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OCR for page 517
24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Numerical Simulation of Wakes in a Weakly Stratified Fluid
James W. Rothmans, Douglas G. Dommermuthl, George E. Innisl,
Thomas T. O'Shea1 and Evgeny Novikov2
([Science Applications International Corooration. USA.
~ ,
~Un~vers~ty ot Aorta, San Diego, USA)
ABSTRACT
This paper describes some preliminary numerical stud-
ies using large eddy simulation of full-scale submarine
wakes. Submarine wakes are a combination of the wake
generated by a smooth slender body and a number of
superimposed vortex pairs generated by various control
surfaces and other body appendages. For this prelimi-
nary study, we attempt to gain some insight into the be-
havior of full-scale submarine wakes by computing sep-
arately the evolution the self-propelled wake of a slender
body and the motion of a single vortex pair in both a
non-stratified and a stratified environment. An important
aspect of the simulations is the use of an iterative proce-
dure to relax the initial turbulence field so that turbulent
production and dissipation are in balance.
INTRODUCTION
The prediction and evolution of the full-scale wakes of
submarines has been an objective of naval hydrodynam-
ics for many years. Full-scale submarine wakes consist
of the wake behind a slender body on which is superim-
posed a number of vortex pairs generated by various con-
trol surfaces. Coherent vortex systems are major compo-
nents of the wakes of submarines. For this preliminary
study, we attempt to gain some insight into the behavior
of full-scale submarine wakes by using LES to compute
separately the evolution of the self-propelled wake of a
slender body and the motion of a single vortex pair in
a stratified and turbulent environment. An important as-
pect of the simulations is the use of an iterative procedure
to relax the initial turbulence field so that turbulent pro-
duction and dissipation are in balance.
In the first section we discuss the studies of the wake
of a slender, self-propelled body without any superim-
posed vortex pairs. The objective of this research is to
be able to simulate accurately a full-scale wake of a self-
propelled body using initial conditions on a cross plane
a short distance downstream of the submarine. The ini-
tial conditions are obtained from laboratory experiments.
In the second section we describe the simulations of a
control surface vortex pair in both non-stratified and sta-
bly stratified and turbulent environments. The ultimate
objective of this study is to develop a subgrid scale tur-
bulence model to accurately represent the turbulence in
the vortex cores and to be able to predict the turbulent
entrainment into and detrainment out of the recirculation
region of the pair and therefore to be able to accurately
compute the motion of the pair at full-scale.
WAKE STUDIES
Laboratory experiments, in the absence of any back-
ground shear, have shown that the far wake behind a self-
propelled body consists of very slowly evolving patches
of vertical vorticity, of alternating sign, whose horizontal
extent is much greater than their vertical extent. These
patches are often referred to by the descriptive name of
"pancake eddies".
The governing parameters for this flow are the
Reynolds number Re = UD/zJ and the Froude number
Fr = U/(ND), where U is the speed of the body, D is
the diameter of the body, zip is the kinematic viscosity of
the fluid, and N = ~—g/pO ~Q/0X34i/2 is the buoyancy
(or Brunt-Vaisala) frequency in which 9 is the accelera-
tion due to gravity, pa is the mean density, and ~Q/0X3
is the vertical derivative of the background mean density
(assumed to be a constant).
In this paper we describe two numerical experi-
ments: the first is an attempt to simulate the formation
of eddies in late wake due to a turbulent flow for the case
with Fr = 2.0 and Re = 105. The second is similar to
the first except that the fluid is non-stratified (Fr = x).
The numerical method is large eddy simulation
(LES). The flow is initialized with a mean wake flow
that includes swirl, which is based on experimental mea-
surements of the wake behind a propeller-driven slender
body, with a superimposed homogeneous turbulent flow
field. A relaxation procedure is introduced to establish
the proper balance between production and dissipation
before the calculation is begun. There are no coherent
structures imposed on the turbulence other than the gross
characteristics of the mean wake flow. The results are
compared with that of a drag wake that is described in
a companion paper, Dommermuth, et al. (2002), and it
OCR for page 518
is found that the two types of wakes evolve quite differ-
ently. The mean swirl generates internal waves. Once the
mean swirl radiates internal waves, the turbulence pro-
duction terms are considerably reduced, especially the
shear stresses. As a result, the wake persists much longer
than a drag wake. Our results show that coherent vortices
appear in the late wake even though the flow is initialized
without any coherent structures.
PROBLEM FORMULATION
A schematic drawing of the flow under considera-
tion is shown in Figure 1 in a reference frame in which
the body generating the wake is at rest. This figure serves
to define much of the nomenclature used here. The wake
is considered to be statistically stationary. Since the en-
tire wake is too long to compute as a whole, we make
the approximation that the flow can be computed within
a rectangular box, with axial dimensions much smaller
than the total length of the wake, that moves with the
mean flow speed U. Within this box the flow is computed
by solving numerically the governing equations for an
incompressible Boussinesq fluid using large eddy simu-
lation (LES).
Large eddy formulation of the Boussinesq equations
We will assume that the fluid is incompressible and
weakly stratified. A large eddy approximation is invoked
whereby the large-scale features of the flow are resolved
and the small scales are modelled. Let hi denote the
filtered three-dimensional velocity field as a function of
space xi (i = 1, 2, 3) and time t. Here, the overbar de-
notes spatial filtering in a large eddy sense. The origin
of the coordinate system is at the centroid of the body,
as shown in Figure 1. x is positive downstream, x2 is
transverse to the track of the body, and X3 is positive up-
ward. The length and velocity scales of the flow are re-
spectively normalized by the diameter of the body (D)
and the free-stream velocity (U).
The numerical method is large eddy simulation
(LES). The flow is initialized with a mean wake flow with
a superimposed homogeneous turbulent flow field. A re-
laxation procedure is introduced to allow the turbulence
field to achieve a balance between production and dissi-
pation before the calculation is begun, as is described in
more detail in Dommermuth, et al. (20021. There is no
coherent structure imposed on the turbulence other than
the gross characteristics of the mean wake flow.
A mixed model (Bardina, et al., 1984) is used to
model the subgrid scale stress tensor. The similarity por-
tion of the mixed model provides an accurate model of
the turbulent stresses, whereas the Smagorinsky portion
provides dissipation. The corresponding mixed model
for the residual density flux combines a similarity model
with an eddy diffusivity model.
Figure 1: A schematic diagram of the self-propelled body and
the coordinate system in a reference frame in which the body is
at rest. The dashed box denotes the slab of fluid that is modelled
using LES. but ~ is the wake deficit.
Following Orszag & Pao (1974), a Galilean approx-
imation is used to relate the spatial development of the
wake to the temporal evolution of the LES. In normal-
ized variables, this approximation results in the relation
x~ = t, where x~ is the distance downstream of the body
in the wake and t is the corresponding time in the LES.
Based on this Galilean approximation, we further assume
that
| dto(t, x = Xo) ~ ~
o
{L
L J dx˘(t = Tome) = (I)) ~ (1)
To
where o is a physical quantity, hat accents denote time
averaging, angle brackets denote spatial averaging, L is
the length of the LES computational domain in the axial
direction (see Figure 1), T is the duration of time averag-
ing, and XO and To are positions in space and time where
the wake of the body and the LES correspond. A tilde ac-
cent denotes the turbulent fluctuations, which are defined
as fib = 0 - (o). As shown in Figure 1, the wake of the
body is modelled as a slab of fluid.
Initialization
The initial velocity field is decomposed into a mean
disturbance and a fluctuating disturbance. The magni-
tude and distribution of the mean and fluctuating compo-
nents are specified based on the measurements of Lin &
Pao (1973, 1974a,b) and Lin, Veenhuizen & Liu (1976~.
The mean axial velocity is specified as
2
OCR for page 519
0.1
( r2 ) ( r2 )
, (2)
where aO is the amplitude of the mean wake deficit nor-
malized by the free-stream velocity, and rO is the initial
characteristic radius of the wake. The cross-stream com- 0.001
portents, (u2) and (U3) are determined from the mean
axial vorticity, which is specified
(Qua = aw (1 - 2 2 ) exp (_ r )
, (3)
where a``, is the amplitude of the mean axial vorticity and
rw is the characteristic radius of the axial vorticity field.
The propeller swirl consists of an annulus of vorticity
of one sign from the tip vortices surrounding a region
of opposite sign vorticity from the root vortices, with a
net vorticity of zero. Interestingly, in another numerical
experiment (not shown) we have found that the swirl is
necessary if the axial velocity is to retain its self-similar
shape. Without the swirl, the axial velocity rapidly disin-
tegrates into small regions of positive and negative flow,
and the rate at which they cancel each other out is more
rapid than with swirl present.
The initial rms velocity fluctuations are also approx-
imated using Gaussian distributions.
~ = at exp ( - 2 2 ) (4)
where ai are the initial amplitudes of the rms velocity
fluctuations and ri are the initial characteristic radii. The
fluctuating velocity field is constructed from a realization
of fully-developed homogeneous turbulence that is pro-
jected onto the rms velocity distribution, as is described
in more detail in Dommermuth, et al.~1997~. The rms
fluctuations are initially uncorrelated and the turbulent
shear stresses are zero. As discussed later, in the Wake
relaxation subsection, an iterative procedure is used to
relax the wake until the production of turbulent kinetic
energy is balanced by dissipation. We assume that the
mean and fluctuating portions of the density disturbance
are initially zero.
Numerical algorithm
The governing equations are discretized using
second-order finite-differences. A fully-staggered grid
is used in the numerical simulations. Periodic bound-
ary conditions are used along the sides of the compu-
tational domain, and free-slip boundary conditions are
0.0001
105
10 100 1000
Figure 2: The kinetic energy for 7 < t < 1000 for Re = 105:
( — ), fluctuations for Fr = oo, ( - - - - - - - - - ),
fluctuations for Fr = 2.0, ( - ), mean flow for Fr =
x, and ( ), mean flow for Fr = 2.0. The bold
solid line represents axial similarity behavior, t-3/2, and the
bold dashed line represents swirl similarity behavior, tat. The
energy is normalized by U2D3.
imposed at the top and bottom. A third-order Runge-
Kutta scheme is used to integrate the equations with re-
spect to time. The numerical algorithms have been im-
plemented using high-performance fortran (PGHPF) on
a CRAY T3E. Additional details and convergence studies
of a similar numerical algorithm are described in Dom-
mermuth, et al.~1997~.
RESULTS
For the stratified simulation, the Reynolds number
is Re = 105 and the Froude number is Fr = 2.0.
The initial mean disturbance and the rms fluctuations are
based on least-squares fits of laboratory measurements
of a cross section of the wake that is seven diameters
downstream. For the mean axial velocity, a0 = 0.10 and
r0 = 0.25 and for the mean axial vorticity an = 0.80 and
ro = 0.20. For the fluctuating portion of the flow, pre-
relaxation, ai = 0.40 and ret = 0.07 and r2 = r3 = 0.05.
We choose a computational domain that is 12D long,
24D wide, and 12D deep. The horizontal and vertical
dimensions of the computational domain are as large as
computer resources allow in order to accommodate the
propagation of internal waves and the spreading of the
wake, which in the stratified case spreads more in the
horizontal than in the vertical. In any case, the horizon-
tal and vertical dimensions of our computational domain
3
OCR for page 520
are larger in most cases than the comparable tank sizes 0.1
used in laboratory experiments. Convergence is estab-
lished using two different grid resolutions correspond-
ing to coarse (128 x 256 x 129 grid points) and fine
(256 x 512 x 257 grid points) simulations. The fine-grid
results are presented here. For the non-stratified case,
we use the same dimensional parameter values as for the
stratified case, except that N = 0 rad/sec. Therefore, all
the nondimensional parameters are the same except that
Fr= x.
Wake relaxation
A relaxation procedure is used to establish the
proper balance between production of turbulent kinetic
energy and dissipation at the beginning of the calcula-
tion. During the relaxation procedure, the mean portion
of the flow is held fixed. The total turbulent kinetic en-
ergy is also held fixed, but the spatial distribution of the
turbulent fluctuations is free to vary. Once the turbulent
production and dissipation are in balance, the relaxation
procedure is turned off and the numerical simulation is
initiated. This is the same relaxation procedure described
in Dommermuth, etal.~2002) and is similar to the pro-
cedure used by Orszag & Pao (1974) in their numerical
simulations of a self-propelled body.
Similarity
In Figure 2 the kinetic energy in the mean
portion of the flow integrated over the volume of
fluid ({v dV (ui) (ui>) and the turbulent kinetic energy
~ ~
(TV dVuini) are plotted versus time for both the strati-
fied and non-stratified cases for Re = 105.
The decay of the mean and turbulent kinetic energy
in the self-propelled body wake proceeds somewhat dif-
ferently than for the towed body, which is described in
Dommermuth, et al. (2001~. Figure 2 shows the decay
of the mean and fluctuating kinetic energy for both the
stratified and non-stratified wakes. Initially, for the non-
stratified case, the energy in the thrust and drag wakes
dominates and the energy decays as x~ 3/2. Eventu-
ally, however, the swirl energy (which is much smaller
initially) dominates and the entire wake approaches the
swirl decay rate of x~ i. For this particular simulation,
the non-equilibrium region is much greater than it is for a
drag wake. The mean and turbulent energies for the strat-
ified simulation decay much less rapidly than the non-
stratified simulation.
For the stratified simulation, self-similarity is not es-
tablished until the very late wake. We believe that once
the effects of stratification disrupt the turbulent produc-
tion mechanism, the wake persists significantly longer
than it would in a non-stratified fluid. This effect is more
significant for momentumless wakes than it is for drag
4
0.01
ant=
0.001 _
0.0001 _ .... . . . ....
10
-
t
100
1000
Figure 3: The total potential and kinetic energies for 7 < t <
1000 for Re = 105. The results for stratified fluids (Fr = 2.0)
are labeled: ( ), potential energy; ( - - - - - - - - - ),
vertical turbulent kinetic energy; ( ), potential
energy plus vertical turbulent kinetic energy, ( - ),
horizontal turbulent kinetic energy, and ( - - - - - ), total
energy. The energy is normalized by U2D3.
wakes because momentumless wakes in the absence of
stratification decay more rapidly than drag wakes.
Energy is redistributed between the kinetic energy
and the potential energy and also between the mean and
the fluctuating portions of the flow. Figure 3 shows the
total energy (E), which includes the turbulent kinetic en-
ergy, the kinetic energy in the mean portion of the flow,
and the potential energy (~ Jo do iV dVp u31. Fig-
ure 3 also shows the total energy in the fluctuating por-
~
tion of the flow (E), which includes the turbulent kinetic
energy and the potential energy. The stratified and non-
stratified fluids establish self-similarity at the same rate
~
for both E and E. The results show a tendency in the
far wake for the energy in the stratified fluid to be higher
than the non-stratified fluid. This effect may be attributed
to the generation of internal waves, which do not decay
as rapidly as turbulence.
The formation of pancake eddies
Figure 4 show time series of the vertical component
of vorticity in the horizontal plane through the wake cen-
terline (X3 = 0~. Part (a) illustrates the results for the
non-stratified fluid, and part (b) shows the correspond-
ing results for a stratified fluid. In this gray-scale fig-
ure, white represents positive vorticity with magnitude
OCR for page 521
As = 4 and black negative vorticity with ~z = - 4. Each
frame has the dimensions 24D in both the cross-stream
and upstream (to the left) directions. Note that the flow
along the streamwise direction (A ~ has been periodically
extended. The centers of each frame are located at (from
left to right and top to bottom) t ~ 6, 22, 54, 86, 118,
166, 230, 294, 358, 422, 518, 614, 710, 806, 902, and
998. Each frame is scaled by the distance downstream
from the initial plane ~ t = 6 ), which is the expected
similarity behavior.
For the stratified cases, coherent structures, in the
form of nearly circular vortex patches begin to appear at
t ~ 100. This corresponds to about the time the mean ki-
netic energy begins to decay at the self-similar rate. Note
that instabilities are evident almost immediately. Further
downstream, the size of these patches of vorticity grow
and the number of patches in a frame very slowly de-
crease.
Over the duration of the simulation, the small-scale
features that are observed in the bulges of the non-
stratified simulation appear to merge to form the large-
scale structures that are observed toward the end of the
stratified simulation.
VORTEX STUDIES
Coherent vortex pairs are major components of the wakes
of submarines. Recently, comparisons with laboratory
and field observations (Delis) and Greene, 1990; Delisi,
et al., 1996; Delisi, 1998) indicate that numerical models
fail to accurately predict vortex pair migration in strati-
fied environments. The reasons for this failure are not
fully understood. A possible reason for this failure is that
the turbulence models employed in the numerical simu-
lations produce inaccurate predictions of the entrainment
and detrainment rates of the recirculating region of the
vortex pair. Large-eddy-simulation (LES) codes, devel-
oped for modelling aircraft trailing vortex pairs in the at-
mosphere (Gerz and Ehret, 1997; Han, et al., 2000), have
not been carefully compared with detailed measurements
of vortex pair motion in stratified environments and may
also not accurately predict entrainment and detrainment
rates. It is clear that a more complete understanding of
the physical mechanisms controlling the motion of vor-
tex systems is needed.
The governing parameters for this flow are the
Reynolds number Re = F0/u, the vortex Froude num-
ber Fr = Vo/(Nbo), and the nondimensional turbu-
lence intensity ~ = (ebo)~/3/Vo of the background en-
vironment, in which F0 is the vortex circulation strength,
z' is the kinematic viscosity, ~ is the turbulent dissipa-
tion rate, be is the initial vortex separation distance and
V0 = Fo/~2~rbo) is the initial vertical speed of the vortex
pair. The nondimensional turbulence intensity is the ratio
of the turbulent velocity at the length scale of the vortex
separation distance to the vortex descent speed.
In this paper we describe two preliminary numerical
experiments: the motion of a vortex pair for Re = 105,
~ = 0.15 and Fr = x and 4. The values of these pa-
rameters are chosen such that the numerical simulations
can be compared directly with aircraft measurements in
an atmosphere with moderate ambient turbulence. These
aircraft measurements are the only data we have access
to for which the values of Re are near to that of full-scale
submarine wakes.
The numerical method is the same large eddy
scheme described in the previous section for computing
the wake of a self-propelled body. However, for the vor-
tex simulation we have modified the sub-grid-scale tur-
bulence model in an initial attempt to deal with the ef-
fects of strong rotation, such as would be found in the
vortex cores, on the small-scale turbulence. The flow is
initialized with an approximation to a measured mean ve-
locity field of a rolled up aircraft wing vortex, and a su-
perimposed homogeneous turbulent velocity field in an
unstratified background. A relaxation procedure is intro-
duced, as described in the previous section, to allow the
turbulence field to establish a balance between produc-
tion and dissipation before the calculation is begun.
PROBLEM FORMULATION
A schematic drawing of the initial flow configura-
tion for the vortex pair simulations is shown in Figure 5,
showing the locations of the centers of each vortex of
the vortex pair. The distance between these centers is be
and the circulation strengths are—F0 for the vortex on
the left and +~0 for the vortex on the right. The self-
induced velocity of this vortex pair is directed upward
and has magnitude V0. The self-induced velocity is up-
wards since we are attempting to simulate a submarine
that is slightly buoyant and so its control surfaces need
to generate negative lift to keep the ship in level cruise.
Superimposed on this vortex pair flow field is a homoge-
neous field of turbulence, whose intensity and character-
istic length scale will be described below.
Large eddy formulation of the Boussinesq equations
We will assume that the fluid is incompressible and
weakly stratified. A large eddy approximation is invoked
whereby the large-scale features of the flow are resolved
and the small scales are modelled. As in the previous
section, let hi denote the filtered three-dimensional ve-
locity field as a function of space xi (i = 1, 2, 3) and
time t. The origin of the coordinate system is as shown
in Figure 5. x1 is along the axes of the two vortices, x2 is
transverse to these two axes, and X3 iS positive upward.
The length and velocity scales of the flow are respec-
s
OCR for page 522
(a)
(b)
Figure 4: A time history of the vertical component of vorticity wz on the horizontal plane through the center of the wake for
Re = 105:(a) Fr = oo and (b) Fr = 2.0. In these gray-scale figures, white represents positive vorticity with magnitude wz = 4
and black negative vorticity with wz = - 4. Each frame has the dimensions 24D in both the cross-stream and upstream (to the left)
directions. Note that the flow along the streamwise direction (x1) has been periodically extended. The centers of each frame are
located at (from left to right and top to bottom) t ~ 6, 22, 54, 86, 118, 166, 230, 294, 358, 422, 518, 614, 710, 806, 902, and 998.
6
OCR for page 523
-
vo
1
rO X3 -rO
5bo ~ ~
( · , ,X2. ~
x1
1' be .1
5bo — -,
Figure 5: A schematic diagram of a vortex pair in a stratified
fluid, showing the coordinate system and relevant parameters,
the vortex separation distance ho, the vortex circulation strength
F0 and the self induced vertical velocity VO.
lively normalized by the initial vortex separation distance
be and the initial self-induced speed of the vortex pair VO.
The numerical model used for this study is the same
as that described in the previous Large eddy formulation
of the Boussinesq equations subsection, except that the
mixed model (Bardina, et al., 1984) used to model the
subgrid scale stress tensor has been modified in an ini-
tial attempt to represent the suppression of sub-grid-scale
turbulence due to the strong rotation in the vortex cores.
Since turbulence is suppressed in the cores, they tend to
diffuse only very slowly. This modified mixed model
was chosen as it is a first step towards implementing a
dynamic SGS model, such as the one described by Ger-
mano, et al.. (1991~. For the modified mixed model, the
SGS stress tensor
Tij = UjUi—UjU
in which Hi is the fluid velocity in the ith direction and
the overbar symbol denotes spatial filtering, is repre-
sented as
Tij= (niUi—U}Ui)
-Cs /\2 ~ Sij - (Sij ~ ~ (Sij
where Si; is the strain tensor,
S -
~ —
- (Sib>)
140Ui Air
_ +
2 Axe taxi
(6)
(7)
The first term, in parentheses, on the right hand side of
(6) is the similarity portion of the mixed model. It pro-
vides an accurate representation of the turbulent stresses.
The remaining term on the right hand side is Smagorin-
sky portion portion of the mixed model. It represents
dissipation and is formulated so that there is no turbulent
dissipation when the strain tensor is well resolved. cs
is the Smagorinsky coefficient and /\ is the width of the
spatial filter, which we set equal to the grid spacing. In
the simulations described here cs = 0.05.
The computational domain is square in the cross
plane with sides of length 5bo and rectangular in the axial
plane with the axial sides of length 1.5bo. As in previous
studies, the smaller length in the axial direction is cho-
sen so as to suppress the Crow instability, Crow (1970),
(which has an axial wavelength of about 8.6bo) and to re-
duce computational costs. Periodic boundary conditions
are imposed at all boundaries. In the calculations de-
scribed in this paper, the grid resolution is (64, 256, 256)
in the (x, y, z) directions.
Initialization
The initial flow field consists of the combination of
a homogeneous turbulent velocity field and the velocity
field associated with a vortex pair. The vorticity distri-
bution within each vortex is an empirical fit to that mea-
sured in large-scale airplane wing vortices.
The background homogeneous turbulent velocity
field is generated by prescribing an initially random dis-
tribution of Fourier modes and then integrating the equa-
tions of motion forward in time, holding the total energy
constant by adjusting the amplitudes of all the Fourier
modes appropriately at each time step, until the velocity
field has reached a statistically steady state. The result-
ing velocity field has a well developed inertial subrange
and is nearly isotropic. After this steady state has been
reached, the amplitudes are no longer adjusted and the
turbulence is allowed to decay. The density field is not
perturbed initially. The strength of the turbulence field
is measured by the parameter ~ = cl/3, where ~ is the
nondimensional turbulent dissipation rate. This is a mea-
sure of the ratio of the turbulent velocity magnitude at the
scale of the initial vortex separation to the self-induced
speed of the vortex pair.
The initial velocity field associated with the vortex
pair that we will use is an empirical fit to observed air-
plane wake vortices developed by Proctor (19981. In this
representation, the lateral components of the vorticity are
zero and the axial component ~ associated with each vor-
tex is given by two functions, one for inside the vortex
core, r < rc, where r is the radial distance from the
vortex center and rc is the radial distance to the peak tan-
gential velocity, and one for outside, r > rc, the core.
OCR for page 524
180
1 4()
3 1m
80
60
40
20
10
x3 6
4
2
o
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 (
r
Figure 6: The initial radial distribution of vorticity for each
vortex of the vortex pair, normalized by their vorticity values at
each vortex's center, plotted as a function of the radial distance,
r, from the vortex center. The initial vorticity of each vortex is
axisymmetric about the vortex center.
For r < rc
Airs= 2 t1—expt—lO(rc/B)3/445 x
expt—1.2527(r/rc)2], (8)
and for r > rc
,,_ _ 7.5/ r \-1/4_~r ares ~ /r~\3/43 (9)
an
OCR for page 525
1
0.8
0.6
0.4
0.2
..
—_ _
larrunar unstratified ' .
rbulent unstratified
turbulent stratified
2 4 6
t
8 10 12
Figure 8: The time series of the total kinetic energy of the
mean flow.
steadily with time in both the non-stratified and strati-
fied cases. The kinetic energy decays most rapidly in the
stratified case since besides dissipation some of the ki-
netic energy is being converted to potential energy and
some of the mean energy is being converted into internal
wave fluctuations.
Time series of the kinetic energy of the fluctuating
motion in the whole computational domain is shown in
Figure 9. In these time series, the fluctuating energy ini-
tially increases steadily as energy from the mean flow is
converted to fluctuating energy. Up to t ~ 1, which is
about a quarter of a buoyancy period, there is no differ-
ence between the non-stratified and stratified cases and
there is not significant until t = 2 or 3, which is about
half a buoyancy period. At this time it appears that the
turbulent energy saturates and begins to decline in the
non-stratified case and to oscillate (with a slight decay)
in the stratified case. The frequency of the oscillations in
the stratified case is approximately equal to the buoyancy
frequency, suggesting that a good fraction of the energy
has gone into internal gravity waves.
The evolution of the spatial distribution of the turbu-
lent kinetic energy is shown in Figures 10 and 11. Early
in the simulation most of the turbulent kinetic energy is
is the vortex cores in both the non-stratified and stratified
cases. A little later, a significant amount of turbulent pro-
duction has occurred at the top of the vortex pair recircu-
lation zone and again this appears to be the same in both
the non-stratified and stratified cases. After t ~ 4 the
two cases start to deviate from each other substantially.
In the non-stratified case, the turbulent kinetic energy re-
mains concentrating in the cores as the energy decays
steadily outside of the cores. In the stratified case, the
turbulent kinetic energy actually increases at first in the
region between the cores and eventually throughout the
entire recirculation zone.
1~
o.s
0.8
0.7
0.6
~ 0.5
I:
0.4
0.3
0.2
0.1
01
I
rl[
:J
turbulent unstratified |
- - turbulent stratified |
, ~ ~
.'
2 4 6
8 10 12
Figure 9: The time series of the total kinetic energy of the
turbulent flow.
Conclusions
We have presented some preliminary results from using
large eddy simulation to compute the late wake of a self-
propelled body moving at constant speed through a non-
stratified and a uniformly stratified fluid at Re = 105 as
well as the motion of a pair of counter-rotating vortices
in both non-stratified and stably stratified fluids.
An important aspect of the simulations is the use
of a relaxation procedure to adjust the initial turbulence
fields so that turbulent production and dissipation are in
balance. Our simulations intentionally have been initi-
ated with different turbulent velocity and density fields
than would be found in laboratory experiments. We have
done this so that the initial conditions would be nearly
the same for stratified and non-stratified simulations as
well as for simulations with different Reynolds numbers.
These similar initial conditions, which would be very dif-
ficult to obtain in laboratory experiments, have allowed
us to study how the wake behaves without the compli-
cations of the initial conditions varying with the overall
flow conditions.
We have observed that the pancake eddies form
without being driven by similarly-sized eddies in the ini-
tial disturbance. In fact, these simulations produced pan-
cake eddies even though the fluctuating component of the
initialization began with random phase. We have seen
that the eddies depend strongly on whether the body is
towed or propeller driven: the propeller-driven body pro-
duces eddies that are smaller and more chaotic than the
nearly periodic pancake eddies in the simulated wake of
a towed body. This difference is consistent with the much
more rapid decay of the mean axial velocity in the wake
in the self-propelled case.
In the simulations of the motion of a vortex pair in
a turbulent background flow in both non-stratified and
9
OCR for page 526
1
-1
1
o
-1
0.4 1
01
0.2 1
-21
o
0.4 1
0.2
- o
2
-1
-21
-2 0 2 -2 0 2
I!
().2
'O
Figure 10: Transverse plane (y, z) contour plots of axially-averaged turbulent kinetic energy at (from left to right and top to
bottom) t = 0.2, 2.2, 4.2 and 6.2 for Re = 105 and Fr = oo.
10
OCR for page 527
o
-1
1
-1
-2 0 2
0.4
10.2
Jo
-2 0
-21
~0.4 1
0'
0.2
o
0.2
910.4
10.2
Jo
Figure 11: Transverse plane (y, z) contour plots of axially-averaged turbulent kinetic energy at (from left to right and top to
bottom) t = 0.2, 2.2, 4.2 and 6.2 for Re = 105 and Fr = 4.
11
OCR for page 528
stratified fluids we found that the ascent rate of the pair
is strongly affected by the background turbulence and by
the stratification, both effects lead to a decrease in the as-
cent rate of the pair. In the non-stratified case this is due
to turbulent diffusion of the vorticity and in the the strat-
ified case it is additionally due to the conversion of mean
kinetic energy into potential energy and to the generation
of internal gravity waves.
The distribution of turbulent kinetic energy in the
vortex pair was studied and found to be quite different in
the intermediate and late times of the simulations. The
non-stratified simulation shows the turbulent kinetic en-
ergy mainly decaying everywhere except in the vortex
cores. The stratified case shows that the turbulent kinetic
energy actually increases in the recirculation zone at late
times.
Acknowledgements
This research is supported by ONR under contract num-
ber N00014-01-C-0191, Dr. L. Patrick Purtell program
manager. This work was supported in part by a grant of
computer time from the DOD High Performance Com-
puting Modernization Program at the Naval Oceano-
graphic Office Major Shared Resource Center. We thank
Prof. G. R. Spedding at He University of Southern Cali-
fornia and Prof. D. D. Stretch at the University of Natal
for many helpful discussions.
References
Bardina, J., Ferziger, J. H. and Reynolds, W. C., "Improved
turbulence models based on LES of homogeneous
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Crow, S. C. "Stability theory for a pair of trailing vortices,"
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Delisi, D.P. and Greene, G. C., "Measurements and
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Delisi, D.P., Greene, G. C., Robins R. E., and Singh, R.,
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study of wake vortex decay and descent in homogeneous
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Lin, J.-T. and Pao, Y.-H., "Turbulent wake of a self-propelled
slender body in stratified and non-stratified fluids:
analysis and flow visualizations," Report No. 11, July,
1973, Flow Research Company.
Lin, J.-T. and Pao, Y.-H., "The turbulent wake of a
propeller-driven slender body in a nonstratified fluid,"
Report No. 14, February 1974a, Flow Research
Company.
Lin, J.-T. and Pao, Y.-H., "Velocity and density measurements
in the turbulent wake of a propeller-driven slender body
in a stratified fluid," Report No. 36, August 1974b, Flow
Research Company.
Lin, J. T. and Pao, Y. H., "Wakes in stratified fluids: a review,"
Annual Reviews of Fluid Mechanics, Vol. 11, 1979,
pp. 317-338.
Lin, J.-T., Veenhuizen, S.D., Liu, H.-T., "Experimental data on
stratified wakes for validation of wake codes," Report
No. 73, October 1976, Flow Research Company.
Orszag, S. A. and Pao, Y. H., "Numerical computation of
turbulent shear flows,"
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in Environmental Pollution, 1974, pp. 225-236.
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Proctor, F. H., "The NASA-Langley wake vortex modeling
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Riley, J. J., Metcalfe, R. W. and Weissman, M. A., "Direct
numerical simulations of homogeneous turbulence in
12
OCR for page 529
density stratified fluids," Nonlinear Properties
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13
OCR for page 530
DISCUSSION
G.R. Spedding
University of Southern California, USA
Summary
The recent development of large eddy simulation
(LES) of stratified wakes flows at significant
Reynolds number (Re) has been eagerly awaited
by the community, not only in ONR-specific
applications (wakes of submerged bodies), but
also as a significant step forward in resolving
some of the conundrums in coherent structure
formation from disordered initial conditions, a
process that seems quite characteristic of flows
in stably-stratified fluids. The paper by Rottman
et al. shows some very interesting results from
simulations of the wake of a self-propelled,
slender body, and for a rising vortex pair. The
selection of these configurations presages a time
when complex geometries, complete with control
surfaces will be simulated at high Re and for
long evolution times. The current results can be
seen as significant milestones on the way to this
goal, and they raise some intriguing and quite
general questions.
Pancake eddy formation from initially
turbulent wakes
The most famous consequence of moving a body
through a stratified fluid can be seen reproduced
in Fig. 4 of this paper, where the lower half of
the figure shows large scale coherence and a lack
of small-scale features that clearly contrasts with
the upper (non-stratified) equivalent. However,
careful inspection of the non-stratified result
shows that it too contains coherent structure at
the same scale as the dominant features in the
stratified case. Perhaps then, the principal
feature of the stratified result is in the loss of
small scales, rather than the generation of the
large-scale structures themselves? How does
this happen? Perhaps vertical fluctuations are
readily transported away as internal wave
motions. Then why preferentially the small
scales? And what happens to the small scale
horizontal fluctuations? Why do they apparently
dissipate so fast? Alternatively, as the authors
suggest, perhaps there is preferential merging of
small scales in a stratified environment. Why
would that be? How are the dynamics different?
If Fig. 4a is filtered through a low-pass filter, do
we get Fig. 4b? The eddies themselves are
described as smaller and more chaotic (in some
sense) than those generated by the mean defect
profile but a direct comparison is not available.
If this is the case, why is it so? Does that shed
some light on possible formation mechanisms?
Which aspect of the initial conditions is
important for this result? While some of these
questions might seem abstruse and of limited
practical significance, they do bear on the
generality of the results. It is only through a
clear understanding of the important physics that
the range of applicability of a set of simulations
can be known.
Importance of initial conditions
A number of quite specific initial conditions are
detailed for both the wake and vortex pair.
Velocity and length scales for mean and
turbulence (and swirl) quantities are taken from
laboratory experiments in the self-propelled
body case and from airplane wake data for the
vortex pair. While the relevant coefficients are
given and apparently copied quite faithfully, it is
not clear how, or if, these values are important.
Some of the external data are quite unusual; the
discontinuous vorticity profile in Fig. 6, or the
large fluctuation magnitudes in the kinetic
energy of the momentumless wake, where
fluctuating quantities are 3-4 times the mean
value. One of the recurrent questions of
stratified fluids research concerns the relative
importance of initial conditions (and, quite
frequently, the search for initial conditions that
are unimportant), so the results can have general
application. Having a correctly functioning
code, the authors would seem to be uniquely
qualified to answer some of these questions, at
the very least with respect to this work. What, if
anything, changes if the ratio of mean
defect:axial:fluctuating velocities is not 1:8:4? It
is to be hoped the authors will have an
opportunity to examine and report the sensitivity
to initial conditions, both for numerical
initialization details and for understanding the
relative importance of the nhvsical field
variables.
Comparing averages
r ~ -
Comparing and interpreting data can be quite
sensitive to particular averaging procedures and
two examples come to mind. The first involves
an unremarked comparison. A general curve
describing the evolution of quantities such as the
mean wake defect in the case of a towed sphere
has been proposed,! whose main features can be
summarized in Fig. D1.
OCR for page 531
log
U. u'
3D
\
-213 \' -1/4
I ·~.
..
NEQ
Q2D
.................. ! -213
. ·-.
log Nt
Fig. D1. A general curve showing the low
decay rate of mean wake defect (and of kinetic
energy) during an intermediate non-
equilibrium (NEQ) regime for a decaying
stratified flow. The lasting effect is that at
late times, when the flow is quasi-two-
dimensional (Q2D), the wake defect or kinetic
energy remain significantly higher than if a
constant 3D power law had been in effect
(dotted line).
Fig. 2 of the paper demonstrates that stratified
momentumless wakes also have a similar region
of comparatively low kinetic energy decay rates,
lasting even longer than in the towed-sphere
case. However, some caution is required
because the data from Fig. D1 above concerns
averages made only over the wake itself- the
averaging box enlarges as the wake itself does,
and the outer ambient fluid is excluded. In the
current paper, Figs 2 & 3 are averages over the
entire domain. The primary effect is to change
the interpretation of the contribution of internal
waves to these measures. Since the wave
motions propagate energy away from the wake,
then this effect by itself might be anticipated to
increase the decay of a locally-averaged kinetic
energy. Since the opposite happens (NEQ decay
rates decrease in Fig. D1) a different explanation
is required. In this paper, the comparatively low
dissipation rates of internal wave modes are
offered to explain the low energy decay rates.
This is possible, and if true, then the mechanism
is different from that of Fig. D1, any similarity
being merely coincidental. In any event, the two
results are not strictly comparable. They could
be if plots like Fig. 2 & 3 were made for local
wake averages, instead of global
averages.
The second possible effect of
averaging is in the comparison
of the kinetic energy
distributions of rising vortex
pairs in homogeneous and
stratified ambients (Figs 10 &
11), while looking across the
vertical axis. Fig. 11, for the
stratified case, has the intriguing
result that the kinetic energy
intensity appears higher in
recirculating regions around the
vortex cores, rather than being
concentrated exclusively in the
cores themselves. The physical
mechanism for this is not obvious, and one
wonders about the role of averaging. In the
presence of the density gradient, vertical motions
are constrained, eventually, by buoyancy forces.
A spanwise, or axial average through some
structure thus might reflect an increased
uniformity in the vertical direction (z), rather
than some change in the structure itself. While
the almost two-dimensional motions of Fig. 10
are still free to vary locally in z, this variance
might be suppressed in Fig. 11, having the effect
of more closely aligning the structures, which
then appear to be more concentrated.
Comparisons with experimental/field data
Of course extensive comparisons are not
practicable in a short paper, but it will be very
interesting to know how these recent and
interesting data compare with other cases, from
laboratory experiment to numerical simulation to
field observations - if there are any. There are
two strains of interest in this: (i) Using
comparative studies to deepen the understanding
of the important physical mechanisms that
govern formation and late-time evolution of
these stratified wake flows. It seems possible,
for example, that wakes with both zero and
nonzero momentum have some period of
surprisingly low kinetic energy decay rates.
Does this point to some universal feature of
flows in a stratified ambient? How is it related
(if at all) to the characteristic pancake eddies that
are observed? (ii) The second theme is that of
obtaining practical, predictive models for real
applications. Is there any evidence from field
tests that submerged wakes either have long
persistence times or higher than expected kinetic
OCR for page 532
energies close to their source?
generally, what are the practical consequences
for the main results reported here?
These remarks apply equally to the wakes and
vortex pair results. The latter, in particular,
ought to be comparable to other data, at least in
homogeneous fluids. Currently, the trajectories
of Fig. 7 show comparisons between a laminar,
inviscid solution and turbulent simulations in
homogeneous and stratified fluids Missing is
the comparison with aircraft wake data for which
the simulations have been set up, or with
laboratory data on either laminar or turbulent
viscous flows.
Or, more
Concluding remarks
It is likely that sophisticated LES calculations
such as those appearing this paper will form an
increasingly important component in making
practical predictions of complex flows. The
combined modeling/direct simulation approach
in principle is a powerful way of making
calculations that do not exceed current computer
capacity, but that can properly account for the
important physical processes. To test and further
evolve the capacities of these programs it will be
essential to keep analyzing them and improving
them in an environment of companion test and
data from direct numerical simulation, from
laboratory data and from field studies. Similarly,
the performance and reliability of the LES model
components themselves will be verifiable if and
only if attention is paid to comparative results
from varying initial and boundary conditions.
The results reported here are intriguing and
thought-provoking, and we look forward to
more.
'Spedding. G.R. The evolution of initially
turbulent bluff-body wakes at high internal
Froude number. J. Fluid Mech. 337, 283-301.
AUTHORS' REPLY
Introduction
Prof. Spedding has raised a number of perceptive
questions about the physical implications of the
results of our simulations. Many of these
questions are of such a general nature that we
will be unable to answer them in any depth here,
either because we have not done the appropriate
simulations, we do not have enough space or
simply we do not know the answers (although
occasionally we will offer some speculations).
The questions that we cannot answer are all
intriguing and form a good basis for conducting
future research.
Pancake eddy formation from initially
turbulent wakes
Prof. Spedding's observations and questions
about the differences between the wakes in non-
stratified and stratified fluids are the main issue.
We cannot fully answer these questions here,
much more research is needed, but we will
attempt some speculations.
We agree that the non-stratified wakes in our
simulations develop large-scale structures that
are similar in some ways to the large-scale
structures in our simulated stratified wakes. We
presume that in stratified wakes the vertical
fluctuations propagate away in the form of
internal waves leaving only the large-scale
structures that have nearly horizontal motion
only. We suspect that similar large-scale
structures exist in the non-stratified wake, but
that they have no preference for being horizontal.
It does seem that there is some preferential
merging in the stratified flow that must explain
the lack of small scales in this flow, but we have
no quantitative theory for this at the present.
The possible formation mechanism for the
pancake eddies is still in dispute. We have found
that the ultimate structure of the stratified wake
is fairly insensitive to the initial conditions. The
initial conditions have a strong influence on the
walce at intermediate times, but for long enough
time we always get the same results.
Importance of initial conditions
Prof. Spedding's questions about the importance
of initial conditions are of very practical
relevance. As stated in the previous section we
found that the more accurate the initial
conditions, the more accurate were our results
for small and intermediate times. This is
particularly true of the turbulent stresses. As for
the eventual or long-time state we found that the
initial conditions were not very important. Of
course, we have drawn these conclusions based
on a rather limited number of simulations and we
hope to do more simulations in the future to gain
further insight into this issue.
Comparing averages
We agree with Prof. Shedding that our averaging
procedure is different than that used to obtain the
A ~ - ~ ~
OCR for page 533
decay laws in his figure D1 and therefore the
comparisons we made with these laws is not
completely fair. Our averaging technique would
include the energy contained in the internal wave
field which would not be included in the local
averaging procedure that Prof. Spedding uses to
analyze his experimental results.
However, we claim that the internal wave energy
will be approximately constant beginning
sometime early in the NEQ regime. If this is
indeed the case, then the slopes of the curves
shown in fig D1 should be the same using either
kind of averaging although the magnitudes will
not. This is speculation, so we are interested in
doing some simulations to see how valid this
assumption is.
Prof. Spedding's speculation about the effect of
our axial averaging on the difference of the
distributions of turbulent kinetic energy in the
non-stratified and stratified cases may well be
true and is something that we will investigate in
the future.
Comparisons with experimental/field data
Prof. Spedding first asks if there is any evidence
from field tests that submerged wakes have long
persistence times and higher than expected
kinetic energies. We have no knowledge of such
full-scale field tests that are available in the open
literature.
He then asks about comparisons of our vortex
pair simulations with laboratory data and with
data from full-scale aircraft generated vortex
pairs. We have begun to make these
comparisons and will present these comparisons
in the near future.
Representative terms from entire chapter:
vortex pair
