to be fiat. and devoid of any un
steady surface flu tuations For this case, the main
f es surface Tact is the requirement that w2 = 0 at
the f es surface, resulting in znisotropy in the near
OCR for page 854
surface P y old stresses The computntions w re
initialized from n volume that had 'core' of unit di
meter and velocity, centered on the jet axis The
computntions were stepped in time until steady state
was reached Five thousand time steps were suit
cient to reduce the residuals -- three to four orders
of magnitude, which was deemed to be a'converg d'
solution As the solution converged, the downstre m
po tion high velocity 'core' decelerated and spread in
space to take on the characteri tics of n jet
The experimental results for free surface jets pro
seated in Hong & Walker (2000) sh w that for Fr =
I O there is very little mean deformation of the f Y.
surface and small surface fluctuations (both are on
the order of n few thousandths of n jet diameter)
Hence, their Fr = I O result are msentinllJ for n
'flat' free surface with no un tends surface fluctuate
tions, and are used here for comparison to the com
putntions for Fr = 0
Figure 3 shows results for the n transverse (y, 2)
plane at x/d = 16 Only the portion of the plan.
corresponding to the region where experimental mean
surement are available is shown in the rest of the
plane the velocity and turbulence kinetic energy Ore
essentinllJ zero For the mean trenmvise velocity
the maximum level observed in the plane for the em
periments (flgure 3c) and computations (flgure 3b'
are comparable, as is the overall shape of the jet cross
section The cross stream, or secondary flow velocity
vectors are shown in figures 3(b) and 3(c) The oven
all pattern of the velocity vectors is simlar; h wever
the magnitude of the velocities at the free su face for
this location is I w by at :ut n factor of two The levy
els of turbulence kinetic energy from the experiment
sh wn in figure 3(e) are comparable to those from the
computntions sh wn in figure 3(1)
A similar set of results for n transverse plane at
x/d = 32 is shown in figure 4 Again, only the For
tion of the plane corresponding to the region where
experimental measurements are available is sh wn
For the mean strenmvise velocity, the maximum level
observed in the plane for the experiments (flgure 4c)
and computations (fig Is 4b) are comparable Again
as at x/d = 16, the overall shape of the jet cross
section for the computations is similar that observed
in the experiments, but the jet appears to h we been
shifted upw rds t ward the free surface slightly At
the free su face, the velocity vectors are comparnblr
in magnitude to those in the experimental data The
levels of turbulence kinetic energy from the experi
ments shown in figure 4(e) are gain comparable to
those from the computntions (flgure 4f)
F om these result, it is clear that the CFDSHIP
re tion to the P y old tressm described above, does
n reasonably good job of predicting the overall evo
lotion of n I w Froud~number turbulent free su face
jet over the range of strenmvise positions examined
4.2 One-Way-Coupled Modeling for
Unsteady Waves
To predict the characteri tics of the turbulence
goner ted surface fluctuations the wave action bal
once model is used To calculate the surface fluctuate
tions, the mean surface velocity field (A and V), as
well at the turbulence kinetic energy k and the dissi
potion e, are required For n fully coupled approach,
the resulting su face elevation variance would then be
used in calculating the apparent stress acting at the
free surface For n one w J coupled approach, the ret
suits from the subsurface fl w are used in calculating
the su face fluctuations, but the su face fluctuations
are not used in calculating the sub su face flow in
this approach, the sub surface fl w is calculated as
suming the fl w is at zero F oude number
In this case, the sub su face fl w behaves eon tlJ as
that described in the previous se tion, and the only
difference is that now, the surface fluctuations are cal
culated, as well This was accomplished by advancing
the w we n tion conservation equation in time in con
junction the P Molds weraged equations The flow
was initialized in n similar fashion as that used above,
with the additional provision that the w we spectrum
was initialized to zero at all locations The calcu
lataions, again converged in 5000 time steps; sines
the w Action equation in the form used hers is lin
ear, it imposes no additional difltcultiss in obtaining
a converged solution
Figure 5 sh w the root mean square (r m s ) sup
face fluctuation level z//d for the one way coupled
computations (figure A) along with those observed
sxpsrimentallJ (fivurs 5 b) for the jet with Fr = 8 0
and Us = 12 700 For both, the peak in z//d is lo
cat d above the jet axis at about x/d = 20 Ths pa k
levels match exactly because the experiment I data
was used to dstsrmins the constant in the turbulent
source term in (31) above (It could he noted, he
ever, that the constant just scalm the magnitude of
:', the spatial distribution is set -- the spectral shape
of the source term and the spatial variations in the
velocity field and k and e ) Ths spatial di tribution
of :'from the computntions appears to be slightly
narr war in the y direction and sligtlJ longer in the
x direction than the experiment I data indicate, but
overall, the grssment is quits good
The r m s surface fluctuation Isvel :' is an integral
~ the predicted w we spectrum While the
OCR for page 855
prediction appear reasonable, complete validation of
the model will require actual measurement of the
directional ape trum of the surface fluctuations This
is an area of n tive investigation
4.3 Two Way-Coupled Modeling of a
High-Froude-Number Jet
The final set of results to be described are for n tw
way coupled computation of n high Froude number
jet where the surface fluctuation variance is used in
specification of the apparent stems acting on the free
surface in this case, the F oude number was set
to Fr = 8 0 and was assumed to be high enough
that there would be no surfac~inducsd nnisotropJ in
the near surface turbulence Hence, the near surface
stress redistribution model w disabled The compu
Rations were carried out in n manner similar to that
described above for the other cases, gain converging
in roughly 5000 time steps The results for y~ will
be presented first toll wed -- comparisons of e Her
imental and computational result for x/d = 16 and
32
The two way coupled computational results for the
r m s surface fluctuation level ~ is shown in figure 6,
gain along with experimental data for Fr = 8 and
I;s = 12 700 The results differ from the on~wnJ coo
pled results discussed above in that the the peak level
of y~ is slightly large and the elevated region e tends
further d wn trenm This is traced to higher turbo
lance level initially in the jet and n higher stre wise
velocity, both due to inaccuracies in the predi ted
sub so face fi w in general though, the gradients in
:~2, which w re used in the specification of the Up
parent stresses acting on the so face, would be equal
to or greater than the correct values
The computational results for transverse planes at
x/d = 16 are shown in figure 7, compared to e Her
imental data for As = 12 700 and Fr = 8 0 (The
experimental data used here is different from that
used above which was for Fr = I O) it is seen that
the computed mean velocity and turbulence kinetic
energy levels (figures 7b an d 71) are higher than those
seen in the experiment (figures 7c and ye) in nddi
tion, at x/d = 16 the jet appears to be slightly farther
way from, and interacting less with the free surface
The most striking difference is in the cross fi w velour
its vectors; in the ccmputaticrls there is at most only
n small outward fi w at the free surface While at
this location the near surface gradient in 42 is non
zero, it is apparently not sufficient, or act over too
small of an area to cause the experimentally oh ervec
outward fi w
Figure 8 shows the computational results for trance
of verse planes at x/d = 32, compared to experimental
data for As = 12 700 and Fr = 8 0 For this trenm
wise location the computed mean strenmvise velocity
is slightly elevated (figure 8b), but the turbulence ki
netic energy level (figure 8f) is at jut the s me those
seen in the experiments (figurm 8c and Be) in ad
dition, the localized sore ding of the strenmvise vet
iocitJ distribution at the free surface seen in the em
perimental data in figures 8(c) is less apparent in the
computations shown in figures 8(b) Again the may
jor difference is in the cross flow velocity vectors; in
the computations there is gain only n small outward
fi w at the free surface
The results in this section sh w that in n two way
coupled approach, the magnitude of the r m s sup
face fluctuations, and the location of the peak are
reasonably well predicted the trenmvise e tent of
the peak is over estimated, with the elevated region
e tending too far downstream Even so, the spread
ing of the jet near the free surface observed in the
experimental data is not captured nccurntelJ This
is mo t likely related to the high F oude number nag
tore of the modeling being used The underlying as
gumption of the modeling is that there is no near
surface tress nnisotropJ, and the free so face eke ts
are confined completely to the apparent stress ret
suiting from the gradient in j2 This may be an
over simplification of the problems, since the results
of Hong & Walker (2000) sh w that, while the levels
of nnisotropJ are smaller in high F oude number jets,
anisotropJ become more impo tant with increasing
strsamviss di tancs, and can still feet the beh A' :r
of the fi w An sfisctivs, rational method for blending
the high and low F ouds number approaches, turn
ing on the anisotropJ e: appropriate, has yet to
be developed
5 Summary and Conclusions
The objective of this study w to develop an ap
Broach to prsdi ting turbulent free surface flows in
the coots t of the conventional PsJnolds aver Red
N. A' '' Stokes equations in the approach develop d,
the exact equations derived by Hong & Walker (2000)
are reduced to an approximate form for small surface
fluctuations This all ws high Froud~numbsr flow
with on tsadJ surface flu tuations to be treated with
a conventional PANS cods such as CFDSHIP IOWA
(Tahara & St on 1996), rather than using a level set
approach Surface fluctuations are modeled via coo
pled solution of a wa:v~action spectrum model their
Tact on the mean fi w are captured using an approve
imats tress condition at the f es surface The e' ts
OCR for page 856
of the free surtace on the sub surtace turbulence nt
I w F oude number nre captured using n nenr su face
nnisotropJ model
The nppronches developed were npplied to turbu
lent fre~surtace jet fl ws, nnd comparisons of the r~
sults to experimental datn were made First n 1~
F oude number jet was calculated, where there was
little wwe generntion nnd nenrsurtace nnisotropJ
dominates the beh wtor of the flow; here the np
pronch performed quite well when compared to low~
F oude number jet datn A on~wnJ coupled predi~
tion of n high F oude number jet was then presented,
where the su face fluctuntions were predi ted using
the wwe n tion model, but the surtace fluctuntions
were not used in the boundarJ conditions for the sub
surtace fl w (i e the subsu face fl w is trented as [p]
n I w Froude number fl w) The predicted surtac~
fluctuntion distribution was in good ngreement with
the observations FinallJ, n two WnJ coupled calcu
Intion w presented, where the su face fluctuntions
were used in the boundarJ conditions on the sub
surtace flow This ngrees less well with the experi
ments; the magnitude of the outward vefocitJ nt the
free su face was substantinllJ under estimated This
is believed to be due mninlJ to n Inck of nn efle~
tive method for switching between the I w Froud~
number nnd high F oude number trentment of the
free surtace
Acknowledgment
This is work was supported bJ the Ofl ce of Naval P~
senrch under Contract Nos N00014 99 M 0082 nnd
N00014 00 C 0057 monitored bJ Dr E P. Pood
References
[1] CHANG, Y C, Hou, T Y. MEEEIMAN, B &
OSHEE, S 1996 A level set formulation of Eu
lerinn intertace capturing methods for incom
pressible fluid fl ws T Comp Phy 124, 449
464
[2] HlNzE, J O 1975 Turbulerce McGra~Hill,
New York
[3] DALM & HAELow 1970 T ansport equations in
turbulence Phys Fluids 13, 2634 2649
[4] HoNo, W L & WALKEE, D T 2000
PeJnoid~averaged equntions for free surtace
fl ws with npplication to high Froude number
jet sprending T Fluid Msch ir prsss
151 KoMEN G J CAVALEEI L DoNELAN
JANssEN, P A E M 1994 Dy cmics crd
Modsiirg of Ocscr WCDSS, Cnmbridge
SHIE, C C 1973 A preLminarJ nemerical studJ
of ntmospheric turbulent fl w in the idenlized
pinnetarJ boundarJ InJer .
1327
[7] P Dl, W 1984 Turbuler~
Appiiectior ir Byd~ulies,
[8]
T Atmos Sci 30,
s Modsis crd Their
, IAHP
STEEN, F. PATEESON, E G & TAHAEA, Y
1996 CFDSBIP IOWA: Computctiorcl Fluid
Dynamics Msthod for Surfces Ship Bourdcry
Lcysrs, Wckes, crd WCDS Fields IIBR Rsport
No S81, low institute of HJdrnulic Pesenrch,
UniversitJ of I wn
TAHAEA, Y & STEEN, F 1996 A large domain
nppronch for calculating ship boundarJ InJers
nnd wakes for nonzero F oude number T Com
put Physies 127, 398 411
OCR for page 857
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DISCUSSION
D G Dommermuth
Science Application h~term~tional, USA
The mfhor hr. derived equctions for Rey o ids-stress
temm s based on c two-phcse approximation A
similar set of equctions cou id be derived based on c
mapping that uses the exact in d mtaneous position of
the free su face Could She mthor please cont Eat He
two approaches from c mmmericcl perspective?
AUTHOR'S REPLY
The Rey olds-a veraged form of She Na vier Stokes
equctions used in this study w re developed by
at ma mg the in d mtaneous equctions et c point m
Euleri m space The cltemative approach suggested
would be to tr msfomm the equctions into c surface-
mapped coordinate syst m, Ed then avenge there
While this would allow the use of turbulence
mo deling approaches based on inst mtaneous distance
from the fiee surface, cdditiorurl terms ..-m~ld he
introduced into the a ~ era ed equctions by the
coordinate h m formation These cdditiorurl terms
would involve the omen era en of the mst mtaneous
coordim3te-t mil:3m anon metrics it is not clear
what the mmmerical implications ..-m~ld he for this
type of approach, but this could complicate other
aspects of the problem signiflc mtly Among the
complications ..-m~ld he She fact that the turbulence
m odelmg would then be carried out in c non-inerticl
reference frame, Ed establishing the relationship
betw en con- emend turbulence measures, Ed those
needed to validate She new modeling approach
DISCUSSION
MC Hymm
Coastal Sy tems Station, USA
This pap r documents m excellent attempt et
under t mding the phenomena occurring et Ed near
the free-surface forced by t rbulence in previous
works, She mthor hr. shown the macro-sccle effects
of free-surface/turbulff~ce interaction Ed successf I
methods of m odeling that interaction for low Froude
mmmber jets By constmction, these models neglect
energy flux due to wave formation Ed propagation,
m assumption that is not con ect for high Frau de no
flows The present paper looks more speciflcclly et
high Froude no flow Ed includes c mech mism for
de ~ mg energy t m port due to wave low an on
To achieve this, c spectral t m port model is coupled
with c flow solver (with appropriate Rey olds tress
modeling) The spectral wave model is forcedby
pressme fluctuations generated by the underlying
turbu lent flow Ed the resulting a ~ era ed free surface
topology is further modified by th near surface me m
flow The turbule t somce term is extremely
import mt it is obtained by as mmmg that the
pressme spectrum is isohopic mdum3lteredbythe
presence of the fiee surface The spechal wave
model c mbe pled back to the me m flow solver
vie the free surface elevation field
Results are provided fu st for She low Frau de no jet
with essentially zero fiee su face deformation Ed
negligible wave dissipation These recohs show the
utility of applying misotropy to the near-surface
turbulence Ed illustrate She formation of c su face
current The results also suggest that the layer over
which the misotropy exists may be thim r that She
model predicts Next, the model is applied to c high
Froude no jet withbodh one-way mdtwo-way
coupling Show in the fomm of free surface
elevation, the recohs seem to suggest that the
coupling is one way This is co mite mmitive Ed
merits some further consideration The results were
obtcinedby neglecting misotropy m the turbulence
field, cssummg that since She flee su face is allow d
to deform, turbulent fluctuations are uninhibited Ed
misotropy carmot develop (use of m isotropic
pressure spech um es c forcing function for the .. a ve
model is consistent with f is assumption) What may
be occuring is chat the free surface defomms due to
larger turbulent length scales but continues to act es c
rigid wall for the mcller length scales The results
show that the su face mu rent et high Frau de no
remains (although is less energetic) but is not
reproduced m She computations in addition, it seem s
unl kely that even the larger cal es remain isotropic
near the su face the ve tick fluctuations me doing
work against gra rotational forces Ed must be
damped to some degree Thus, I would I he to ask
the mthor ff the possibility of imposing c scale
dependent misotropy model has a chance of success
Such a model would act m the spectrum in nearly She
opposite maimer fi om equilibrium t rbulence in Nat
it would reduce isotropy at the smaller scales
AUTHOR'S REPLY
While it is possible that there is a length- scale
dependence m the turbulence misotropy, it is not
clear how Nat could be implemented m a Rey olds-
averaged model The averaging process would
effectively t lend' the two effects together md recall
in some intemmediate, but rep~eiem3n- e, level of
misotropy A more-lik Iy possibility, in the author s
opinion, is Hat even at highFroude mmmber, Here ale
regions of She flow where the t rbulence is not
OCR for page 862
ener en enoughtodist rbthefieesurface Indhese
regions, he near-surface turbulence will be
misoh opic md will affect the development of the
me m flow
Representative terms from entire chapter:
surface fluctuations
