Reynolds stresses and Schwarz' inequality for turbulent shear stresses. We find that the present ..... f_/S. - -0.5 the ske model does not show the effect of rotation on turbulence as it gives ..... 22 Samuel, A.E., and Joubert,. P.N., "A boundary.
/'//--._/_' NASA
Technical
ICOMP-94--21;
Memorandum
106721
CMOTr-94--6
A New k-e Eddy Viscosity Model for High Reynolds Number Turbulent Flows-Model Development and Validation
T.-H. Shih, W.W. Liou, A. Shabbir, Z. Yang, and J. Zhu Institute for Computational Mechanics in Propulsion and Center for Modeling of Turbulence and Transition Lewis Research Center Cleveland,
Ohio (NASA-TM-106?2I)
A NEW k-EPSILON
EODY VISCOSITY REYNOLDS NUMBER
MODEL (NASA.
MODEL FOR TURBULENT
DEVELOPMENT Lewis
AND
Research
N95-I1442
HIGH FLOWS:
VALIDATION Center)
Uncl as 32
P G3/34
August
0022322
1994
Ico__,.)Mr'l National Aeronauticsand Space Administration
A New
k-_
Eddy
Turbulent
T.-H.
Viscosity Flows-Model
Shih,
W. W.
Center and
Model
NASA
Liou,
A. Shabbir,
Reynolds and
Z. Yang
of Turbulence
for Computational
Lewis
High
Development
for Modeling
Institute
for
Research
and
Validation
and
:l. Zhu
Transition
Mechanics
Center,
Number
in Propulsion
Cleveland,
OH
44135
Abstract
A new tion
and
model
k-s eddy
a new
lation
Schwarz'
a set
are examined
include:
a mixing
layers
flows.
model
with
The
model
from
the
standard
that
the
present
model.
shear
(i) rotating
homogeneous
predictions k-E eddy
and
a pressure
round
viscosity
is a significant
model
in this
The
positivity
that
shear
flows;
(iii)
a channel
with are
improvement
and
(iv)
available
the
backward experimental
the
present
and
stresses
model
The
plate step
data.
k-s
with
flows shear
flat
facing
new
formu-
Reynolds
for comparison. standard
The
viscosity
of flows.
flow,
equa-
mean-square
(ii) boundary-free
also included over
paper.
eddy
of normal
We find
rate
of the
new
well for a variety
jets;
dissipation
equation
number.
gradient;
are compared
dynamic
stresses.
perform
model
is proposed
the
can
planar
of a new
the
Rcynolds
coefficients
without
model
on
constraints;
for turbulent
layer,
and
turbulent
consists
formulation,
is based
realizability
inequality
of unified
including
the
which
viscosity
equation
at large
on
model,
eddy
rate
fluctuation is based
and
ary
realizable
dissipation
vorticity
viscosity
that flows
bound-
separated The
results
It is shown eddy
viscosity
1.
Introduction
The eddy
main
viscosity
viscosity well
task
formulation
model,
for
viscosity
does
not
known
anomaly
to the
model
viscosity eddy
not for flows
the
about
but cases
appropriate
rate
to predict
viscosity
model
formulations
for both
significantly
improve
exact
the
be
rate
The
dissipation
rate
equation
can
standard
model
shear
rate
a round
these
purpose
of this
viscosity
be written
as,
rate
equation the
well-
jet is mainly
due
of the
in the
study
k-_
eddy
existing
k-e
is to propose
and the eddy
k-_ eddy
standard
ability
deficiencies
quite
or a massive
For example,
the
eddy
by the
dissipation
to improve
k-e
performs
is overpredicted
rate equation
of the
mean
jet versus
flows,
an appropriate
dynamics,
for turbulence.
In order
performance
a high
of a planar
turbulent
The
fluid
standard scale
removed.
model
dissipation
the
is to provide
equation.
viscosity
length
complex
rate
with
eddy
equation.
should the
the
spreading
model
in computational
In addition,
the
dissipation
model
The
flows
in these
give
dissipation
used
formulation.
always
viscosity
is widely
layer
because
a k-6 eddy
and a model
which
boundary
separation, eddy
in developing
viscosity
new
that
can
model.
2v
where
e = _,uijui,j,
t and
x_.
term
ve,li,
used
for
scales not
All
terms
are new
of turbulence,
energy terms kinetic resulting
equation. which
are
energy model
- 2L'2U_,jt, Ui,:ik
2vui,kuj,_uij and
on the
hand
side
Thus,
they
must
is extremely
difficult.
That
equation which
is, the
assumed
has
dissipation
rate
eddy
equation
(la)
for the
derivatives the
viscous
be modeled
before
this
equation
with.
which
in the
similar
one
to that also
time,
related Eq.
creates of the
k/E.
and
in the
this
following
be
small
is usually
a simple turbulent
dissipation
With
can
to the
and
to
diffusion
(la)
has generation
to the production
be written
are
literature,
Instead,
equation
can
respect
except
unknowns,
turn-over
with
(la),
Therefore,
rate
- 2L'_-_,,]-u_,kU_,k
of Eq.
a structure
to be proportional large
new
to work
dissipation
by the
( ),i stand
right
of these
equation
divided
( ),t,
Modeling
as a useful rate
-
unknowns.
applications.
dissipation
- 2v_,ku_U_,_
e' = uuijuij
the
considered
- 2v_U_,k_
model kinetic
destruction of turbulent
assumption,
the
form:
g2
+
=
-
-
e
-
Ce2-_
(lb)
Eq.(lb) used
is the in
Eq.(lb)
standard
various have
form
turbulence
also been
closure
proposed
lent
flows 1-4
and
rate
equation
based
on the
between
eddies
interactions scale
has
form
from
from
large
free
also
for the
inal
_ equation
(lb).
dissipation but
This
is achieved
mean-square
vorticity
a model
dissipation
at large
Reynolds
The
eddies
rate
and
was
his new
the
more
can
which
statistical
energy
is not only
physically
more
Once
equation
the
be readily
the
standard for the
dynamic obtained
a new
related
dynamic
the
model
to the
orig-
equation
equation
for _
by using
time
turbulent
dissipation
equation
to the
transfer
of some
of deriving
than
due
is of a different
possibility
robust
turbu-
for an inverse
prediction
of
a dissipation
transfer
the
a model
wiwi.
energy
of the
widely
versions
s proposed
equation
in the
been
in near-wall
E equation
physics
we explore
developing
equation
spectral
successful
which
and
fluctuation rate
mimics
equation
by first
Lumley
has
modified
for example,
A new transport with
which
several
applications,
sizes.
study,
simpler
equation
addition,
of non-equilibrium
model
present
also
In
in conjunction
to small
rate
flows 5. Recently,
of different
This
fiows 6. In the
form
concept
of Eq.(lb).
eddies
schemes.
turbulent
suggested
dissipation
for different
in rotating
been
that
shear
of the model
of the
is modeled,
relation
E = vwiwi
number.
standard
eddy
viscosity
formulation
for incompressible
turbulence
is
2
-u uj
=
(2a)
+ T(U ,j +
c. = 0.09 It has
been
known
mean
strain
rate
can
become
strain
(e.g.,
negative
realizability, mean
for long
the rate.
this model
Schwarz' coefficient
of C_, is quite
sublayer
of a flat
0.09
in the
0.05
in a homogeneous
new
formulation
this
paper.
inertial
shear
for C,,
following
which
sections,
flow was
not
different
boundary of Sk/e suggested
stresses
layer
in each layer
by Reynolds
the
and
the
must
7 and
example,
above Shill
development
To insure
be related
= 3.3, and
to the
of large stresses
be violated.
For Sk/e
case
normal
and homogeneous
case.
in which
= 6. According
describe
in the
can
be a constant
on boundary
we will first
non-realizable because
for shear
C_, must
value
the
become
S = _),
inequality
the experiments
show
will
> 3.7 where
In fact,
also
In the
that
Sk/c
and
model
that
(2c)
to the
shear C,
flows
is about
C_ is about
considerations,
et aI. s, is adopted
of a new
model
a in
dis-
sipation The
rate
performance
rotating
of the
jets),
backward
new
a channel
facing
step
equation
The
equation
exact
and
mean
boundary-free
boundary
new
dissipation
viscosity
with
flows
of flows
(e.g.,
a mixing
and without
pressure
formulation. which
include
layer,
planar
gradients,
and
rate
equation
wiwi
"
TT. ( OJiOJi _
•
+ v -y-m
the fluctuating
vorticities
eddy
in a variety
shear
layers
new
for wlw_ is
--y-),t
Ui are
of the
flows.
for
OJiOJi
ui and
development
will be examined
flows, flow,
of the
the
model
separated
Dynamic
where
then
shear
Development
2.1
and
homogeneous
and round
2.
equation,
which
are
•wiwi.
+
-
+ wiwjuid
-
and
defined
1
-
= v(--5-
mean
+ w- U ,j
(3)
uwi,jwi,j
velocities,
and wi and
fli are
the fluctuating
by
wi = eljku_,j,
(4)
_i = eiikUk,j
and 1V. Tennekes first
two
transport
and
Lumley
9 clearly
terms
on
right
vortex
9li_i
with
the
the
fluid,
Reynolds
The
it appears
source
production
hand
same
sign,
fourth with
produced due
and
opposite
by
mean
to fluctuating
respectively. numbers,
the
sixth
and the
meaning
of each
viscous
transport
the
term
is the
vorticity.
This
it will
either
the
sign in the
vortex
and
third
represents
vortex
Tennekes
physical
represent
mean
hence,
term
the
side The
stretching
the
simultaneously. because
described
of bJ_w_, respectively.
fluctuating for
the
1
equation
have
also
shown
terms
sixth
dissipation that,
7./,3 _3/2
._ O[,--_.ut
seventh
are the \
)
is produced
by
equation and and
term
wiwi _'li_i,
represents
terms
to the
largest
The
turbulent
wiwi
fifth
due
the
_i_i
at sufficiently
in Eq.(3) ---
uwi,jwi,j
The
in Eq.(3).
in the
between
and
of order: wiwju_,j,
which
or decrease
for _i_i.
the
and
appears
exchange
The and
seventh
term
vorticity
stretching
term
increase
stretching.
Lumley
source
term
are
the
viscosity
high terms
of
turbulent and
are
All the
remaining
of order Rt
-
and
(u3/l
terms
on the
_) or (u/1)_R_.
is the
turbulent
length
scales
of turbulence,
in Eq.(3),
then
As pointed
out
Reynolds
the
(_i0)i
hand
except
analysis,
number,
TT. ( Wiogi
and
Lumley,
the
l are the of order
be described
1
term,
denotes
terms
j,_+ v_- V),_ = -_(_),j
by Tennekes
second
u and
If the
of _viw_ would
the "O"
and
respectively.
evolution
_
side,
In the above
ul/u
kept
right
are
smaller,
order
either
of magnitude,
characteristic (u s/l 3)Rt
by the
velocity
or larger
following
were
equation,
+ w,_ju_,j- .w_.j_,,j
at very
large
Reynolds
numbers,
(5)
Eq.(5)
becomes,
w_wjui,5 = vwi,_w_,j Or
equivalently,
wlwjui,j
is always
created The
by the
vortex
eddy
production
sizes.
equals
positive.
vortex
In
tends
However,
this
process
effect
of viscosity.
microscale
which
corresponds
2.2
This
can
which
verified
Modeling
of the
dynamic
Modeling
of
from
wiwjui,j.
that
to the
end
length
and
at a certain
that
the
scale
indicates
there
terminal
for the
the
a broad
of eddy eddy
size
size
derivative
term
length
of fluctuating
to create
level
that
is a new
derivative
size of eddies
We expect
be easily
relation
it indicates
the
must
to the
This
is related
to reduce
smoothing
wi,j.
addition,
stretching
stretching
dissipation.
(6)
scale
vorticity. spectrum
because
is the
of
of the
Kolmogorov
of fluctuating
vorticity
Eq.(6).
equation
for
We first
define
wiwi
a fluctuating
anisotropic
tensor
bit using
wiwj wiwj
1_ .
b_ = _,:,.,,_ -_,_
(7)
,_iwju_,_ = b'5w_wkui,j
(8)
then
We
expect
that
that
the
anisotropy
the
anisotropy
the
b_
vortex
stretching
b_ is mainly may
be assumed
tends
due to the
to align anisotropy
to be proportional
vortex of the to the
lines
strain
sij
where
s_j = (u_,j + u_,_)/2 5
the
fluctuating
b,3 _ --, 8 s = (2s_js_ff/:,
with
strain
strain rate
sij.
rate rate;
That
and
hence, is,
(91
This leads to wiwyui,j
o¢ WkWk iiSi Ui"---'---_ 0¢ w_w_
S
(10)
8
If we further
assume
that
w_wk
and
(2sijso)
1/2 are well
correlated,
we may
write
(II)
w_w_ui,_ 0¢wkwk ¢_ Noting
that
wiwi
= 2_
at large
Reynolds
wiwjul,j
Eqs.(ll)
and
should
(12)
both
numbers,
o¢ wkwk
indicate
that
_-_Wi
the
we may
also
write
-- -WkWk WiWi
model
for wiwjui,j
(12)
is of order
(u3/t3)R_/2
as it
be.
Modeling must
be
terms
in
of
of order
wiwjui,j
-
(u_/£3)Rt,
Eq.(5).
because
Therefore,
the
iv_wk
wiwi/_)
in such
wkwt,
w_w_/_)
by an order
be related
to the
following
_'wi,jwi,j. that
model
a way
two
Eq.(5) is the of
that
their
of p_/2.
This
indicates
order
of the
-uwidwi,j difference suggests
that
magnitude
must
cancel
is smaller that
wiwjui,j
the
than
sum
for
_'wi,jwid the
wi, wk
¢_sij
(or
w_wk
_sij
(or
of these
two terms
wkw} -wiwi
k(_
both
the
ratio
of s to
u 2) denotes
the
turbulent
As a result,
the
dynamic
can
terms:
w_w_ S, since
other
S and
the
equation
V
ratio
kinetic
of k/u
energy
and
for fluctuating
wiwi. U / wiwi _ --5-- )'_+ j'--T ''j
(13)
_ + to
_
S is the vorticity
mean can
1 = --2(_)'_ WkWk
are
of order
strain
R_/2.
rate
be modeled
Here,
(_). as
+ Cl_-_S WiWi
(14)
V
Note
that
the
denominator
number
turbulence
it there
in case
This
also
reflects
since k vanishes the
fact
of the the
term
last _
somewhere that
the
term
in Eq.(14)
is negligible in the
parent
term
flow field of the
should
be k/u
compared
to k/u._However,
to prevent model,
for large
unnecessary
Eq.(12),
shows
Reynolds we keep singularity.
no singularity
anywhere
in the
in Eq.(14) model
2.3
flow
models
for either
It should
the last
two terms
individual
Modeling
of the
Noting readily
field.
that
obtain
also
be pointed
in Eq.(5)
out
that
as a whole
the
sum
of last
and should
not
two
terms
be viewed
as a
term.
dissipation
at large
a modeled
rate
equation
Reynolds
number
dissipation
rate
_ = _'wiwi
and
multiplying
Eq.(14)
by _, we
equation, g2
e,, + uj e,j = -(uje'),j The
model
as the
coefficients,
Reynolds
body
number
rotation
ing,
wiwjuij,
the
other
the
calculation
the
evolution
example,
rotating
of e through,
in a decaying must
shear
both
the
flow, the
"source"
two
The
term
types
model
stresses
do not
will
more
robust
used
in conjunction
with
than
the
posed that
by
Reynolds
the
present
stretching
and
than
s which form
present
is based
of the model
dissipation
terms
on the
the model
rate
This
rate
appropriately.
right
rate
especially
of spectral
compared
case
rate
energy describes
For
side of Eq.(15) of homogeneous with
time
be positive.
C2.
Eq.(15),
term.
and the
The
dissipation equation
Reynolds rate when
behaves with
is similar transfer.
so
In fact,
C1 and
the
in
determined.
S normally
term
affected
will also affect
increase
for cases
to
k, as shown
hand
equation,
model
since
equation
be easily
"source"
dissipation
"production"
concept
7
4.1.
coefficients
present
schemes,
of the
dissipation
the
is the
calculations,
form
say
C1 must
dissipation
closure
more
hence
Eq.(lb),
standard
field,
For the
and its dissipation
model
in numerical the
be positive.
Consequently, the
on the
for determining
equation,
second-order
stresses
In addition, Lumley
rate
term
be positive,
present
in Eq.(15).
tion
conditions.
be
dissipation
energy
be used
the
last
C2 can
weak
stretch-
be substantially
flows in section
C2 must
must
turbulent
by solid
vortex
is rather
will first
number
be affected
of fluctuation
effect
of C1 and
only the
kinetic
will
between
appear
shear
hence
in Eq.(15)
of flows l°'n
difference
standard
turbulent
this
of the Reynolds C2 may
reduction
of the
signs
turbulence,
C1 and
stresses
change
homogeneous
be negative,
the
Reynolds
say, k. The
grid
that
however,
in a substantial
of the
and
We note
by BardinaS;
(15)
k+
to be independent
through
For example,
result
is non-zero
these
shown
mechanisms. and
large.
on turbulence
as was
E - c2
C2, are expected
becomes
imposed
by rotation
that
C1 and
+
poor to that
equait is better initial pro-
We believe
turbulent
vortex
Eq.(15) the
turbulent
turbulence
can
be applied
transport closure.
be described
term
Here,
in the
in conjunction
next
(_-_ui),i
we apply section,
with
needs
any level
of turbulence
to be modeled
Eq.(15)
differently
to a realizable
and where
eddy
(6-_ui),i is modeled
closure;
however,
at different
viscosity
model
levels which
will
as
(16)
= The
3.
model
coefficients
Realizable
Shih form
C1, C2 and
eddy
et al. s proposed
represents
ae will be determined
viscosity
later.
model
a realizable
an isotropic
eddy
Reynolds
viscosity
stress
algebraic
equation
model.
Its
(17.1)
= _,T(U_,_+ U_,_)- -_k&_ k2
UT
the
inertial other
coefficient sublayer
hand,
experiment
C_ is not of a channel
a constant.
and
shear Corrsin
uau_2
n.
Shih
et al. s proposed
experimental
layer
flow
as well
suggest
C_ = _____/_k ov /e _y
Based
on the
(a
-- 1,2,3)
(a
= 1,2,3;8-
that
which
realizability
as DNS C_
data
on the
=
0.09.
On
the
is about
0.05
from
the
conditions:
(18)
< 1
_ and
The
flow,
u__>o
Reynolds
(17.2)
or boundary
for a homogeneous of Tavoularis
_ Cf * g
--
linear
model: 2
-_uj
Here
of
the
following
1,2,3)
formulation
for the
coefficient
of Cu:
1 Cu = Ao + A_U (*)k-
(19)
E
In the
formulation
of Shih
et al. s,
U(*) = _f s_i&i + _ _'lij = flij
-
2eijkwk
flij
-
eij_w_
-- flij
(20)
m
where
_ij
velocity
is the
wk.
mean
The
rotation
parameter
rate
viewed
in a rotating
A8 is determined
reference
frame
with
the
angular
by 1
A8 = V_cos ¢,
¢ = _arccos(v_W)
(21) W-
SijSj_Sk_
_3 Calibration
of the
of Eqs.(17),
(19),
(20)
a realizable
model.
Ao
such
as a homogeneous
layer
flow
is a constant,
Cu -
in hope This
For the
and
The
that
sublayer.
which
C_, = 0.09.
only
the
the
value
the
model
1 which
shows
that
to the
standard
form
can
be
closer
to the
of the the
and
form
boundary
layer
homoge,
shear
us go back
to the
value
of C_, also
the
of the
log-law in the
hence
is
for simplicity simple
we choose
= 0.09
Corrsin
flows,
a boundary
of the inertial
inertial sublayer.
11, Eq.(19),
with
A0 = 4.0, gives
of 0.05
that
of the
for
than both
produces
component
the
reasonable
flows
standard is listed
in
b12 compared
coefficients
present
-0.149 -0.274
-0.149
-0.149 -0.142
modeled
k and ¢ equations,
b12
b12
standard
b12
e,t + uje,j = (
•
one
the
b12 (_-_/2k)
1. Anisotropy
k,, + r_jk,_= (_k,j),j
and determine
by
to reproduce to C,
and
If we assume
Here,
exp.
let
is A0.
formulation
of C_. Table
Now
viscosity Eq.(18),
flow.
experimental
present
eddy
constraints
layer
corresponds
anisotropy
new
be calibrated
able
flow of Tavoularis
component
Table
realizability
or a boundary will
The
coefficient
of A0
to A0 = 4.0 which shear
A0.
undetermined
flow
is much
The
satisfies
shear that
leads
coefficient
(21)
then
homogeneous
0.06
model
-0.18
(22)
- u-_V_,j - E
(23)
e,j),j + ClS _ - 6"2k + v_
in Eq.(23).
~
Calibration bulence
at
large
of the Reynolds
model number,
coefficients the
equations
C1,
C2
and
for turbulent
a_.
In decaying kinetic
energy
grid k and
turits
dissipation
rate
6 are g2
Let
the
following
equations
can
be obtained
from
the
a=n+l,
k and e equations: n+l
C2=_
(24)
n
Experiments choose data
1° show
that
C2 -- 1.9 which of homogeneous
which
is found
strain,
the
decay
corresponds shear
exponent
n varies
to n = 1.11.
After
flow n and boundary
to be a simple
function
from
C: is chosen,
layer
of the time
1.08 to 1.30.
ratio
of the
we
we use the experimental
flow to determine
scale
In this study
the
coefficient
turbulence
to the
C1 mean
7: C1 -
max{0.43,
(25)
5 + rl }
where Sk rI = _,
S = _/2Sij
S_j
g
The
value
relations
of ae will be estimated hold
in the
inertial
using
the log-law
1
-
_r
dissipation
flow.
The
following
rate
log u__yy + C
K
---'l.KO
the
layer
sublayer: U
Analyzing
in a boundary
,_
V
2 U.r ,
equation
(26) _u---_OU
,_,
oy
in the
log-law
region,
we obtain
/¢2
ae =
where
the
von Karman
constant
c2
_ = 0.41.
cl The
model
= 1.20
coefficients
(28)
are summarized
.
Table
Uk
Ue
1.0
1.2
2. Model
coefficients
C2
C1
1.9
Eq.(25)
10
Ao
Eq.(19)
4.0
in Table
4.
Model
applications
The shown free
results
in this shear
dients,
section.
flows,
and
different initial
are made
cases
(which
conditions
value
k0, with
the
present
the
able
for
kinetic
to pick
evolution
trend
of these
and
f_/S
-
which with
4.2
the
growth
is already the
LES
the
other
rate
of turbulence
as it shows
Boundary-free
a planar profiles
and from
the
the
flows
the
present
a round model
jet.
not
the
Figures
predictions
gra-
standard
LES
not.
more
rate
the case.
between
hand
decay
the
of the
and the
and
the
11
rate
of the
of the
turbulence
present
model
is able
present than
the
model
the
_2/S
the
For
to pick
as
case
of
the
with
the
a result
agreement
of the for
up
is not
time.
for a mixing
comparisons
measurements
= -0.5.
case,
is in reasonable
performed
the
as it gives
no rotation
energy
is
kinetic
LES
on turbulence
kinetic
were
and
ske model.
of rotation
turbulence
= 0.5 and
model
model
1 (d) compare
rate
present
various
shows
present
1 (c) and
as it did for the
4 show
1 (b)
growth The
both trends
growth
ske models
2, 3 and
case.
case the
of _2/S
effect
energy
show
Figure
the
= 0.296.
this
cases
is a lot better the
that
Figures
The
the
data.
rate
The
by its initial
For
models
LES
shows
_/S=0.50).
and eo/Sko
-- 0.0.
hereafter)
to the
does
and
et al. 5 for four
normalized
of f_/S
no rotation
that
show
other
shear
using
The
kinetic
On the
the
of Bardina
energy,
case
closer
shows
it still
does
are
pressure
and
turbulence
kinetic
for two
agreement
model
data
LES
cases,
model
(ii) boundary-
without
_/S=0.25,
by ske
ske model
no rotation
ske
known.
Calculations
the
flows,
present
simulation
St for the
over
energy
the
although
as it is for the -0.5
cases over
good
same
kinetic
of the
to isotropic
= 0.25.
the
with
results
_/S=-0.50,
model
f_/S
while
and
eddy
e (denoted
is increased
trend
is decreased
time
present
case
energy
up this
first
energy
the
the
layers
of turbulence
k -
the
of turbulence
For the
this
with
large
correspond
evolution
standard
LES,
comparisons
turbulence
the
non-dimensional
and by
the
shear
turbulence
and experiments.
_/S=O.O,
cases
homogeneous
The
LES
new
flows
with are
in all these
compares
exhibited
DNS,
the proposed
boundary
flows.
shear
comparisons of _/S
and
step with
using
(i) rotating
flow
homogeneous
1 (a)
the
a channel
are compared
Figure
the
include
backward-facing
Rotating
The
flow calculations
These
(iii)
(v)
k - _ models
4.1
of turbulent
layer,
self-similar
mixing
layer,
planar
and round
Reynolds
shear
the
results
jets,
respectively.
stress
and
are shown
In these
the turbulent
kinetic
in a self-similar
that
y0.1, Y0.5, and of the
velocity agree
free
profiles well
with
predictions
ske model.
jet
are
are
of the
mixing
the
present
For the
model
round
the experimental
spreading
rates
while
of these
model
yields
round
is always the
anomaly
3. The
Case
2 shows
present
model
model,
with
The
the
is slightly
made
16 and
15.
by the
current
predicts
a much
wider
of the turbulent
level
distributions
than
and
the
overall
for the
the
model
given data.
the
measured
in Figure agree
distribution.
and 4. The
well
distribution. stress
by The
predictions
model
shear
planar
measurements
than
l:todi lr and are shown
predicted
prediction
better
experimental
the
mean
gives
predictions
lower
between
the
to
or the ske model
predictions with
velocity
that
however, stress
compared
well
centerline
Fielder
The
Hekestad
agree
are
shear
levels. are
mean
with
Significant profile
over the
The calculated
with
measurements
and are shown
in Table
predictions
than
the ske model;
especially,
well-known
than
measurements)
layer,
are compared
better
smaller
mixing
jet
the centerline
of planar
Table
model
the ske model
flows
the
local
Figure
present
peak
14, and
velocity
in the
the
For the mixing
of the
and the Reynolds
comparisons
mean
The
predictions
ske
and
the ratio
by either
for their
at the
the
of both
rate
contradicts
level
of the
spreading jet
energy
the
is also achieved in terms
present
and
jet,
data,
ske model
The
of Pate112.
of Wygnanski
distributions
improvement
data
model
velocity
as
respectively.
predicted
The
are presented.
mean
y0.5
where
layer
1_, Bradbury
energy
measurements
profile
3.
for the
Y0.9 - y0.1
and 0.9,
true
profiles
r/defined
y -
locations
0.5,
kinetic
Wyguanski
kinetic
values.
0.1,
is especially
in Figure and
turbulent
the
experimental
This
shown
of Gutmark
the
stream
of the turbulent
the
both
Yo.9 denote
the
energy
coordinate
rl =
where
figures,
and that
round
(i.e.,
the
measured
jet,
but
the
of a planar
is removed spreading
jets
the
spreading
model
prediction
completely.
rates
of turbulent
free shear
flows
measurement
ske
present
layer
0.13-0.17
0.152
0.151
planar
jet
0.105-0.11
0.109
0.105
round
jet
0.085-0.095
0.116
0.094
12
rate
3.
of a
usually
4.3 Channel flow and boundary layer flows Turbulent were the
channel
calculated present out
down
to y+
values
were
were
The shown
used
profile
in Figure
5. This
the present the
number
up
Wieghardt
skin
19.
Both
flow
formance gradient
7 shows
the
flat
and
model
Since
the integration
was
calculations.
channel
At
y+
= 80,
channel
flow and
flow
Re_-
direct
boundary
ske
model
the
give
a slightly
the
layers
with
wall
-- 395
is
simulation.
DNS
data.
with
the
Figure Reynolds
experimental
good
better
at
numerical
well with
is made
gives
flows.
flows.
plate
the
wall,
turbulent
reasonably
comparison
present
in the
turbulent
agree
model
the
by Kim is using
for the
Here,
present
the
results
favorable
to that
of the
studied pressure
better
for the
pressure
gradient
The
21 and
studied
Herring
and
gradient.
ske model.
by Bradshaw
in Figure
results
agreement
prediction
of
with
for
the
boundary
and
flow 2°, which
present
model
gives
layer
under
boundary
turbulent
by Samuel
8 and Figure
The
turbulent
the
Norbury
boundary
Joubert
9, respectively.
layer
22 were
In both
is a boundary
compariable adverse
under
the
pressure
increasingly
also calculated.
cases,
per-
The
present
adresults
model
gives
predictions.
Backward-facing
The through
calculations
other
24) with
calculations finite-volume
discretized terms
by
flows
of the present for two
(KKJ-case
conservative
other
step
performance
to benchmark
the
the
under
are shown
were
coefficient
layer
gradients
for wall bounded
from
wall,
pressure
development.
layer
the
calculated
model
for the
developed
ske model
-" 16000.
Overall,
Figure
4.4
and the
friction
to Reo
experiments.
verse
model
to the
boundary
fully
with/without
away
conditions
turbulent
flow was
flows
than
boundary
for 2D
flows
of the present
= 80, rather
for the
layer
for turbulent
as the
velocity
6 shows
layer
used
boundary
performance
is proposed
functions
Both
to test the
and
model
carried DNS
flow
for complex
backward-facing larger
step
of separated procedure.
a second-order
by the
model
standard
step
flows,
one
both
of which
flows.
The
calculations
accurate
convection and
differencing
13
terms
bounded scheme.
flows
(DS-case
height,
The
central
recirculating
have
2_) with been
were of the
is demonstrated
extensively performed
governing
differencing Sufficiently
smaller
scheme fine
and used
with
a
equations 25, and grids,
all with
201x 109 points in the DS-caseand 199x91 points in the KKJ-case, were usedto establish numerical credibility of the solutions. The computational domain had a length of 50 step heights, one fifth of which was placed upstream of the step. The experimental data were used to specify the inflow conditions, the fully-developed flow conditions were imposed at the
outflow
boundary,
viscous
sublayer
Figures
10-14
the
near
velocity
quantities
free-stream
standard
the wall.
compare
and the mean All
and the
the
Table
were
velocity
coefficients of the
A new
normalized
of the
reattachment
point
to bridge
the
the
lengths.
bottom
downstream
wall
locations.
experimental
reference
locations
ske
present 6.02
6.35
7.50
of the
size
of the
separation
that
the
overall
performance
buble,
the
skin
friction,
and is better
of the
present
model
in this
paper.
It consists
the
pressure than
that
Remarks
k-_ eddy
viscosity
model
is based
eddy
viscosity the
in various
effect
of mean
backward
standard
anomaly
step
equation
especially,
and
The
jets
is expected when
dynamic
rotating layer results
in almost
round
viscosity
it is used
for fluctuating
3 ensures stresses.
show all the
with that
The
cases
is completely to enhance in conjunction
and
The
new
model
vorticity.
The
tested.
The
numerical
stability
with
advanced
more
gradients;
performs
well-known
In addition,
is tested
boundary-free
pressure
model
contains,
model
flows;
without
present
removed. the
model
and
present
shear
the
of a new
realizability
homogeneous flows
14
formulation.
equation
on turbulence
boundary
flows.
eddy
in Section
including:
k - e model
of planar rate
rotation
and flat
facing
on the described
flows
channel
is proposed
a new realizable
formulation
benchmark
flows;
calculations,
the
4.99
equation
dissipation
h and
6.26
rate
rate
at three
7=t= 0.5
dissipation
the
height
along
DS
and
than
distribution
KKJ
equation
and
step
used
of the reattachment
profiles
measurement
rate
shear
pressure stress
by the
4. Comparison
dissipation
as well,
the
comparison
26 was
model.
5. Concluding
new
the
approach
U_f.
suggest
ske
4 shows
skin friction,
Case
comparison
function
as well as the turbulent
Table
The
wall
the
better spreading
new
in turbulent closure
model flow
schemes,
such as
second
dissipation that
rate
the
initially
order
initial
closures.
equation decay
isotropic
We have
into
the
behavior
rotating
also just
finished
LRR 27 second
of k and
order
s and
homogeneous
shear
implementing
the
closure.
effect
flows
the
present
Preliminary
of rotation
model
results
on both
show
k and
6 for
are well captured.
References
1 Jones, with
W. P. and Launder, a two-equation
Transfer,
2 Chien,
16, 1973,
K. Y.,
"Predictions
turbulence
3 Yang,
Z. and
lence,"
T.-H.
application
5 Bardina,
TF-19,
eddy
s Lumley,
W.C.,
simulation,"
T.-H.,
model,"
9 Tennekes,
lo
M.S.
NASA
TM,
and
Vol.
scale
pp.
Stanford,
TM
W.
k -
phenomena
of Heat
and
Mass
a low-Reynolds-
33-38.
for near
of near-wall
wall turbu-
Karman
turbulence
and
its
105663.
C.,
"Improved
turbulence turbulent
models
flows,"
based
Rept.
No.
1983.
Phys.
Fluids,
of turbulence
J. L.
pp.
with
6 model
incompressible, CA.,
for Von
20, 1982,
behavior
on turbulence,"
and Lumley,
J.C.
Journal
layer flow
based
NASA
"Fundamentals
1994,
number
1191-1198.
of homogeneous,
H. and Lumley,
Mohamed
time
modeling,"
Notes
J.,
Journal,
J. H., and Reynolds,
1987,
Zhu,
boundary
J. L.,"Kolmogorov
comments
Lecture
International
and
31, 1993,
University,
J. L,"Some
7 Reynolds,
8 Shih,
Vol.
simulation
Stanford
AIAA
"A new
in turbulence
of low-Reynolds
1119-1130.
of Channel
and Lumley,
J., Ferziger,
on large
pp.
T.-H.,
Journal,
calculation
of turbulence,"
model,"
Shih,
AIAA
4 Shih,
model
Vol.
number
B. E., "The
Institute,
, "A new
for Agard
Reynolds
Vol.
4, 1992,
turbulence Report
stress
pp.203-211.
modeling No.
and
755.
algebraic
equation
to appear.
J. L., A First
Larue,
"The
Course
decay
15
in Turbulence,
power
law
The
MIT
in grid-generated
press
(1972).
turbulence,"
Journal 11
of Fluid
Tavoularis, flow
with
104,
1981,
12 Patel,
S. and
"An
E.
73, 1976,
Vol.
15 Hekestad,
study
of a plane
Journal
turbulent
o.f Fluid
shear
Mechanics,
Vol.
layer," AIAA
mixing
Journal,
Vol.
Mechanics,
its evaluation
Wieghardt,
and
Vol.
serf-preserving
in a plane
"The
two
Journal
of Fluid
Me-
jet,"
Journal
of Fluid
Me-
turbulent
jet,"
Journal
o,f Fluid
dimensional
mixing
region,"
Journal
o.f
pp.327-361.
of analyzing
jets,"
Disa
hot-wire
signals
Information
1975,
in highly
No.
turbulent
flow
and
17.
boundary
University,
Coles,
layer
at constant
AFSOR-IFP-Stanford
pressure," University,
Computation Coles,
D.
J., of
"Equilibrium Turbulent
D. E. and Hirst,
"Equilibrium of Turbulent
55undary Boundary
boundary Boundary
E. A. edt.,
layer
in mild
Layers-1968 Vol.2,
negative
pressure
AFSOR-IFP-Stanford
pp.249-258.
in moderate
Layers-1968
16
layer
positive
AFSOR-IFP
pressure Stanford
of
E. and
2, pp.98-123.
Norbury,
Computation
Computation
41, 1970,
Layers-1968
gradient,"
P.,
of the
H. E.,
"Equilibrium
E. A. edt.,
_1 Bradshaw,
jet,"
pp.721-734.
method
Boundary
H.
turbulent
communication.
K.,
Turbulent
planner
measurements
in round
J., private
"The
structure
Fiedler, Vol.
"A new
I.,
pp.31-64.
32, 1965,
I. and
W.,
"The
"Hot-wire
Vol.
16 Wygnanski,
so Herring,
gradient,"
homogeneous
pp.465-495,
23, 1965,
G.,
Mechanics,
Hirst,
in nearly
temperature
experimental
L. J. S.,
chanics,
19
pp.195-214.
"Experiments
and Wygnanski,
Vol.
14 Bradbury,
18 Kim,
mean
1990,
pp.67-71.
x3 Gutmark,
17 Rodi,
S.,
219,
pp.311-347.
29, 1973,
Fluid
Vol.
Corrsin,
a uniform
R. P.,
chanics,
Mechanics,
gradient," University,
Coles,D. E. and Hirst, E. A. edt., Vol.2, pp.241-248. 22 Samuel, A.E., Adverse
J.,
Thermosciences
25 Zhu,
J.,
Methods.,
2s Launder, Comput.
27 Launder,
Div.,
Vol.7,
B. E. and
and
App.
B.
Reece,
Mechanics,
Flow
of Mech.
Vol.
1985,
Stanford
oscillation-free
shear
and
step",
University,
convection
pp.481-505.
turbulent
of separation
a backward-facing
Eng.,
66, 1974,
layer
pp.163-171.
"Investigation
over
in an Increasingly
of a reattaching
Vol.23,
J. P.,
Developing
reatta_h-
Rept.
MD-37,
1978.
scheme",
Comm.
App.
Num.
pp.225-232.
Spalding,
Meths.
E.,
layer:
Dept.
1991,
of Fluid
Journal,
Johnston,
shear
"A low diffusive
Reynolds-stress
AIAA
S. J. and
Layer
H. L., "Features
flow",
of a turbulent
"A boundary
Journal
Seegmiller,
channel
Kline,
P.N.,
Gradient,"
D. M. and
in divergent
ment
Joubert,
Pressure
23 Driver,
24 Kim,
and
Mech.
Turbulence
G.
D. B.,
"The
Eng.,
Vol.3,
J.,
and
Closure,"
numerical 1974,
pp.269-289.
W.,
"Progress
Rodi, Journal
537-566.
17
computation
of Fluid
in
of turbulent
the
Mechanics,
flows"
Development
of a
Vol 68,
pp.
1974,
5.0
5.0
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18
energy in various
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2.
Self-similar shear
profiles
0.5
1.0
1.5
kinetic
energy;
(c)
"q
11
for
a plane
mixing
layer.
(a)
stress.
19
mean
velocity;
(b)
turbulent
1.2 (a) 1.0
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Figure stress.
3. Self-similar
profiles
for a plane
jet.
(a) mean
velocity;
2O
(b) turbulent
kinetic
energy;
(c) Reynolds
shear
1.2 (a) 1.0
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4.
shear
stress.
Self-similar
profiles
_.,_°
I
0.20
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ylx
for
a round
jet.
(a)
mean
21
velocity;
(b)
turbulent
kinetic
energy;
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Reynolds
1.30
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5.
Turbulent
shear
stress;
(d)
channel
dissipation
flow
at Re,
=
395.
(a)
mean
rate.
22
velocity;
(b)
turbulent
kinetic
energy;
(c)
turbulent
0.005
1.20
1.00
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Figure6. Zero pressuregradientturbulentboundarylayer.(a)mean velocity at /?_e=8900;(b)skinfriction coefficient;(d) displacement
thickness; (d) momentum
thickness.
23
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Figure 7. Favorable pressure at x=4.0 ft.; (b) skin friction
3'.0
4'.0
5'.0
6.0
x (ft)
gradient turbulent boundary layer (tterring and Norbury coefficient; (c) momentum thickness; (d) shape factor.
24
flow).
(a) mean
velodty
0.005
1.2
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Figure 8. Adverse pressure (b) skin friction coefficient;
5.0 x (ft)
61 ...... 0 7.0 8.0
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4
io'i'i 50 60
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25
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ft.:
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Figure x=1.76
9. Adverse pressure gradient turbulent boundary m.; (b) skin friction coefficient; (c) displacement
layer (Samuel and Joubert flow). (a) mean thickness; (d) momentum thickness.
26
velocity
at
3
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2 0 (:D (:D (:D I"
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27
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the
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wall
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(b) KKJ-case .
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11. Static
pressure
coefficient
along
(legend
as in figure
10)
28
the bottom
:
40
x/h
Figure
l
wall
_,_,1,,.,1_
.
(a) DS-cas_ 2
t
,'t .
_,,,i,,,_1,,_11,
I
w g
t-
,4
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x/ 0 -0.5
, .
0.0
0.5
1.0
1
,! .
L==lr.
,
.
p I
,
,
,
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,
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1.0
1.0
f.... .....t i-I
t-
x/h=_
"
.... _I,JP,,, .... , , _L,... ,, .... , -0.5 0.0 0.5 1.0 -0.5 0.0 0.5 1.0
0 -0.5 0.0
U/Uref
Figure
12. Streamwise (legend
mean
velocity
as in figure
29
10)
U-profiles
3
I
x/h=2
2
x/h=5
t-
/ / / /
0 -0.50.0
0.5
Figure
1.0 -0.50.00.51.0 - 100uv/U ref**2
13. Turbulent
sheax
(legend
stress
proKles
as in figure
0.51.0
-0.50.0
in the DS-case
10)
,,,i
,,,i,,,i,,,1_,,
.'1 I
''l'''l'''
I'''
I'''
.1
x/h=20
2
1
o
,I,,,I,,,I,ll
012345012345
01
2345
lO0(uu+vv)/Uref**2 Figure
14. Turbulent
normal
(legend
stress
as in flguze
3O
profdes 10)
in the
DS-case
REPORT
DOCUMENTATION
Form Approve# OMB No. 0704-0188
PAGE
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USE ONLY
(Leave blank)
2. REPORT
DATE
August 4. TITLE
AND
3. REPORT
TYPE
AND
DATES
Technical
1994
SUBTITLE
COVERED
Memorandum
5. FUNDING
A New k-e Eddy Viscosity Model for High Reynolds Flows-Model Development and Validation
Number
NUMBERS
Turbulent WU-505-90-5K
6. AUTHOR(S)
T.-H. Shih, W.W. Liou, A. Shabbir, Z. Yang, and J. Zhu
7. PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)
8. PERFORMING ORGANIZATION REPORT NUMBER
National Aeronautics and Space Administration Lewis Research Center Cleveland,
Ohio
E-9087
44135-3191
9. SPONSORING/MONITORINGAGENCYNAME(S)ANDADDRESS{ES)
10. SPONSORING/MONITORING AGENCY REPORT NUMBER
NASA TM- 106721 ICOMP-94-21 CMOTF-94-6
NationalAeronauticsand Space Administration Washington, D.C. 20546-0001
11. SUPPLEMENTARY
NOTES
T.-H. Shih, W.W. Liou, A. Shabbir, Z. Yang, and J. Zhu, Institute for Computational Mechanics in Propulsion and Center for Modeling of Turbulence and Transition, NASA Lewis Research Center (work funded under NASA Cooperative Agreement NCC3-233). ICOMP Program Manager, Louis A. Povinelli, organization code 2600, (216) 433-5818. 1211. DISTRIBUTION/AVAILABILITY
Unclassified Subject
13. ABSTRACT
STATEMENT
12b.
DISTRIBUTION
CODE
- Unlimited
Category
(Maximum
34
200 words)
A new k-e eddy viscosity model, which consists of a new model dissipation rate equation and a new realizable eddy viscosity formulation, is proposed in this paper. The new model dissipation rate equation is based on the dynamic equation of the mean-square vorticity fluctuation at large turbulent Reynolds number. The new eddy viscosity formulation is based on the realizability constraints; the positivity of normal Reynolds stresses and Schwarz' inequality for turbulent shear stresses. We find that the present model with a set of unified model coefficients can perform well for a variety of flows. The flows that are examined include: (i) rotating homogeneous shear flows; (ii) boundary-free shear flows including a mixing layer, planar and round jets; (iii) a channel flow, and flat plate boundary layers with and without a pressure gradient; and (iv) backward facing step separated flows. The model predictions are compared with available experimental data. The results from the standard k-e eddy viscosity model are also included for comparison. It is shown that the
14.
present
model
SUBJECT
TERMS
is a significant
improvement
over the standard
k-e eddy viscosity
model.
15.
NUMBER
16.
PRICE
20.
UMITATION
OF PAGES
32 Turbulence
modeling
CODE
A03 17.
SECURITY CLASSIFICATION OF REPORT
Unclassified NSN
7540-01-280-5500
18.
SECURITY CLASSI RCATION OF THIS PAGE
Unclassified
19. SECURITY CLASSlRCATION OF ABSTRACT
OF ABSTRACT
Unclassified Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. Z39-18 298-102