NASA/CR-2002-211449
ICASE
Report
No.
Uncertainty
2002-1
Analysis
for Fluid
Mechanics
with
Applications
Robert
W. Waiters
Virginia
Luc
Polytechnic
Institute
Research
Institute,
and
State
Universi_,
Huyse
Southwest
February
2002
San Antonio,
Texas
Blacksburg,
Virginia
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NASA/CR-2002-211449
ICASE
Report
No.
2002-1
V
*
===========================
Uncertainty
Applications
Robert
Analysis
for Fluid
Mechanics
W. Waiters
Virginia
Polytechnic
Institute
Research
Institute,
and
State
Universi_,
Luc Huyse
Southwest
San Antonio,
Texas
ICASE
NASA
with-
Langley
Hampton,
Research
Center
Virginia
Operated
by Universities
February
2002
Space
Research
Association
Blacksburg,
Virginia
Available
from the following:
NASA Center
7121 Standard
Hanover.
for AeroSpace
Drive
MD 21076-1320
(301) 621-0390
Information
(CASi)
National
5285
Technical
Port Royal
Springfield.
Information
Road
VA 22161-2171
(703) 487-4650
Service
(NTIS)
UNCERTAINTY
ANALYSIS
FOR
ROBERT
Abstract.
problems
This
in fluid
probabilistic
and
implemented.
(Interval
Results
Key
laminar
words,
Subject
in the Computational
tbr
Fhfid
Dynamics
HUYSE
methods
their
convection
sensitivity
equation
flow over wedges,
were:
Journal
Code
reasons
devoted
Verification,
uncertainty,
for the increased
The structures
community
whereas
the computational
infancy
of the
computing
discipline
few years,
with
expansion
there
addressing
power
Consequently,
interest
the
primary
to review
1.1.
and
E'r'ro'r:
controls
large
a model
discipline
is to serve
and
recognizable
some
discussed
in this area
1998
issue of
section
of the
design
primary
methods.
:in uncertainty
analysis
due in part, to the relative
However,
decades,
simulations
in that
One
in risk-based
simulations.
stochastic
with
CFD
guide
the
increase
is coming
for engineers
in
of age.
with
an
methods.
discussion
and uncertainty
deficierzcy
key issues
appearing
CFD
the May
of Uncertainty.
as an introductory
analysis
methods,
In fact,
have a long history
two
papers
of credible
is its application
over the past
fbr error
the
is a newcomer
of uncertainty
the basic
A
term,
and an airfbil:
has been mnnerous
Sources
cost, of CFD
of this paper
definitions
non-
are described
a source
corners,
the subject
Among
and
community
to the
applications
topic.
management
improvements
purpose
the AIAA
DEFINITION
in part
software
proceeding
we adopt
in uncertainty
fluid dynmnics
in fluid dynamics
Befbre
paper
and
on the
Certification
and dynamics
and
derivatives)
and
Mathematics
literature
section
Validation,
interest
Chaos)
error
In the past
(CFD)
a special
Code
to fundamental
Polynomial
[1], [2], [6], [7], [10], [21], [30], [33], [351, [36], [38], [39], [41], [42], [49]).
the AIAA
application
methods,
using
APPLICATIONS
t
and
Moment
a model
and Numerical
Motivation.
WITH
flow.
probabilistic,
and
LUC
of error
supersonic
layer
Applied
analysis
Propagation
equation;
classification.
AND
(Monte-Carlo,
Analysis,
boundary
MECHANICS
WALTERS*
uncertainty
are presented
stochastic,
Introduction
W.
Probabilistic
convection-diffusion
two-dilnensional
(see
reviews
dynamics.
methods
non-linear
1.
paper
FLUID
on nomenclature
is warranted.
In this
[3], namely:
i77, a_zy
o7" activity
phase
o.f" 'rn, odelirz9
a_zd
si77zv, latioT_,
that is not due to lack o.f knowle@e.
DEFINITION
1.2.
Uncertainty:
A potential
deficiency
in any phase
or activity
of the modeling
process
that is d'ue to lack of knowledge.
These
definitions
Uncertainty
can be further
uncertainty.
Further
for describing
methods
*Virginia.
under
and
NASA
categorized
propagating
Institute
No.
are
nature
aleatoric
possible
and
NAS1-971)46
Sta|:c
This
uncertainty
while
Ulfivcrsity,
research
SUl)l)orted
stochastic
uncertainty)
in [29].
in models.
Del)artm(mt
was
and the
discussed
condition
was
the. a uth()r
of error
(or inherent
and
form and boundary
walt('.rs((2_m(m.vt.cdu).
Colltra.(.:t
into
parameter
with model
P(_13'techl_ic
(elna.il:
the deterministic
categorizations
for dealing
24()61-()203
VA
recognize
This
nature
and
epistemic
report
focuses
In a companion
report
uncertainty
for the non-linear
of
and
Aerosl)a.(:e
1)y tile
in r(;sidc.n(:e
National
at ICASE,
O(:eml
Langley
and
(or model)
on methods
[29], we describe
Burgers
Engineering,
A(nOlmuti(:s
NASA
of uncertainty.
Bla.cksl)ulg,
Space
I_.eseal(:h
equation.
Center,
Ha.ml)t(m_
23681-2199.
*S(mthwest
This
whih:
r(_s0.a.r(:h
the
author
R(:sear(:h
was
Institute,
SUl)l)(_rte(l
was
by
in residen(:c
P,,(,,lial_ility
the
National
a.t ICASE,
and
Engineering
Aer(mauti(:s
NASA
La.ngley
M,;(:lmni(:s:
;tll(l
Sl)_l(;c
R(:searcll
San
Ant.(mi(_,
Administrati_m
(',elltel,
Haml)t(m,
TX
,re(let
\:A
78228
NASA
(email:
Colltl';i.(;t
23681-21!0!).
V'A
A(lministra.ti,m
llmyse((rswri.e_lll).
N(_.
NAS1-97()4(i
5'o,,.'rce
of
U',.cc'rt,.i',,.t'
0 a.',M
E',r',,',"
i',,. CFD
Sim',d,
TABLF
1.1
ti,,_, ......
s',,.',,.'m.,'ri:_,d
.fwm
SollrCC
Physical
and
Inviscid
Flow
Viscous
Flow
Incompressible
in the PDE)
Chemically
R,eacting
Equation
Physical
Gas
of State
properties
"lTi'ansport
ModeL_"
Chemical
properties
models,
reactions,
Turbulence
\Vail,
Conditions
Geometry
Representation
spatial
In some
where
instances
special
the simulation
1. Physical
and
error
(see
grid.
The
probabilistic
there
sources
for much
e.g.
equation
that
used
a more
and
Blottner
Oberkalnpf
phenomena
genera]
definition
of uncertaillty
that. ilmludes
error
goverr_ed
[36] group
by PDE's
sources
of uncertainty
into four broad
and
error
arising
categories:
studied
impact
methods.
approach
that
discretization
many
error,
in modern
scatt.er
observed
extensively
and
Grid
of geometric
some
errors
1.1 shows
of uncertainty
of the
[40], [41], [43]).
has been
arithmetic
error
of Table
believed
has been
precision
errors
An examination
largest,
Representation
arises.
solution
round-off
4. Programming
account
dependent
modeling
3. Computer
the
issue;
- time
state
Error
we have
of physical
2. Discretization
It is generally
report,
t.hat no ambiguity
In the AIAA
from
&: User
in this
it is clear
Finit,e-
temporal
- steady
convergence
Geometry
Programming
and
convergence
Iterative
Error
Surface
error-
Iterative
& Solution
Round-Off
e.g. far-field
Fret
Truncation
and rates
model
e.g. rougtmess
Open,
Discretization
/:¢5]
Flow
Thermodynamic
Boundary
R,:f.
Flow
Transitional/Turbuleltt
Auxiliar'.v
B/,,tt,,.,",'.
Examples
Modeling
(Assumptions
OIw.'rk,,.,,pf
Although
in this
to rank
the
relatively
area
relative
of error
and uncert.ainty
uncertainty
t_eynolds-Averaged
has been
little
have
frequently
st.udied
1:)3:Darmofal
models
importance
recently,
of the
Godfrey
closme
and
tbr
coefficients
collectively
this error
improving
model
the
blades
base
ltsing
uncertainty,
the continuous
in three
are
[2]. Discret.ization
t'or modeling
turbulence
[19] used
simulations.
uncertainty
[13] tbr compressor
has been done tbr quantifying
[8], [20]. More
data
been proposed
use these
model
simulations
and computational
of' techniques
schemes
arise in CFD
and turbulence
Navier-Stokes
experimental
a number
uncertainty
work
geometric
between
adaptation
sources
turbulence
sensitivity
models:
theBaldwin-Lomax
algebraic
model,theSpalart-Alhnaras
one-equation
modelandthe\¥ilcoxtwo-equation
k - co turbulence
With
the
model.
present
state
example,
for essentially
sufficient
hardware
certainty
as the only
encompass
trated
a large
uncertainty.
layer
2.
Analysis
a range
contain
maximal
error
(IA) is that
of error
Analysis.
bounds
(i.e.,
operations
However,
can result
pointwise
input.
Methods.
using
without
simple
negligible.
For
flows, a user typically
has
zero,
leaving
will continue
further
additional
model
effort
concenconstant
power.
for dealing
problems
un-
eventually
relatively
computing
simulations
model
and
research
will likely remain
with
In this section,
Two deterministic
sensitivity
values
derivatives
Chaos
with error
and ending
worst
to the user.
implement
it should
interval
be pointed
in different
interval
To demonstrate
and three
probabilistic
with
and
laminar
Thus,
analysis
a way that
out that
the output
widths
even if the
this,
consider
the following
f(:c)
m
-- 1 + Z
1) Monte
interval
that
Interval
Analysis
systems
are
such
that
simulation
that
the
details
tool,
such as
by the programming
an interval
mathenmtically
two expressions
of all
about
for computing
expressions
consists
represent
it is supported
expressions
intervals
results
things
computing
on input
interval
one can take an existing
different
and
- 1) Interval
methods
operations
Consequently,
appealing
provided
deterministic
methods
below.
on the input.
way on modern
review
analysis
is to perform
One of the most
in a systematic
are transparent
in such
performed
case results).
analysis
we briefly
uncertainty
are summarized
of the input
of the operations
code, and immediately
quantity
with some
The basic idea in interval
it can be implemented
environment.
starting
trend
uncertainty
be reduced
can be made
essentially
this
can be used in CFD
and 3) Polynomial
of the result
of the interval
a CFD
that
time,
However,
model
can simply
analysis.
the set of all possible
values
that
of error
or laminar
low levels,
Over
sinmlations.
Analysis
for uncertainty
methods,
Interval
possible
of uncertainty.
methods
of Uncertainty
methods
2) Moment
2.1.
error
sources
inviscid
to very
and managernent,
of problems
and 2) Propagation
Carlo,
error
of three-dimensional
we review
some
steady,
flow are presented.
Review
probabilistic
source
estimation
section,
resources,
two-dimensional
discretization
significant
class
Next,
boundary
the
it is not a known
In the next
that
to drive
on uncertainty
over time since
of computational
all one- and
output
equivalent
are equivalent
for
for point
values,
(2.1)
1
g(:c) -- 1 +- I "
and
2C
Table
2.1 shows
defined
the
in terms
results
of performing
of the interval
interval
midpoint
value,
(2.2)
in the table
9(z),
correspond
is substantially
software
bounds
can take
may
interval
- union,
may
through
results
to carry
smaller
immediate
be possible
investment
be obtained
used
for these
two
functions
for input
intervals,
at,
c, by
z = ¥[1 - s, 1 + e].
Values
this
analysis
T, and uncertainty,
to e = 1/10.
than
by careful
complement
and
since
analysis.
out the operations
found
of interval
design
not be prudent
can be obtained
intersection,
the width
advantage
interval
Note that
width
by evaluation
analysis,
construction
of the
methods
exmnple
also illustrates
to the precision
Different
are affected
interval
associated
of f(z).
siglfificant
probabilistic
This
is not related
exactly.
the interval
This shows that,
improvement
operations
provide
1)3"the st,ructural
occurs
the second
relation,
although
existing
in the size of the error
within
much
one other
of the calculation.
output
with
the software.
more
point:
information
the fact
Here:. rational
because
sel; theoretical
.[bT'm of the operations.
However,
than
that
can
different
numbers
were
operations
TABLE
l'll l e'r ml l a.'lm.l!l._i._
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Moreover,
modification
solution
interval
process,
of correlation
examples
2
"T
_
further
inside
between
independent.
and nlust
Since
compute
Propagation
derivatives
may
the widest
variable
fluid dynamics
approach.
point
or smaller
rnethod,
each
codes
iteration
into
and
in the
that,
in the case
it cannot
varying
anahsis.
The
depending
where
take advantage
sortie
for the
modeling,
account
indicate
than
without
rely on iteration
In a probabilistic
be taken
this
be larger
growth
on the
all \-ariables
of this
are
informatioll
bounds.
using
with
in error
can readily
6.2 will illustrate
Sensitivity
in use for man 5, years
is the i t_' indeI:)endent
results
use of this
_]
[5
Since many
is a deterministic
of Error
has been
3).
variables
and
analysis
loops
fl'om the
the uncertainty
interval
necessarily
iteration
random
5.2
t 103'
[5;"]
(see Sect.ion
detracts
of Section
of the correlation,
error,
i-7
arithlnetic
this
practical
2.2.
,,5-6
to the base algorithm
degrees
value
7
Derivatives.
(see e.g• Dahlquist
error
A_i
associated
Error
and Bjorck
with
it. then
propagat.ion
using
[9]) • If ,, = t,.({,
a deterlninistic
sensitivity
c ) where
, ....
c
%.i
%`,,
approximation
to the
by
A'u, is given
I
(2.3)
A_, _
_
L¢=
A computat.ional
which
fluid dynamics
the laminar
conductivity
flow of corn syrup
model,
the geometry,
sensitivity ......equation
results
to emphasize
that
one of their
inputs
2.3.
history
applying
bomb
research•
Carlo
in which
Brush
and
performed
Shortly
equation
modern
this with
tbcus
in this
had
Monte
Laboratory
Carlo
a particular
Handsco:mb
\_;ar II, Fermi,
woblems.
By 1948, Monte
Sometime
thereafter,
who subsequently
techniques
Ulam,
appear
in connection
problems,
methods.
here.
around
primarily
,,;on Neumann
estiinates
both
A briefly
Arguably;
1944 when
as a tool for
and others
began
of the eigeiwalues
to the attention
a 1901 paper
with
(for example;
e'nti're uncertainty
simulation,
summarized
This
\,_,_ wish
Carlo
sinmlation
Carlo
data.
2.4.
this work came
unearthed
the continuous
interval
the
to have its start
transport
the thermal
in Section
of Monte
in [241 and
is considered
of neutron
used
input
framework
applications
model,
eTZ,.,°"
in Eq.(2.3).
contains
is on probabilistic
Carlo,
the end of \Vorld
been obtained.
Monte
and
they
et. al., [11], in
the experimental
T 4-2(_),
a probabilistic
report,
work,
derivatives,
to bound
that
different
simulation
to deternfinist.ic
sensitivity
temperature,
are many
name.
the
in the viscosity
In their
was shown
assumption
measured
there
direct
after
methods
at the Livermore
remarkably
on the
in the work of Turgeon
to uncertainty
conditions.
work
by Hammersley
of the method
of the SchrOdinger
Stephen
the
is given
and Ulam
Monte
is based
Although
is presented
4) to evahmte
in their
We will contrast
probabilistic,
method
development
which
approach
Carlo.
and
of the
Section
At,,
i
subject
t.he boundary
was an experimentally
Monte
yon Neumann
atonfic
this
was analyzed
.
A_!,i2
0 %.i
of' this technique
and
(see
estimate,
due to this variable.
deterministic
the
method
in an error
interval
example
1
t)3 Lord
the Boltzmann
of Dr.
Kelvin
[31]
equation.
i000
samples
I0000
350
300
250
200
150
i00
50
80
60
40
20
samples
....
0.2
0.25
0.3
0.2
0.25
0.3
FIC:. °.I. T!Ipicalh.istog'ra,'m,._
obt.a.i',,edby .sa.'m,
pli',,g.fro'm,o, No,r,m,,,ldLs'l'ribt't/.lio'n,
,wilh,a 're,ca'n, of (1.25 a.'nd_I_.stw/l.dwrd
dcviat'io'l_,
of 0.025co'r'rc.>'po',.di',,.q
to ,, coc./.]i.cic',,t
of _a.ri,.tio',,.CoV = 10%. L_.:.fl- J.l)Ol)._rm.l,l___,
i_i.qh.t10,01]0._a.m.ph._.
Apparently,
the methods
were obvious to Lord Kelvin and consequently
prior to this application,
there are isolated accounts
Here we demonstrate
one of the simplest
Monte Carlo. In this approach,
3. determine
The statistics
deterministic
statistics
from their known or assumed
of the output
distribution,
(mean,
density function
c
c
C
that describes
some event or process and
distribution
(also referred to as
/
=
IE[(_-_)"]
skewness, and kurtosis
=
are related
can be evaluated
problems
deviation
from the
the mean is given by
(2.6)
In the application
can be determined
variable, _, say g((), name,15_
of the distribution
=
some cases, the integrals
density function
the origin) is
(2.5)
The r th moment about
(or basic)
skewness, ...
...)
is over the support of the PDF. The mean of the probability
the first moment about
and standard
/
=
c
The variance,
skewness, kurtosis,
of a random
(2.4)
the integration
(.joint-) probability
e.g., mean, variance,
variance,
value of a function
where p(_) is the probability
referred to as crude
is:
output for each sampled input value(s)
of a distribution
definition of the expected
Even
[23].
of all Monte Carlo methods,
the basic procedure
1. sample input random variable(s)
2. compute
of the method
his focus was on the results.
to the 2"d, 3_d, and 4th moments
analytically,
to follow, we sampled
c_. The probability
-E,"
c
,,) p(_.)d_.
'(_-
in others, the integrals
fi'om a Normal
density function
(Gaussian)
about
the mean.
In
are replaced by discrete sums.
distribution
with a mean #
(PDF) of the Normal distribution,
PN(_), also
denoted N[#, c_],is given by:
C
(2.7)
2o.2
pN( ) -
Two typical samples are shown in Figure 2.1 in which the Gaussian
Frequently,
it is convenient
to use the standard
normal variable.
and a variance of 1. it,s definition follows directly fi'om Eq. (2.7).
shape of the underlying
N[0, 1], i.e., a Gaussian
PDF is evident.
with a mean of 0
Tile
Monte
Carlo
method
has
the
number
of samples,
n --+ oc.
However,
standard
deviation
of the mean
scales
property
that
convergence
inversely
it. converges
to the
of the mean
with
error
the square
exact
stochastic
estilnate
solution
is relatively
root, of the t,he rmmber
as the
slow since
tile
of samples:
(7
(2.8)
c7,, =
Subst, antial
reported
basic
ef[icienc3;
in the
literature
procedure
2.4.
consider
Methods.
CFD
fl'om truncated
a function,
expected
value
basic
[22]: [24]. [25]. [32]).
and correlation
A number
of" applications
expansions
expanded
about
about
the
known
The
as variance
two primary
reduction
nlechanisms
techniques:
are
for improving
the
methods.
using moment
value,
methods
Moment
expected
the mean
have appeared
method
value
of the
approximations
input
4. The first-order
in the literaare obtained
parameter.
accurate
For example.
approximation
for the
of u. is:
(2.9)
_Fo[_,(_)] = _,(_).
N-ot.e that the first-order
first moment
ministic)
at the mean of t.he input.
The
scheme,
(see [26]: [27], [28]: [37]. [44]).
series
u(_).
the
sampling
simulations
Taylor
s over
(see e.g.
are importance
Moment
ture involving
improvement,
_,
value evaluated
second-order
improve
first, moment
(FOFM)
(S()F_[)
apl)roximation
is nothing
Frequently,
requires
more than the t)ointwise
this is referred
t.he computation
of the
(or deter-
to as the deterministic
second
(sensitivit.3')
solution.
derivative,
t.o
the estimat, e of the mean.
(2.10)
Es'o['u.(_)]
For some
whereas
problems
we investigated,
tbr other
approximation
= ",.(_) + __Var(._)
1
O2'u
0--7
due to the higher
the iml)rovement
problems
it was not.
to the variance
of 'u is:
.
Estimates
of the
variance
order
correction
are obtained
terrn was significant
similarly.
The
first
order
9
( )c)_
(2.11)
VarFo[u(#)]
The second
order
estimate
of" the variance
(2.12)
Varso[_,,(_)] =
•
For discussion
as FOSM:
Taylor
moment
ofu
purposes,
SOSM
series expansions
approximat,
of functions
involving
ions to the expected
V_r(_)
approximations
random
variables.
and variance
)'2
O_2
variables
For example,
respectively
to the second
is straightfbrward
moment
through
c
if i_,= "_z
(<{l, s2). the first-order
are
n; ro [,,(_ ,.._,)]= _(_. __).
c
(914)
c-.
VarFo[U(_l
where
the covariance
(a measure
_2 can be defined
in terms
(2.15)
to mult.iple
nmltiple
value
-_
first.- and second-order
E.xtension
(2.is)
and
Wr(_) +
2
_&- _
respectively.
"v_r(_).
0--_ 7
is:
a_,,
.
we shall refer to these
methods
=
•s.,)]
- = _ __
)
.
ere, W
to the extent
of expected
Covar((,,
-that
values
c
<.-2)
c
rr_ + O
( )( )
&.j
_-
-#)G g
sc] and _2cvary., .jointly)., between
as
c
c
IF, [,,_,2
- IE((,)E(_2)]
•
Covar(<l
the random
_'))
variables
_1
TABLE
Q,.,a.'n,t.i,h_._
Frequently,
it is useful
to define
2.2
o.f th, e Sta,_ut.,'rd
No'rm..l
c_ (%)
Quantile(_)
67.000
0.97
90.000
1.64
95.000
1.96
99.000
2.58
99.900
3.29
99.990
3.89
99.999
4.42
and use the correlation
The extension
of Eq.(2.14)
in the standard
error
to n independent
estinmte
_2)
C [-1,
cry, cr_
random
bu, tion.
coefficient,
__ Covar(_l,
P_ ,_ --
Dist'ri,
variables
1].
with standard
deviations
(cry,, ..., cry,, ) results
for cr_....
I
or,, =
(2.16)
_.,
.
ki=l
Note the similarity
Eq.(2.3).
interval,
e.g.
within
to be contained,
knowledge
specified
of the quantiles
fraction
of the point
confidence
intervals
by Turgeon
and hence
than
are shown
and
the
in Table
probabilistic
2.2.
available,
then
substantially
highlighted
estimates
less) than
in Section
measurements
which
can then
design
activities.
representations
the width
deviations
be used to generate
Chaos.
of uncertainty
deviations,
accurate
Recently,
interval,
function
and reliable
several
papers
practice
to distinguish
individual
that
99.99%
intervals
of the location
is the quantile
between
of the interval.
that
probabilistic
have appeared
output
the
input
Eq. (2.;3)
uncertainty
contains
more
data
is
will be less (and possibly
This distinction
repeated
statistics
in the literature
concept
is further
experimental
of the distribution
for multi-disciplinary
(see [14], [15], [16], [17], [47], [48]). An important
a
for various
If experimental
Certainly,
describes
such that
estimate
deviations
fl'om
the location
distribution
of the
-t-3 standard
problem.
of a point
is
mean
can be obtained
the interval
interpretation
smnple
the population
measuring
normal
A(_,, used in Eq.(2.3).
that
(i.e., single)
in which
cr_, can be obtained
convection-diffusion
density
is the
Note
contains
It is common
of the standard
difference
Eq.(2.16),
of the uncertainty
4 for a non-linear
Quantiles
confidence
gives a bound
Confidence
is a measure
of error formula,
a specified
interval
intervals
the median
derivatives.
of the input, standard
will give rise to a probability
Polynomial
probability.
with
a confidence
drawn,
confidence
The important
approach,
-+-4 standard
probability.
A quantile
lie to its left.
that
a randomly-
and mean
with the propagation
uncertainty
Recall
a certain
a prescribed
t,he scale .fi_ctor for the sensitivity
and
interval.
lies to its left,. For example,
50% of the data
99% of the interval
2.5.
with
data
uses an input
in which
of tim distribution.
of the data
such that
intervals
probability
again
uncorrelated
confidence
to lie with
confidence
a pre-specified
assuming
one commonly
is expected
prediction
to lie with
is expected
used
a parmneter
single
expected
estimate
approach,
T ± 1.96c, T, a 95% mean
which
between
of this FOSM
In a probabilistic
investigating
risk-based
spectral
of this approach
is
thedecomposition
ofa randomfunction(o1"
variable)int,oseparable
deterministic
andstochast,
ic components.
Specifically.
fora velocityfieldwith rarlclom
fluctuations,
wewrite.
P
i=0
where
tLi.(;r) is tile deterministic
Effectively,
_i(:c)
is the amplitude
modes represented.
of random
possible.
P -
and
g2i(_) is the random
Here,
random
space
A convenient
The
Hermite
to 1"el)resent. akhough
fbrm of the Hermite
H,,({i,,
polynomials
..-._,i,,)
where ( = (_,:, : ..., s,i,, ) is the n-dimensional
orthogonal
between
the functions
set. of basis
functions
H
= e_, e"'_ (-1)"
random
,_(sic
....
_
in t,he random
(2.19)
,_,,
space.
defined
functio11
I¥(_)
taking
the
basis functions
are
--' : e"e
vector.
As discussed
in [47], there
polynomials
In terms
of the inner
is a one-to-one
torln
,_
a complete
product.,
.fd_'c
(s).q (s)
c I"1';(_)
C'_'
Gaussian
distribut.ion
with
unit variance
1)3;
1
__, _,_
_e
-'" "
Vs(.2rr) ''
l,I,"(_) -
t,he inner product,
of the basis functions
is zero
(2.21)
respect
with
(q2i_9.i)
6Li is t.he I(ronecker
of' t.he distribution
The
other
to span
'- c
• . . 0<,i,,
t.he form of an 'n.-dimensional
(2.20)
where
of
by
.
(fc(_,)f_/(s
),c\=
the weight
basis functions
) and k_i(_i). The Hernfite
,
with
'
over the number
chaos, l)- and the number
use of many
--e
0"
variable
c
1 t
sum is taken
polynomial
the
is given
8_,i
correspondence
discrete
to the i th mode.
corresl)onding
of the order of the polynomial
we use nmlt.i-dilnensional
that_ we wish
(2.18)
basis function
of the i Lh fluctuation.
(n,+_,)! which is a function
dimensions..n.
n-dimensional
part
funct.ion.
can be readily
_: = 1 ..... 'n, rhodes
int.eractions.
delta
are the
The variance
Once
evaluat, ed.
Gaussian
< _9)
mean
estimates
v,a., of the
of the
random
of the variance,
is given
i.e..
(q-,"ij.
t.he modes,
The
of" the distribution
=
to each other,
solution
are known,
solution
is given
all higher
modes
then
statistics
by IEpc,['u]
provide
= uo.
non-Gaussian
1)5
P
(2.2_o)
= E
i.=l
3.
Linear
of int.erval
equation
Convection
analysis
with
of a chemical
with
for both
a source
designed
7- +
where
l//cf,
equilibrium,
time scales
and
to mimic
(3.1)
in which
Term.
time-dependent
term
time scale•
a Source
com;ective
reaction
that
cause
rates
wave
speed
are essent.ially
numerical
::stiffness".
steady-state
focus of' this example
calculations.
chemically-react.ing
l_:f is the fbrward
/)t
o is the
The prilnary
(taken
_ o &,:
This
problem
in which
rat, e. The governing
the
scalar
T plays
equation
wave
the role
is
7,
t.o be o =
inst.antaneous,
\Ve consider
flow problems
reaction
is on the apl)lication
i.e.,
1 tor
all cases
l,:f +
.:_ and
7- _
studied
1)y Godfrey
has been
considered).
Near
0. restflt.ing
chemical
in disparate
[18] in connect.ion
O2
1
1
U
0
1
with
implicit
preconditioning
t,o uncertainty
in measured
algorithms.
reaction
Here,
rates.
(3._9.)
we consider
The
exact
uncertainty
solution
to
in the
this
equation
and
boundary
chemical
time
scale,
7, due
is
V(t- -,:': r)e-xl_.
_(., t):
O,
In
order
taken
to
to
complete
the
definition
of
this
problem,
the
u(x,
(3.4)
_L(0, t) = 1
these
conditions,
the
function,
(3.5)
0) = 1
the
chemical
time-dependent
examine
the
3.1.
right
(3.1).
scale,
from
time-dependent
V t > 0.
x r)=(
a"
-
1.
e -(t-'_/_)/_
t> :±
0 < t <
accuracy.
stage
Runge-Kutta.
outflow
Three
= 0 can
time
and
the
wave
1 is shown
at z =
1 (seen
In
terms
be written
a deterministic
consider
to be
interval
problem,
the
case
upwind
- condition
integration
of the
speed,
a =
in Figure
along
the
o"
1, a graph
3.1.
front
For
side)
the
and
of the
exact
uncertainty
the
steady
deterministic
analysis,
state
we will
solution
(seen
differences
(:c =
methods
steady-state
1) was
were
were
used
prescribed
, R(u)
Euler
-
the
spatial
by approximating
implemented:
residual.
to approximate
a°"
T/z +
explicit,
,u the
7,
&'
_
Euler
Euler
z--1
derivative
=
0 to first
implicit,
explicit
and
method
4for
as:
,L(n+_)
However,
(3.7)
0.9,
First-order
boundary
(3.6)
the
=
0 to t =
behavior
Analysis.
The
order
For
r
t =
_-:
--
edge).
Interval
ut + R(u)
time
solution
the
in Eq.
are
:c C [0, 1],
•
along
respectively,
9, is
9(t
Taking
cond.ition,
be
(3.3)
With
initial
that
in which
= 'tt ('_) is all
there
there
AtR(_t
(''))
is to this
is uncertainty
r = _[1
(Euler
method
in the
Explicit-I).
(referred
input
- c. 1 + c],
to hereafter
chemical
time
as Euler
scale,
r.
Let
Explicit
- 1).
r be defined
where
than
_ is the midpoint
of the interval
a point value at any x location.
of interval
operations
in Figure
3.2. this
One can consider
during
Moreover,
the previous
method
results
alternatives,
(3.9)
(13.10)
midpoint
from
t.he previous
interval
is uncertain.
Since
uncertaint.y
interval
estimates
Method
2 uses the interval
any point
in time,
and hence
bound
The
step
It is common
a vector
norm
maclfine
midpoillt
residual
converge
to the
One of the
However,
uses
the
to the
and lower interval
bounds
stun to time
residual,
methods
norm
in Figure
t.he solution
in t < 2 yet
steady-state
common
hence
reduced
As seen
simulation.
accounts
the exact
than
it gives
time
scale
local
in very
small
t.o be on top of one another.
growth.
from Method
In fact; at
4 is equivalent
"u.(") to evah.mte
intervals
(provided
along
of' the
onh" the
results
for cumulative
values
In this case.
process
however,
we monitored
the
the
the residual
will grow without.
a steady-state
._,- azi,s
{-1.79769
in CFD
and
exists).
using
1. the
of R(77(")).
interval
× 10 a°s, 1.79769
useless.
steady-state
The
is achieved
orders
sirmflations
a time
in terms
_(_l. (''/) is an interval
L2 -'norm
For Method
three
the chenfical
this
of the outpllt
points
the midpoint
3.:2.
is completely
more
appear
of a solution
3.3.
on this COlnputer
although
steady-state
R('_ (')).
values
are shown
in Figure
convergence
since
use the interval
grid
with
derivative,
visually
to steady-state
81 equally-spaced
the
time
problem,
The magnitude
1
residual
t of the uncertainty
1 and 3 both
_:,: = 5, are shown
vector
of zero
a time accurate
For this
and hence
3 converge
with
to monitor
in the
in r.
Method
the
is still an interval
value
uncertainty
growt, h of uncertairlty.
a,_l,
4 evaluate
R('E (''))
midpoint
(integrated)
representation,
most
2 and
at time t. Methods
at t = 1 have been
are close to their
Methods
step.
strictly
to I =
point-wise
the residuals
interval.
four methods
of the tour
maximum
Explicit-4).
of the steady-state
for obtaining
histories
(Euler
"u/") in the time derivative
practice
rather
as a consequence
has no uncertainty.
during
v,(''+_) = 77('') - AtR(_/'_))
1 and yet with
At corresponding
intervals
Explicit-3)
the cumulative
of these
condition
(Euler
the upper
2 uncertainty
in Method
g_'owth of the
is now arl interval
the first iteration
,l/(,zq-1) = 77(,,) _ AtR(u(',,))
due
allow fbr teml)oral
results
even if the initial
= ZL('') -- AtR(-g (''))
4 also
in which
to the Method
step
after
Explicit-2)
time
step
time
in Eq.(3.6)
(Euler
of the
Method
at. each
v.(') is also an interval
in exponential
2Z('n-l)
E is the
a. The residual
namely.
(3.8)
where
with uncertainty
size rapidly
three
methods
at. t = 1. However,
of magnitude
not suitable
The
× 103°s } which
other
and
the values
of
residual
reaches
the
results
in a
smoothly
in all cases.
at tiffs time
values.
4-stage
Runge-I<utta
methods
(and
the one implemented)
can be writ.ten
as
72 = R("a('') + 7_/2).
Y3 - R(u<'') + 72/2) •
(3.11)
74 = R,(u (") + 7:3).
,'__ku
('_) = -- ("q-,
_ (71 ÷ 272 -- ....
z,i3 -F 74).
u(,,+l) = u(,,) + A_t('').
Again,
we face the same
"u,('") vs _('").
the update
Numerical
st.ep yields
For many
CFD
issues as before,
experiments
the most
simulations,
useflfl
namely
confirm
interval
implicit
time
the residual
that
the
analysis
integration
1(I
evahmt.ion,
R('u) vs R(g).
use of /_,(7i) fbr the residual
and the update
evaluations
step,
and
u,('') in
For demonstration
pur-
results.
met.hods
are
preferred.
0.9
08
500
-
Euler
-'
';
i
.....
Euler
Explicit
-
Explicit
-
2
1
0
25O
-
-
'
,
"
,
05
°'6I_
0,3
02
0.2
03
04
-
0.5
0
,,,,,,,,
,,,,,,,
--
0,5
I
Time
1
.
0.9
,
0.8
!
O.7
,
15
2
Time
1
0.9
0.8
Euler
Explicit
-
3
--
Euler
E'<plicit
-
4
0.7
06
=
0,5
I
04
-
I
05
_
"m
+T
o,4
0,3
0.3
-
0.2
01
i'
\
02
Ol
o
i
0
0.5
, _1
)
1
1.5
+
2
0.5
0
1
Time
FI(;.
3.2.
I'n,t,+.."rm+,l+m,a,l'fl.s'L; r_:_._u,
lt.s ,,.si,'n.fl d'_t]'_:'r+:'n.!l;+l,r'i++,t'i,_,.s +m, t,h,_ E,,l_?"
10 2
10o
10
15
2
Time
E:rp/ic'it
Euler Explicit
Euler Explicit
-1
-2
Euler Explicit
-3
Euler Explicit
-4
'm,+;th,od.
\
.2
\
:3
10 4
\
\
tv
,,_ 10 +
\
0
E
__ 10 +
\
\
0
Z
_J_'l 01°
1
0 12
1
0 -14
Y,
I
I
i
I
1
i
i
i
r
I
2
Time
FI(;.
3.3.
C'+m,t;+_ql(:ll,+:_:h,'],slo'ri_,.'_.fo't" l,h.+:'.four
¢'m,pl_m_,c'n,ta,l'i+m,.,," of l,lm Eu,l+"r E:q, li+',il 'm,+_ttmd .f,r
1]
'i'n,L_,'r,t:+da.'nn,//l.,,"i.,,.
poses,
we implemented
the
Euler
implicit
(3.12)
This
time
integration
A--t,
1 ÷ &,-OR)
results
in the
tri-diagona]
bl
o.2
set
scheme
which
in delta
fbrn_
can
be written
as
('')
of equations
c1
/N?I-
b2
A u2
c2
/2(lll)
1
R,('.,e)
(3.]3)
0
(In-1
bn
1
C__l
(in
where
tbr
the
general
case
of wave
speeds
of either
o j ---(3.14)
bo =
C.j
At.
the
inflow
For
positive
by
a fbrward
boundary
boundary:
wave
cl
speeds,
substitution
conditions)
)
__- , _:_.
o.-I_,1
1
,
1
,
I C'l
/
.j
==
2 .....
"I'1 --
I.
2",:r
=
R.('ul)
=
the
above
matrix
reduces
sign,
c,+lal2,.,,._.
=
pass.
/)n
0 and
Likewise,
bl
=
1 and
reduces
to
for negative
to an upper
hi-diagonal
at.
the
a lower
wave
speeds,
matrix
which
outflow,
o,,,
hi-diagonal
the
-1,
b,,. =
matrix
nmtrix
can
=
which
(with
be solved
1:/2(u,,)
cart
appropriate
=
be
0.
solved
change
193, a backward
in
substitution
step.
Using
explicit,
to
the
Euler
see.
the
is only
case•
both
are
Since
the
linear
problem
and
results
steps
that
governing
for large
example,
is linear,
we refer
to as
the
yield
due
to
of' the
1 is shown
identical
the
solution,
tirne
the
the
time
history
in Figure
results
increased
steady
and
in some
3.4.
in terms
from
of
become
available,
the
Although
of interval
number
cases,
met, hod,
solution
is obtained
method.
in Eq.
(3.1.5)
options
implicit
March
derivative
other
steps
Euler
a Time
time
at. a:
residual
Euler
difficuh
size.
evahmtions
The
and
method.
stable
by eliminating
methods
intervals
steady-state
For
a comparisor,_
methods
nmlti-step
in the
equation
which
method
evaluations,
explicit,
in larger
are
applicable.
residual
Runge-t(utta
Euler
in the
interested
of algorithms
for
4-st.age
implicit
update
If" one
March
and
niet.hod
intermediate
use
midpoint
Euler
Runge-I(utta
the
interval
implicit,
The
(3.1),
Eq.
space-marching
(3.12)
in one
other
prirnarily
can
step
alternative
be
through
methods.
used
1)3 solving
with
the
is to implement
In this
At
-+ >c.
resulting
a Space
i.e.,
o....
()d'
Discretizing
this
equation
with
first-order
upwind
(3.16)
T
diffhrences
11.;-
(t, aking
o. > O) yields
_u.j_ ]
1 + ,,x._:
(1 T
where
3.5
u.j -
where
_L(3A:r).
the
advantage
The
results
of' space
of" the
nmrching
Time
March
versus
is evident.
12
Space
March
approaches
can
be seen in Figure
1
0.9
0.8
0.7
Euler Explicit
Euler Implicit.
4 stage Runge-Kutta
.......
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
Time
F IC.
3.4.
T'i'm,_"
_t_;ctl,'nd,
c
"in,t_'rml.l
c_dc'_t,
lM,'io'n,,_"
mith,
Eu, lf::'r
c:l:pl'ir:'it;
R'tm,
fle-I_"u,
1
tl_l,.
_m,d
£tt,
l_l"
'i'm, pl'ic'i/
lim,c
'i'ntcfl'r_H'ion,
'm, eth,
od.s.
1
0.9
0.9
0.8
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
i
F
I
o
FtC.
1,o
_
i
I
,
,
,
0.25
3..5.
St, emt'!l
.s'p_z, cc-'m,(l,'rch,'i'n,!l
._'t_l.Lc
I
I
0.5
X
_'H,t <_'rmd
,
,
sohl,
.
T=ta'4-J-4-J-±$
0.75
t,'io'n,
'u,._'m, fl
I I I
1
t,h,c
EM,m"
,
"i'm, pl'ic'it,
Input
I
0.25
,m,
_
,
t
"i'n,.fi'l_,'ite
Output
,
I
0.5
X
_mJ,_::
,
,
,_l_q_
,
t
('T'i'm,e
I
0.75
i
i
,
Ma,'n:h,)
_
I
1
cO'm
lm.'r<'d
h'
Ca,'rlo
Uncertainty
50
40
30
4O
20
20
10
0.15
3.6.
'mith
=
70
60
60
tio,n,,_.
od
Uncertainty
80
FIC.
'm,c,t,h,
,
'm, cth.od.
100
,s,m_,_tla,
Exact
0.8
Exact
0.7
H'i._tO!l'ra/m,.s
0.2
of
the
,i'n.p,ttt
0.005
0.25
ch,_;'m,i_;a,l
l,im,
c
._c_z, le:.
7.
a,'n,d
l,h,c
o,M,
pu,
l
mlriahlc.
_z
_zl
:r
:
I
.[_'_'t_
1000
Mon,
1
--
x=1/10
0.9
}
x=1/2
_,
o
_\
--
Deterministic
0.8
\\
x=
....
1
95%
Single
CI
0.7
\
\
.........
_,o
0.6
__
ccc
".
_oooo
occ
coo
oo
coo
_-.o'..;o.c
oo_
oo_oooo
.................
0.5
0.4
\
\_\
_
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&_
_
.............
0.3
\
0.1
0.2
\
I
0.25
0
0.5
0.75
1
Time
Fie;. 3.7. Ah'u',. _1l,.,._ ,,.d
.95'.'_;1
._'m.flh"])r(dicl'hm
c
-
0.8
.
O. 7 [
-
95%
....
.......
:,,_
0035.
MC
-
.r h,,tUm._.
Deterministic
-0.9
cou./ffdc'uc_ m lcT't_ol.,,'.f'r(,'m li,m.c-dW)c'udc_J.lAlo'n.lc C_lTlo at lhl_
95%
95%
'
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CI
I-
Single
CI
CI - FOSM
0.03
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I
Mean
_'
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|
....
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a - MC
--
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0.5
0.75
O0
1
0.25
0.5
x
x
FIe;. 3.S. Co'mim'vL_o'n of lh+ Fh'._l-()'nh"v
3.2.
also
Monte
Carlo
implemented.
random
fl'om
For
variable
1000
with
Monte
reached
here
the
for
Monte
±o
clarity.
simulations
"
7 =
The
95%
The
and
mean
coefficient
be seen
are
the
confidence
sensitivity
we
a 5%
from
from
FO
derivative
fl'om
95%
confictence
_[onte
agreement.
Carlo.
intervals
The
from
distribution
the
FOSM
are
of standard
die
at
are
time
(CoV).
three
n_.ethod
tO
'
moment
scale,
Figure
is clearly
:z: locations
of "u, =
roughh'
Cu'v/o ._huu./alhm.._.
First-Order
chemical
condition
moment
37
and
t.he output
compared
required
Or -
Carlo
of' variation
that
initial
_u.
The
took
intervals
the
(3.17)
Monte
it. can
calculations
m decays
results
3.8.
The
methods,
where
Carlo
at. any
Steady-state
in Figure
0.2
simulations
Mom.c'Hl "m,cth<m: ,,'/.lh. ld()O Alo'u/c
Method.
probabilistic
a mean,
solution
at, t =
hloment
these
Carlo
Time-dependent
problem,
m_d
0.75
30
are
implement
1 until
times
the
r.
to be
a Gatlssian
3.6
shows
histograms
non-Gaussian.
in
Figure
its
steady-state
narrower
compared
FO
were
methods
3.7.
and
with
moment
this
value
are
1000
For
not
shown
kionte
Cmlo
method
is
-:r/o.r
(,r 2 e
comparable
deviation
to the
and
single
coefficient
prediction
95c/c confidence
of variation
are
in good
intervals
overall
is
1
0.9
/""
0.8
\
///
0.7
\_A
/ /
0.6
//./
0.5
O. 4
//
U
du/dp
\\
-0.2
-1
-0.4
-2
,,,
4.
Non-linear
to mimic
such problems
exist
proposed
meters
by Rakich
which
(ldl,
2
ttl"tll.'i'll.'i.h'_'L(;
-3
is particularly
and referred
to as the general
the
exact
flux,
stationary
f,
is a non-linear
solution
function
is given
is the
Burgers
model
or non-linear
the
uncertainty
analysis,
are important
to viscosity,
the higher
For all numerical
domains
in CFD
took
in Figure
derivatives
need for high-order
we
Ox
c9 (c9_)
of tt.
the
in the analysis.
tL, are shown
is commonplace,
there
variables.
A more complete
study
simplified
but at lower cost and
on specific
We consider
the
Many
problem
choices
steady
prob-
effort.
convection-diffusion
[4]. Based
behavior.
-
We took
of para-
form
of this
=o
the specific
external
are consequences
rises
oscillatory.
we limited
to
solution,
The solution
or finer grids
discussion
viscosity
The exact
are increasingly
methods
However,
input
(4.1).
computations,
examples.
frequently
non-linear
equation
"u(:c) = 7 1 + tanh
O"u/Off',
the
situation
-50
2
dm".i.m,',/:t.u(:.,,',
fluid dynamicists
complicated
useful
._c.n..,_zl.i.c'ilfl
1
case of f = "u (71 _ u) for which
the
by
(4.2)
that
./_:'r.,'t ff_,'r(:_:
a,',,d
0
x
-1
law form is
L?.f
Oz
where
-40
-2
but one that
in conservation
-30
J I
-1
.ffo/'//,/,'Loll,
linear
\
3
of a more
one can obtain
I
-4_
Computational
featm'es
(4.1)
For
....
Equation.
certain
in the equation,
equation,
_-#_':l:o,cl
Burgers
lems designed
-20
\
,
1
_.1.
-10
/ /
_A
FIC.
10
_
/ / /
A/A
-I
3O
0
-0.8
-2
40
1
-0.6
-
3 /
....
f_'"
/
0.
d3u/du
! "_
2O
=,_
"0
"-.
0
/I
0.3
2
3
0.2
',/
......
d2u/dp
0.4
_
....
4
0.6
_,_
-"
5O
" 0.8
V
"
be
stochastic,
u(z),
to numerically
estimate
the computational
of this with
of this topic
regard
wings,
15
sensitivity
derivatives
from 0 at -oc
consequence
those
domain
and
to boundary
and a random
in [29].
the
and its first three
monotonically
One practical
flow over airfoils,
hence
derivative.s,
with
respect
to 1 at +oc.
of this that
Note
we found
was
derivatives.
to z c [-3, 3]. Truncating
aircraft,
condition
field analysis
configurations
treatment
of this problem
infinite
are
a few
of stochastic
can be tbund
IO:_
0.9
....
0.8
A
9 points
e ....
33
.._,"
points
10 -_,
_f/
0.7
-_ 105
0.6
_10"
r¢
0.5
"0 10"
0.4
m0 10 _
A'
0.3
10""
0.2
10 _:
0.1
_-_
:----
3
Etc.
_.]_ l_((
_
-2
4.2.
A_um,¢.'ri,'ul
h,'L_lov!l
u,._in..q
4.1.
This
-1
....
0
X
in the
'
'
'
= '
'
I '
2
'
'
'
I
3
'method
.for
,
._cu,'ud-o'nh"u
/he
guuc'r,,1
Problem.
linearized
,ccu'r,h
system
(cn,
of' equations
conditions
were
tri-diagonal
specified
.
i
_. ._,Cb.._,..._..'q.-_.-.'q,_
10
of
arm
at
the
set. of equations
were
u
on,
lhrcu
20
"m.¢._h
h'vcl.,(Tcfl).
7!qp,.,I
c,,rcr-
the
used
to solve
update
the
non-linear
problem,
Eq.(4.l).
step
A_z('') - -R(u(')).
used
endpoints.
to soh;e
.,'+'IJcu,
15
iterations
('V_!lhl).
was
_(,,,+1)
differences
lc'n.d
_:q,ol'io,
method
_
centered
.
5
LY,qff'r
Newton's
(4.3)
Second-order
,
10"
Number
.,,olMio?l.,-u,.'&W]
j%TEvll'lO'll',h"
Deterministic
results
'
I ,
1 '
to
apl:_roximate
:z: = 4-3.
fbr each
= _z(.) + A.u(,,).
Dora
Newton
the
the
spatial
exact
iteration
deterministic
(fbr
i
derivatives.
details,
Dirichlet
solution.
see
This
boundary
results
in a
[29])
n('_l
)
ll 1
A'u,2
C "2
(4.4)
0
t --
Ot
_)_ _ --
1
n(-..,,_ :l)
C,u -- 1
1
(1,, n
n0,.)
b n
where
O.j
(4._,)
--
2,",:_:
by
--
.'XT2
(l/2--ui-i)
(1/2-"i-Cj
bl = b,, =
1, and
Numerical
grid
points
three
129
to
solutions
and
different
mesh
machine
Several
common
c_1 = c_ =
mesh
grids
point
= c,, =
R(_I_)
generated
on
spacing
is given
is shown
in the
is on
in the
approaches
approach
were
solution
epsilon
o,,
same
have
is based
the
=
right..
4.1.
Indepertdem
of' iterations
proposed
on Richardsol_
_a:-'
R(u.,,)
of Figure
fbr
i: ....ZL_
_)
'2/k._
"
= 0.
a sequence
in Table
left
number
been
--
Aa:-'
of
uni:[ormly-spaced
The
numerical
4.2.
A typical
of the
level
grids
solution
for
convergence
of' grid
the
for
which
dependent
history,
refinement,
the
mnnber
variable;
"4L,on
corresponding
the
residual
of
to the
converged
(4-1).
establishing
extrapolation
truncation
although
16
other
error
estimates,
estimates.
Probably
such
as mult-grid
the
most
estimates
GCI estimate
Exact
lO -._
10"
10 _
Mesh
,,.. 10"
o
33
%
10 s
129
.N
10 '_
FI
I
._
Q
1o ._
loq
1o•,
........
Fie;.
4.3.
di.scc'rti:-:a,tio'n,
can also
L+',f!
spacing,
!
til_T
;o .....
T
i
-1
+:'rro'r "n,o'r'm, m"r.+'u,.+ 'm,c,.+h,.+'tm,ci'n,fl.
I
1
'
0
X
r
r
r
i
I
2
i
i
i
1
l?i.qh, l - D'L+l'r_bu, tio?+ o.[ +_:/:acl a,'n,d c._'l,hn,ah,d
_r_. th,'rc_.:g'rid._.
Roache
[40] has used
and this technique
we require
I
2
h
- Co'n,,l;+"ql+.:'n,cc o.[ t,h.c L2
c'rro'r
be used.
estimator
that
F
10 0
Mesh
513
10 _
Richardson
extrapolation
will be used here for demonstration
can be obtained
by series
E1 ---
(4.6)
expansions.
E2
ar?'l,d
1 - _'_
modifications
purposes.
The result
_
with
to obtain
The discretization
error
an error
estiinates
is:
--
1 -'r'P
where
El, E2 --=error
estimates
c -- tL2 -_,1;
on the fine grid
the pointwise
':1" and coarse
difference
between
grid
"2", respectively
successive
solutions
]z9
r = /-771> 1; the ratio
p = the order
The
Grid
Convergence
Index
of the numerical
by
used
spacing
R,oache
as
GCI _ & IE, I
F_ is a factor-of-safety
numerical
every
of accuracy
(GCI)
(4.7)
where
of mesh
experiments.
in the domain.
this analysis
to mixed-order
Note
examined,
that
schemes
error
and
co!
error
estimat,
the
GCI
estimate
are available
refinements
or
is
defined
by
_ .F_levi
value of F_ = 1.2,5 (which
estimates
further
discretization
o_.
with a recommended
In all cases
point
a
method.
bounded
was used here)
the
true
on all grid levels.
are possible
tbr high fidelity
discretization
error
at
Roy [43] has extended
using infor:mation
from
additional
grids.
Measures
of accuracy
L.2 - _zorm of the actual
of the
discretization.
distribution
Index
(GCI)
and
error.
Table
of the discretization
discretization
The slope
4.1 shows
error
the
error
are
shown
in Figure
of the line is well known
computed
is shown
values
on the right
estimate.
17
4.3.
The
plot
to be a measure
of the
of Figure
slope
of each
4.3 along
on
the
of the order
line segment.
with
the
Grid
left
shows
of accuracy
The
exact
Convergence
TABLE
S'wm,'m,o,'r!/
Grid
of
grid
co'n'ccrgc'H,c_
Number
Level
Grid
4.2.
P._I(#),
solution
Spacing,
and
0.00350755
1.97940
3
33
0.1875
0.00095693
1.87398
4
65
0.09375
0.00023901
2.00129
5
129
0.046875
0.00006023
1.98851
6
257
0.0234375
0.00001505
2.00001
7
513
0.0117188
0.00000376
2.00000
Problem.
Stochastic
Solution.
t_ = 0.25
and
in terms
expression
following
The
viscosity,
a coefficient
#. was taken
of variation.
of its probability
is rather
wittfin
the definition
intuitive
a specified
of' the Gaussian
PDF.
is shown
It can be seen that
art([ peaked
but
approaches
deterrninistic,
i.e., the
field models
may produce
it. is simply
is constant
Interval
Analysis.
numerical
Pu varies
method
described
stochastic
b_'
that
the
a transfbrmation
probabilits'
ofa
of variables.
can be analytically
evaluated
Using
to obtain
_,,,,+7) _
(u--l)ut.anl_
t(l--2u)'-'
markedly
by the enlarged
with
:r. Near
as :r --, 4-1 flom
which
different
\¥e
is given
a statement
under
this expression
4.4 followed
is zero,
quite
which
exact
variable.
s_
a Gaussian
variance
-- __ = 10_/_. The
function,
2"_
in Figure
of' the PDF.
randoln
p,,: (#).
since
range
to be a Gaussian
CoV(#)
density
t)L'_
("(:_')) =
A plot of' this flmction
the
all random
results,
applied
interval
boundaries,
nearest
boundary.
models
near
analysis
The
input
view in Figure
the
variable
particularly
previously
the PDF
both
predict.
interval
the
object
is highly
At .r = 0, the
the centerline
to
4.5 with more resohltion
skewed
solution
As a side note,
is
random
[29].
exact
solution
was the
and
viscosity,
the
which
CFDwas
to be
(4.10)
/_. = F[1 - c. 1 + ¢ii.
Numerical
results
c = 0.1, are
The bounds
within
with
4.6.
about
interval
results
the
from
tile interval
became
by evaluating
in Figure
things
analysis
evaluated
obtained
shown
attractive
interval
bars
Segment
0.375
(4.9)
defined
of Error
17
(., .....,,_'i.,
oriented
of
2
e
4.2.2.
Slope
-
for this
Eq.(4.1)
L2-norm
Ax
p_.,(_(a:))=l_'u(:c:t_.)-'
variable
cqu_l,tio'n.
0.01383131
can be expressed
random
Bv,'tgc'r',_"
0.75
a mean
Justification
for
Mesh
Points
(4.8)
most
rr:._Ml,._
9
Exact
with
of
a,'n,d
1
Stochastic
4.2.1.
4.1
pwrwmcl(:r._
so large
display
area.
The
the
error
analysis
the nmnerical
midpoint
values,
for the bottom
exact
bars
deterrninistic
indicate
is that
method
Figure
two cases
the
solution,
width
it is very simple
in which
4.7,
both
of the
the
a (-hange
Jacobian
estimates
in scale
(4.2)
output
to implement.
the uncertainty
that
Eq.
for an input
interval.
However,
matrix
are
was required
and
One
value,
of the
as seen in tile
residual
unacceptably
were
large.
to keep the error
X
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0.4
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Interval
Analysis
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4.2.3.
the
0
x
Moment
random
Methods.
variable.
The
In
this
example_
sensitivity
equat, ion (CSE) approach
techniques
tbllows:
Co'n, tiT_m_t.s
1-)ere-written
Se't_.sitivit_q
in quasi-linear
methods
we
(C'5'E).
require
colnputed
and the discrete
Eq'_m, tioT_
tbrrn
moment
The
the
sensitivity
numerical
aI)proximations
adjoint (DA) method.
conservat.ion
law
derivatives
form
with
from
the
A brief description
of' the
Burgers
respect
to
contimlous
of these two
equat,
ion
(4.1)
can
as
1
(4.11)
The
Au,:,:= H_-:,::,:
continuous
,_z = "t_(:_';if).
for the
sensitivity
Defining
sensitivity
s,
derivat.ive,
equation
is derived
_= _)_L/_)tl, and
solve
carrying
,\ = -_ - u,.
differentiating
out.
the
Eq.
different
(4.11)
iat.ion
yields
with
respect
a linear
to
/.L noting
different,
ial
that
equation
namely,
(4.12)
Given
by
where
(A.s,_,):,. = t_(,s,,)_.:,: +.t,.:<:,..
a numerical
Eq.
(4.12).
solution
\\;e
used
of' the
a spat, ial
Burgers
equation
discretization
as
input,
consistent
2()
one
can
use
a variety
of' munerical
methods
with
the
flow
solution
method
a 2 ''t
(i.e..
to
order
accurate
centered
finds
the solution
CSE
equation
CSE since
scheme)
to machine
is identical
one merely
Discrete
problems
on the discrete
precision
Method.
version
(_
10-16).
design
cost function
of the governing
F*=
denoted
is next
_i. After
differentiated
some
with
respect
rearrangement,
(4.14)
the vector
yield
a linear
of Lagrange
system
is arbitrary,
be evaluated
of Lagrange
optimization
denoted
F('u; ;,_) where
multipliers
operating
i.e.,
or more generally,
any generic
variable
term
in parentheses
may
be equated
to zero to
nmltipliers
a-
is known,
0u
the sensitivity
of the cost
function
with
respect
to ;i can
from
--cgF-AT(0R)basic
compute
implementation
is widely
the flow sensitivities,
can be readily
total
of the
for A,
(4.16)
This
the coding
AT(0R)
the first
-_u
the vector
of the
in aerodynamic
of Lagrange
(4.1),
variable,
ATOR.)c)u
(4.15)
Once
used
a vector
Eq.
matrix
ATR(_t; g_:).
to the design
(OF
multipliers
of equations
This simplifies
ahvays
one obtains
OF*
Since
F(u; _i)+
Jacobian
The cost function,
with
in this case
the
iteration
software.
is commonly
is augmented
one Newton
that
equation.
original
is minimized.
equations,
(4.13)
This equation
side in the
method
is linear,
mentioning
of the Burgers
adjoint
variable,
the problem
It is worth
right hand
discrete
a user defined
Since
matrix
the
The
generic
i th
method.
to tile Jacobian
([34]) in which
the
Newton's
has to change
Adjoint
q_ represents
and
number
obtained
of grid
i.e,.
used
c)tL/_)q4,, directly.
by defining
points.
in a design
the
Extending
In this
set of cost
Eq.
environment
(see
problem,
functions,
we require
F = {Fj
(4.16) to a system
[34])
and
precludes
those
derivatives
_ w IJ = 1,. 2, ...n}
of functions
the
and insert.ing
need
to
and they
where
))_is the
the definition
of F
yields
(4.17)
-_u
where
I is the
required
reduces
identity
sensitivity
matrix
derivatives
and
A is a matrix
can now be found from Eq.
c9_
that
Eq.
derivatives
provided
can be solved
be computed
it may
both
because
to machine
(4.17)
can then
Although
occur
consisting
of a set of column
(4.14),
which upon
vectors,
substitution
Aj for each
j.
The
and rearrangement
to
(4.18)
Note
a = -I
methods
not
for all right
hand
by the single matrLx-vector
be readily
are fornmlated
of differences
precision,
efficiently
=AT(OR)
in coding,
the discretizations
apparent,
both
the
and implemented
convergence
CSE
multiply
and
the
consistently.
level, etc.
are consistent,
sides by LU decomposition.
Here,
in Eq.
DA methods
In practice,
however,
and consequently,
2]
indicated
(4.18).
yield
identicM
this generally
the equations
the sensitivity
All sensitivity
results
does
are always
derivatives
not
solved
obtained
/
0.5
0.8
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_rn
on
the
result,
s for
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are
identical.
left
data
three
(COV(/.I.)
_--
in
4.9.
Figure
second
the
the
relative
SOS_I
maximurn
error
order
as
of magnit,
4.2.4.
99%
,rl,Hfl
_'.','.
derivative.
of the
t)y Monte
mean
of the
the
at the
method
ude
reduced
the
input,
FOSM
approximation
variable.
#. into
equation
for
the
time
Consequent.ly.
(_I_.i_.)vI_a.).
to
achieve
Newton's
loli_,t_
o.f
by
half
(:ontains
(77 = 0.2,5)
that
of knowledge
is
shows
of' the
error
input
error
the
input
the
atoo_t
and
relative
in the
standard
first-
first-order
moment
t.he second
derivative
(-orrection
less
As
Adding
results
second-moment
in substantially
deviation
One
and
error
the
4.1.
n|ethods
figure
100%
assuming
advantage
both
of the
300
times
MC
the
three
residual.
6%
simulations
cost.
and
Increasing
the
Eq.(4.1).
expansions
and
t,aking
tor
additional
inner
be seen
the
the
to obtain
on
the
F()S__I
in roughly
number
shown
follows
term
can
2_,
t'or the
the
are
closely
results
at, considerable
chaos
n_ethods
method
error.
is approximately
thousand
deviation
the
and
same
of simulations
cost.
dependent
product.
variable.
{-. @_:} yiehls
I,/_' mode.
\Ve
iteratively
the
machine
method
right
uncertainty
The
in l)oh-nomial
R_, -- 2 Orr
e,i.ia. -
c_._mlmV_fl
h_l_qpr_
output.
b3; approximatel3;
input
of the
1) the
samples).
to viscosity
The
the
at roughly
error
respect
and
standard
the
CPU
m,"ll_mt.';
tm_'.,;)
from
boundaries.
SOSM
exact,
P
the
._C'H,._'H/'_";I'!I
(uv'/o/
error
mean
Sul)stituting
where
('(._rlI./'/'II.[I.()H,.'_
Carlo
in Figure
in estimating
residual
with
the
that
in the
of the
estimate
random
derivative
mean
the
(4.19)
I
0
-1
H.c,n-l.,v_bo.l._il'i.'_l'ic
with
error
Chaos.
the
mrl.Frknl
/h_c.Q
first
in the
Polynomial
t_.. and
an
error
(_
shown
respectively
the
uncertainty
predicted
distribution
moment
estimates
_l/.s_'l_:l_
2) a probabilistic
reduced
distribution
order
lh_
(flo..',h,r:fl
assuming
and
absolute
10(_,)
derivative
second
right,
an
--
The
of
compared
deviations
of the
_
b!/
]ri_,_im.bil'L'.;li_,
A plot
and
(2.3))
in the
A comparison
c(*mt*u.lcfl
o.f
uncertaint.y
standard
seen
fl,,:v/mllim
4.8
the
- Eq.
is clearly
-2
-3
0].
of Figure
(non-probabilistic
0
3
X
(..,'mn./m./rL,.,.on
2: E
estimating
represents
-
I,,,,I
2
1
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methods
shown
0
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was
i=0 j=0
solved
zero
this
equation
steady-state
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P
O2,u.)
-- _
_
eij_:l'i
i=0 j=0
by
solution
by
'90
using
the
the
Euler
was
excessive
linearization
0:,: 2
explicit
relative
of' the
method
but
found
to implicit
residual,
which
that
methods.
can
be
0.0015
First Order
Second Order
MC - 1,000
MC - 10,000
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shown
to reduce
(4.20)
ARt,. -_
O'llt,(!'lLt
_
....
(left)
studied
were obtained
2. the mean
and
the remaining
and
modes
and skewness
One advantage
with
mean
function.
c.'r'ro'r "mth, c ._l,(l,ll.¢t(l,'l'(] dc-cia, tio'n
1
('rigM,)
2
3
u,._"i'n,gAJo'ntc
C,'rlo
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conditions
fl'om the exact
were specified
up to a user-specified
order.
The three
bom_dary
stochastic
conditions
(BC2)
BC1,
(BC3)
implies
that
are set to zero on the
of
the
exact
and solves
solution,
Chaos
For a single
standard
deviation
chaos
the output
variable,
the
u0 is set to the mean
The
second
the
remaining
problem
distribution
is that
I,
1)oundary.
algebraic
random
lUl
the first mode,
respectively,
a simple
of the polynomial
of Polynomial
obtained.
u0 and
_=o j=o
The boundary
and skewness
condition,
variance
BC3 sets u0 to the mean
can be easily
_
0
x
j=o
of the distribution
variance
variance,
The first boundary
variance
1
by matching
(BC1)
3. the mean,
delta
moments
1. the mean
mean
-2
'llL¢:LhodN.
(Szjis the Kronecker
the
_0.06_3
a',,d 'rcla, tivc
MC - 1
MCSC)SM'o00
10,000
-- _----
10
Auz = 20x
by matching
to
I
to
solution
and
005
_
i=o
where
-0.0
_.
_
i
boundary
match
the exact
PDF's
which
first-order
condition
modes
to find the values
are
set.
to
solution
sets u0 and
zero.
ul
Similarly.
of Ul and _2 such that
the
solution.
depend
chaos
of" the exact
on the order
yields
a Gaussian
of the
chaos
distribution
i.e.,
e-(_ ....,,.,,):_/2,,i
--
(4.21)
The
higher-order
second-order
modes
chaos,
account
the result
(4.22)
for non-Gaussian
interactions
and
are reflected
is
PPc._(u) =
V/2_- lu 2 + 4u2 (u - u0 + ,_)1
Further
analytic
representations
for higher
order
chaoses
23
are possible
but lengthy.
in the
output
PDF.
For a
0.0015
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4.10.
Th.c
Numerous
boundary
from
condition
an
order
Note
solution
in
Figure
maximum
129
point
increases,
converged
on
In
the
terms
discrepancy
condition
absolute
513
the
i.e.,
of the
grid
grid.
estimates
of
refinement
standard
similar and very close to the SOS_[ method
standard
deviation,
Carlo
ahnost
all
A further
the
the
that
no
results.
The
the
in the
standard
first-order
and
gives
on
third
using
error
chaos
chaos
the
is grid
in
error
the
exact
order
a slightly
method
mean
(i.e.,
subsequently.
deviation
chaos
polynomial
Ibm" modes
of the
mode
reduction
in the
the
third
different
value
be discussed
first-order
seconcl-order
first
derivatives
will
.._,lul;,'u.
with
expected
sensitivity
appreciable
reduct.ion
deviation,
these
the
3
_::1_1_1
order
first-order
the
lh.c
varying
and
errors
of
2
of
is between
mode
for
makes
larger
converged
nor
is there
is
clearly
the
finer
513
on
both
grids
results
not
grid
point
are
(see Fig. 4.9). The 2 ''_zorder chaos has very small error in the
order-of-magnitude
examination
reduction
of the
rate
error
issues
that
simulations,
there
does
not
the
use
the
convergence
where:
preclude
on
the
left,
convergence,
more
increases
(see
accuracy,
boundary
on
the
of the
reduction
Table
are
and
4.2).
curve
more
utility
smaller
due
convergence
as the
are
than
This
must
domain.
to the
of the
also
curve,
Note
be
L2 norm
the
estimate
order
of the
higher
obtained
shows
consistent
content,
in order
the
boundary
match
grids
the
the
require
are
from
10,000
inherent
level
5Ionte
In
In
as
the
solution
this
the
of accuracy
order
higher
to
order
statement
of the
a given
sought.
statistics
stay
on
chaos
level
Thus.
at the
the
modal
this
"theoretical"
order
to
theory,
practical
although
supports
to achieve
required
in the
conducted.
be observed.
unachieval)le
exact
higher
also
was
4.12
that
with
chaoses
finer
rates
Figure
at
to
PC
should
convergence
supplied
order
that
is increased
inethod.
in order
specified
frequency
such
chaos
to be
that
also
the
chaos
to make
needs
means
increased
of the
polynomial
behavior
simply
versus
order
likely
of the
information
convergence
computational
rate
or
conditions
exponential
error
of the
CFD
aries
this
1
simulations.
exponential
stay
seen
for
grid
is a substantial
the
be
the
-o.oool
........
dc/",;_.,_lhi_:c.,_
0 represents
with
show
However,
There
mode
relative
the
Computations
re-distribution.
point
and
Refining
grid.
fltrther
can
mean
._un.._ili_'H:q
chaoses
modes
treatment
in the
/h,
we summarize
that
higher
that
grids.
coarser
just
129
errors
point
Here
Recall
of the
to
polynomial
grids.
4.10.
shapes
Co'mtm.r_
Hermite
on different,
only
mesh.
significant
using
to boundary
and
than
C'ha_._.
in Figure
of the
The
shows
on
129
and
shown
4.1.
is due
error
the
are
Pol/lnO'mLul
generated
similarity
4.11
2 PC
.flollJ
treatment
the
which
1 and
'modes
were
3 PC
Figure
derivative
grid.
.fou'r
results
mean).
on
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3o
x
X
Fir..;.
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bound-
exponential
solutions.
of
to
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4.11.
A l,,_ottl, tc
Provided
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PDF
the
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that
modal
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and
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that
low-order
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25
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4.13
chaos at z = -2;
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correspond
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exact
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pressure
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the normal
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relation
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Note
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Mathematica
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numbers.
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root
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[46]).
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This
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As can be seen.
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some
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solutions
occur
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finding
procedure
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2_
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to the weak
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for the value
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to physical
the latter
for this
[5].
in the variable
Frequently,
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one generally
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polynomial
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relation
the subscripts'_:l
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5.1.
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displayed.
shock
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46
albeit
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rise plot
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were Monte
computed
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with
with
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The results
of the sensitivity
linear
until
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This
is just
another
although
derivative
the maximum
example
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0/3
are shown
flow deflection
of the highly
non-linear
M,
before
by A. Taylor
on
finite
is shown
the exit
volume
in Figure
0 = 5 ° are
pressure
visually
and wedge
of 10%.
uncertainty
derivative
angle,
The
analysis
the first-order
behind
plane).
the
On 31
indistinguishable
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sensitivity
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ANSERS)
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numbers
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the
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shock
Mach
shock
moment
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at
than
of Figure
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on the right
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oblique
Other
this problem
subsets
with
calculations
from the exact
associated
and
code,
rise.
governing
is shown
to solving
wedge
value from
CFD
of the Mach
(see [44]). A smnple
calculations
the
shock
equations
methods
the CFD
Problem.
uncertainty
arise in the
CFD
over the
of
component
of the pressure
approach
contours
to be the cell-averaged
on the pressure
which
an oblique
discretization
pressure
the
values
the two-dimensional
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grids,
gas)
the Navier-Stokes
upwind
for the normal
the deterministic
rise across
general
we used
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perfect
pressure
A far more
To that
one solves
29
Mathematica
including
numerical
in Figure
5.4.
angle at, which
nature
by hand.
derivatives
Note that
point, the derivative
of fluid mechanics.
version
4 [46]was
could
be used.
the derivative
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Figure
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relations
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mean
two
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,
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of oblique
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0°
[16 °, 24°].
deflection
Clearly,
be drawn
angle
with
for this
Figure
be seen.
The
center
and
M
0 =
20 °, respectively
=
number.
5,
graphics
with
many
and
transport
[-oo,
easier
co].
the
rather
random
distributions
can
shows
In this
the
on
5.6
input
and
to
see,
properties
One
with
are
approach
finite
22
159
143
23
67
73
24
23
24
25
6
8
26
1
2
randolnly
than
hence
for
support
one
may
bounded
to
the
drawing
1000
samples
23 ° (see
Table
5.1)
M
with
histograms
for
that
r
_
r
deflection
_
I
15
angle,
r
t
i
I
20
_
O
Draw
292
= 2, 0 =
mean
are
which
exmnple,
but
variables
Random
309
histogram
academic
I
10
> 0
Samples
21
greater
right
of
Values
492
number
can
_
]
500
when
this,
5.
20
values
Mach
_
d,.,,.t ,..,.c _/.,.c,.l,cr th.a..,,0 (,,.,.t of l r)(]o).
Number
Exact
will
z
O A,la_.
N.,..,,,.l,r,.r of .,',.'/,,.:n/c._f,,,',,,. N/2().2)
smnples
_
Flow
TABLE
is roughly
I
5
0.02
0
all
case,
this
in fluid
from
below
this
(such
Beta
as the
was
are
in the
by
zero
problem
distribution)
is to
samples
well
input
below
sampling.
Such
for
example,
either
truncate
provided
that
than
=
2.
23 °
0 =
for each
purposely
Mach
to
occur
thermodynamic
has
distribution
use of such
support
or
to
a distribution
be justified.
For
demonstration
purposes,
correlations
between
the
upstream
Mach
number
and
15 °
to make
is likely
distribution
the
the
M
flow
illustrate
greater
0M,,_
variables
a Gaussian
numerous
maximum
To further
conditions
applications,
whereas
to the
rejected.
numerous
in non-physical
mechanics
a distribution,
for input
angles
variations
resulted
arise
overcoming
sampled
large
point
output
such
corresponds
of 20 ° where
pressure
of the
we chose
20 ° data
angle
the
from
which
0 were
considered.
use
i00
8O
6O
4O
20
16
20
7060
50
8o!
i
30
2O
40
i0
20
401
h
24
1.8
025
2.2
[]
- - ,_, - -
M=3, 0=5 °
M=5, 0=5 °
----_----
M=5, 0=20 °
2.6
5
...........
7
9
j--V'"
f"_'"
....._...--
0.2
/._""
. 0.15
/"
%
O
./"
_Z_
0.1
....
I
I
-0,5
'
0
I
I
05
1
Correlation coefficient,p
The
result
co,related,
of correlating
these
the combined
model
uncert.ainty.
a rather
rnean
Likewise,
substantial
3!
and
mean
t.o have knowledge
\:ariables
uncertainty
negative
variat.ion
0).
is shown
correlation
with
Clearly,
increases
reduces
correlation
when
of the (-orrelation
in Figure
among
since
the
increasing
the variance,
coefficient
considering
5.T. As expected,
variables
variance.
:_2
particularly
random
in order
A/ and 0 are positively
:l! or O independently
since one tends
oc(:m's
multiple
when
input
increases
to off:set the other.
for the
variables,
to get reliable
stronger
Note
cases
the
that
(larger
it, will be imi)o,'tant
output
estimates
of the
Expansion
Fan
M1
Pl
T1
FIE;.
6.
Prandtl-Meyer
ibr compressible
expansion
output
fan emanating
upstream
6.1.
Prandtl
the
from
number
to be the
Problem.
Mach
for this
line
convex
pressure
u.'n c:rpu.'n..','io'n,
waves
the
form
another
fundamental
flow of a calorically
cornel" as shown
drop
co'ln,c'r.
through
in the
the
perfect
sketch
expansion
building
gas across
of Figure
fan which
6.1.
block
a centered
\hze take
is a flmction
the
of the
angle.
The
solution
by Meyer
given
o'l,cl"
Expansion
we consider
and flow turning
Deterministic
equation
Waves.
a sharp
in 1907 and subsequently
rearward
o.f .,,'u],(_'n,_o'n.'ic ]-to'tt,
In this case,
of interest
Mach
,S'L:ctch.
Expansion
flow theory.
quantity
G.1.
to this
supersonic
in 1908 [5]. The basic
flow properties
ahead
of the
flow problem
problem
expansion
was first
is to determine
corner.
The
presented
properties
governing
by
behind
differential
flow is
dV
(6.1)
dO = V/M 2 - 1
where
V is the
equation
flow velocity
over the entire
where
the
Meyer
function,
notation
and dO is an infinitesimally
expansion
A0 implies
which
(6.3)
the upstream
problem
yields,
total
turning
for a calorically
u(]ll)
Given
the
angle
Mach
perfect
= _/_+1
V _/-1
number,
All,
small
angle
that
expansion
the
Integrating
flow experiences
and
the
differential
L,(]_I) is the
Prandtl-
gas, is
tan-
1 _/__]
V7_-1
(2V/2 - 1)-
and
the flow expansion
tan -1 v/k12angle,
1.
[A0], the
procedure
for solving
is:
1. compute
_(M1)
from Eq.(6.3)
2. compute
_(z1h)
from Eq.(6.2)
3. compute
-/1"I2by soh,ing
4. Use isentropic
For exmnple,
relations
the pressure
Eq.(6.3)
as a root-finding
to compute
decrease,
problem
flow properties
P2/Pl,
can be obtained
P2=
+
[_
2-''2
7--1
53
or find ;I2
in region
froxn
_..A_
i6.4 )
angle.
,_,/-2
2
froln
tabulated
values
this
PIP,.(
0 65 0.67
0.69
0.71
0.73
0.75
077
0 79
081
0.5
The
earlier.
second
approach
Simple
fl'om
a Mach
consisting
of 31
over
a 5 ° expansion
value
and
Monte
Carlo
analysis
the
the
flow
The
solution.
On
numbers
the
approximately
angles_
this
to the
(see
Figure
increases
However.
occurs
at
We
implemented
statistical
quantity.
sense
it is not
that
any
25 -
error
estimates
200
for
{M.
the
The
bootstrap
case
and
bootstrap
estimate
has
the
to a specific
interested
reader
samples
to obtain
mean.
of Math
estimate
with
expansion
by
the
we
the
a coefficient
were
shown
to
the
be
codel
the
upstream
the
Mach
of variation
implemented
in Figure
CFD
refined
of the
of t.he
of 1()_.
using
6.3
closely
we found
grids
both
fl_llows
discrepancies
to 31
standard
an_les.
of
inviscid
6.3.
Nia.ch
sl:andard
in terms
exact
of Figure
increasing
in the
turning
the
with
x 46
for
which
For
Note
deviation
but
of
quite
solu-
turns
of less
at higher
is directly
of the
coefficient
that
moderate
number,
deviation
the
Prandtl-Mever
turning
proportional
pressure
variation
distribution
monotonically
substantial
variation
confidence
intervals
in
the
0} pairs.
nmthod
restricted
of the
the
higher
numbers
contours
comparisons
we took
method
decrease
right
with
uncertainty
Pressure
For
behind
described
solution.
in the
is reflected
FLO\\;.f
simulations.
6.2.
example:
variables
filfite-difl'erencing
first-order
behavior
relative
_kiach
The
estimator.
takes
numbers
the
the
this
wave
used
code
plane.
moment
exact
increases
for the
pressure
exit
pressure
0.99
analysis
in Figure
Consequently.
shown
sensitivity
the
increasing
are
0 95 0.97
flow
used
shock
grids
to the
using
the
the
random
angles.
closely
of
first-order
x 31
093
2
shown
of the
of the
21
turning
Since
a_
_,
the
value
sensitivity
reverses.
6.4).
and
were
value
oblique
the
0.91
was
are
to
computed
pressure
sensitivity
with
output
were
points
ahead
original
compared
0.89
t .5
corner
to be Gaussian
case)
the
and
grid
surface
expected
10 °. pressure
trend
pressure
per
0.87
problem
discrete
Similar
discussed.
{!_I, _} pairs
for
the
angle
this
x 46
the
cell on
(1000
derivatives
Results
than
last
took
expansion
Mach
at all
Sensitivity
tion.
the
simulations
higher
again
Problem.
deterministic
solutions
tbr
at
methods
at. the
we
Stochastic
number
t.o solve
h-meshes
res_flt.s,
cell-averaged
we used
0.85
1
X
3 simulation
Prandtl-Meyer
6.2.
that
0.83
standard
5 flow
of the
advantage
distribution,
is referred
a standard
deviation,
over
standard
t.hat
an
and
expansion
error
it is easy
e.g.
to
to compute
and
efficient
a C.aussian,
[12]
error
coefficient:
cornel'
tbr
and
details
estimate.
of
Table
of variation
with
a mean
to implement:
it can
the
be completel3:
method.
6.1
for
shows
a range
angle
fbr
general
In
automated
practice,
bootstrap
one
standard
of bootstral_
of 20 ° and
an\
in rlte
salnple
a coefficient
Symbols-
0.7
Lines
FLOW.f
-0.01
- P randtI-Meyer
L
_"
""
_
_"_* _--'_" _--'_""_
relations
0.6
/I"
0.5
0.4
-o.oV ..y
---- ,.o,,
....
._c._
-0.06
.......
M a ¢h4
.%//
0.3
\_r_
"'",,.
"" _
_
_
M = 2
0.2
_'_._
""_--...
0.1
"
0
_
L
,
,
I
5
=
_
_
l
I
10
Lines
_
_
_
15
Flow
Symbols-
J
expansion
oo i/
M = 4
M=5
""-!;>
I
I
/".//"
-oo_L//
"" _'k M=3
_I_....._...2_-_..V
k/"
-0
t
09
10
20
angle,
20
30
e
0
FLOW.f
- PrandtI-Meyer
relations
._------
Mach
2
Mach
3
.........
Mach
4
......
Mach
5
/_"
E]
/
/
/
/
/
0.3
//
._-'_
_/
CL
/
_,_
/
/
\_._
"_,.
/
o
0.2
0
7"
/,
>o
0.0_
/°/-I
/
\\
,/
,"
[]
__A__
....
_'
....
Mach
Mach
2
3
Mach
4
Mach
5
\x
\
0,1
\
----I_----
=
i
t
I
10
r
i
Flow
FIC.
f'rom
6.4.
1000
P'rc,_._,.,'n,
Mo'n,t._'.
of variation
estimate
6.6.
over the
coordinates
airfoil
_
I
20
I
5
0
,
_
_
I
10
_
Flow
d_ucia, l i(rn.
6.6 shows
The
Results
trends
are
a,'n.d ('ri!lh/
Airfoil.
This
shape
are given
exmnple
described
intervals
that
) coc._;ci(;'n.!
o.r .ca,,riatio,n,
the
_
_
_
_
expansion
.,ii.h.
I
15
angle,
96 _
,
_
,
;
I
20
0
"_r_],'_
(:O'n.tidc',C_'
i'n.t.(_'r_..l._
similar
to the
This
variable
oblique
steady,
and
correlation
shock
since
the Mach
theory
has been
= _
COSIer(
1
- _)],
surfaces,
y,(x)
35
the
bootstrap
results
increasing
as evidenced
by
{M, 0} pairs
are
among
the
in that
positively
the
expansion
used
extensively
adiabatic
y,,(z)
to model
angle
correlated
results
= -y,(x).
supersonic
flow of an inviscid,
and _(x),
by
y,(x)
from
in
number.
two-dimensional
lower
obtained
is non-Gaussian
p.2//pl
assuming
is intuitive
increasing
6.5 were
of pressure.
random
analysis
by its upper
in Figure
of variation
Shock-expansion
considers
shown
of the
uncertainty.
as does independently
Supersonic
airfoils.
_
in the coefficient
right.
in the greatest
uncertainty
over thin
error
tail to the
result
greater
_
95% confidence
in Figure
in Figure
variables
i
0
(lc.f_ ) S/.a,'.,da'rd
The
histogram
presented
I
15
r
angle,
._i'm,'../,.,l.io'n,,_.
of 10%.
the significant
(7.1)
o...Lt)'.,l."
Ca,'rlo
i
expansion
of the standard
The
7.
,
_>
x E [0, c].
respectively.
perfect
The
flow
gas
airfoil
Number
of
Standard
Bootstraps
Error
in the
Standard
Mean
Error
Standard
in G
Error
in CoV
100
0.000547105
0.000751298
0.0144571
200
0.000598759
0.000673770
0.0138239
300
0.000557259
0.000678446
0.0137223
400
0.000581007
0.000674247
0.0133587
500
0.000600424
O.000691841
0.0134268
0.2
Mach 5 Expansion Flow
Mean
0.15
.....
CoY
/_
950/°
c_
_ _
_
_..-
_-_-_
>0.1
O
0.05
I
6
I
7
Flow
,
I
8
expansion
angle,
,
,
I
9
I
10
e
0.8
120
::
M ach 5
i
110
-
--
100
_
i
0.7
..... ""-
_
.
M=3,
e=5 °
M=3,
M=5,
e=20 °
0=5 °
----
M=5,
e=20
i_ i(i:
:
t_1_ --
iiili:i_
60
.._..-t_,"
/1_. I
0,4
0
/
40 _
:
02
i
°
•
50 -
30
--
0.5
7O
,
.4>"
.....D_ "
::ii:.ii!
::
0.6
80
[]
-- --A--....
_ ....
/
._"
M
//z_
20
0.1
0.3
""
_
._,_,..._._,._..._._._,.-.-_'
lO
0
I
0.025
,
I
ii
0.05
0075
r'Tr_
01
,
_1.
0.125
1
-<-'_--_'_'o'_:'
FIC.
6.6.
L_.fl
];',qlh, l -
Hi.,,loflr..'m,
cm:./_fr'ir"nl
of
- ()..lpu,
vf
mrrmlhm..
th.c
l prc._,,,'_u_vMpu.l
:¥vlu
coC_n.H
p'rcs.,.'u.'r_
th_l
' o''''
Correlation
PiP,
lhu
_l
v.f
di._trZh.lhm.
di..drH_ul_<m
I'(I,'l'HI][(kll
.t
dr."
CO'l"l'('/lll',iO'll
,'II.H_
i..._ .,J<cu:u/
lv
5
the
:36
th,'r_qlh
'r41hl.
vf.f'r_<'-.,d're.m
_.
c.rtm'H,.,_ion
_.lh.h
.,ith.
'o'.5....
coefficient,
.
H,.'ml.
"m.c.'n.
I
p
_ ...d
.WII_
o.f
.,cdfff
20 °
.n_l h . O.
..d
.
IH_;"
0.1
1.7
1.6
" ,,
1.5
"
Upper
Surface
Lower
S urface
0.05
1.4
%
%.
1.3
.%
%
o
_,
_
1.2
_
1.1
.%
0
1
0.9
-0.05
0.8
0.7
0.6
i
-0.1
i
_
i
I
0.25
0
FK;.
7.1.
I
r
i
Stt'/fa.cc
i
I
0.5
X/C
_
r
coo'rdi'n.atc._
I
I
I
0.75
(left)
K
r
a.'n.d
i
A
&
A
I
1
0.25
p're.,'.s'u.'rc
0.0751
A
I
di._t'rit)u.tio'n
_:o'rrc.,'po'n.di_Jg
0.0084
J.
[]
[]
[]
[]
to
.h/_
:
0.75
3
a.'n,d
c_ =
I
I I I IIII
3 °
(rtqh.t).
10-
_,..._,&
9.9-
0.0083
0.0750_
O.5
_C
9.8-
[]
c/c,
[]
0.0082
9.70.0081
0.075
-
[]
A
0.07495
9.6-
C_
C d
0.008
9.5-
o.o079
9.4-
9.30.0078
9.2-
0.0749
0.0077
-
9.1
0.0076
r
0.07485
0
102
7.2.
The shape
actually
three,
C_'i(l
of the
co'n,
Spacing,
ve_fle_J,('c
airfoil
,rc.,'u,
is shown
is due to the aspect
at an angle
Grid
of attack,
convergence
ratio
varies
from
lt._
fo'r
65 grid point
grid point,
process
2 panel
For the stochastic
variable
with
were generated
value
in Figure
a mean
7.1.
,.i'rfoil
Visually,
of 0.5 to the
CI/C_,.
corresponding
the solution
calculations,
the
distribution
prior
[]
r i lilT[
Monte
I
10 _
minimum
airfoil
a,'n.d
o.'n.
stochastic
of 0.000976563.
ratio
and
the
37
frst.-
and
of
five times
2
duprcu._'.
thicker
number
65 grid point
Ten grid
The mesh
levels
were
size. ' h = _'_
c
7.2 shows
the
occm's
fairly
(starting
from
it
(h. = 0.016)
solutions.
"
than
was fL'ced at
on the right.
Figure
'n.--1
results
of this
quickly.
On the
a ver_j coarse
purposes.
was assumed
of variation
I I
is shown
level of refinement
for all practical
,,tt,,(:h,
Mach
on a uniform
g = 0.05,
r
h
o.f'
is displayed
t
ratio,
to the sixth
Spacing,
the free-stream
obtained
I
10":
o,'n.gle
As can be seen, grid convergence
is converged
method
2
from n = 9i+
1.' i = {1,, 2, " ..10}
"
the thickness-to-chord
Carlo
]ll,,(:h,
to generating
value of _(: = 0.05 and a coefficient
1)3' the
,/
For all calculations,
pressure
_z., determined
•
mesh
airfoil),
th, in.._'upc'v._'o',.i(:
of the plot.
on Cz, Ca,, and
(h _ 0.016)
the
were performed
of points.
a maximum
grid refinement
[]
i
Mesh
ct = 3 ° and a thickness-to-chord
studies
used with the number
[]
i
h
(2/.I_ = 3). The non-dimensional
mesh
9[_-
8.%b_
0.0075
10 _
Mesh
FIe_;.
rz
-
to behave
of 1%. Solutions
second-moment
as a Gaussian
random
on a 65 grid point
me, thods.
The
mesh
sensitivity
3
9.2
98_"_
._
1 ,t deriv
7 F_:
_,
6 _-
\
_
("..
5I
_
\\\
"'"'"X
e.=O
40
2"_ deriv
c_=0
.........
1"
c_.=3
.....
2_
deriv
deriv
o:=3
//
•=
35
"E
30
O
"_
__ _
9.15_
_ ....
---Otto
[]
mean
9.1
0
9.05
E
3
\'_\
.7:o
"'X'_X
"
,,
x tt, 'X
I
• -'
\t,
-2
\\
I
'
r
]
,,;,"
"N.
,
,,
.,;
,, "'_..J._
_
,
_
.
{j_
20
_•
._
15
mm
o9
lO
_
5
(n
.o
"-..
9/"
_ _
0
- CI
qt. ----D-_---C]
8.858._
i
DL_h_r!l.
- CI
SOSM
8.9
,
('o'uccqlc',.<:<"
CI
FOSM
8.95
;
,
.....
I
.......
10_
Number
of Simulations
075
_.'(I'FI()
- ........
Cl
single
- - D .....
......
I
0.5
X/C
7.3..4hm.h
03,
,9"
_-.. / /
0.25
FIe;.
25
/5 /
\
-5
i i
mean
The
FO,S'3I
u.',,d
S()S.4/
'r('.+,/l._"
a'r<
._Im,",
./'o'r
I
10
r(:.fcrc,(<.
9.2
9
8.5
9
8
A
8.9
7.5
//
d
b" 7
_8.8
(..)
I]-A7
6.5
///
_
Mean
Cl/Cd
.........
Mean
Cl
_r/
--
.-
Single
2/"
_..
/._._/t
FOSM
-''°
•
Cl
.//>"
8.6
=-
f
8.71
6
Z
t"
t"
_
/_/
CI
8.5i
5.5
,
,
,
,
I
2
....
I
3
t
r
,
I
I
4
....
I
5
8._=
_25'
_
'
'
I
,
2.75
,
,
L
I
3
,
,
,
Deter
Mean
Cl/Cd
CI/Cd
Mean
Single
CI
CI
FOSM
CI
=
I
....
3.25
I
3.5
(7.
Fic.
7.4.
E.rt,<:t<d
derivatives
finite
_;a.lux
o.f lifl-to-dr_ql
with
respect
differences.
The
r'ou.tfi<:ic'nl._
c
1)y this
procedure.
comput.e
directly
vahie
sensit, ivities
t.he differencing
of C_/C<_ and 95% mean
Finally,
attack
the
performance
was varied
angle-of-at.tack
from
correspondence
seen.
The
enlarged
moment
since
7.3. Three
in a single
Figure
and single
airfoil
across
between
view
with
the FOSM
on the
method
results
right
produces
results
output
quantity,
the
confidence,
shows
interval
that
virtually
:c/c
Monte
Carlo
for
thbl.n(._.,'.
_I'LI:/'I,'I/
second-order
accurate
different
two
attack
quantity
can
convergence
angles
of
are required
is no need
l:)e
to
der.ermined
of the
expected
The
angle-of-
intervals.
fl'om
of attack
100 Monte
method
and t.he single
identical
using
i'll
sucli as C'_/C.'<z, there
moment
FOSM
t,armlLo'n,
per angle-of
of the out.put
of angles
the first-order
confidence
clearly
qf
versus
evaluations
derivative
a range
c(,,._:ciu'nl
were con-iput.ed
of 1/4 °. Results
fl'om
<_ J/',/
deri,,;atives
7.3 shows
prediction
to
function
the sensit.ivit.x,
procedure.
of the
sensitivity
1° to .5° in increments
are compared
good
second-order
in Figure
It' one is only int, erested
the pressure
from
pressure
d_u'
methods
"
surface
att, ack (o. = 0 ° and 3 °) are shown
u.'n.c<'rhrh_l!/
by the moment
to L required
upper
u,_lh.
slightly
was studied.
Carlo
in Figure
prediction
underestimates
to FOSM
simulations
for each
7.4 where
confidence
a_:ain
interval
the variance.
and were omitted
for clarity.
is
The
8. Laminar Boundary Layer Flow. Theboundarylayerequations
for steady,two-dimensional
incompressible
flowwith constantproperties
canbewrittenas
(8.1)
0u
Ou
(8.2)
subject
to the boundary
+N0v =0
O_L
U dU
+ 0y =
02u
conditions
(8.3)
It is well known
ordinary
that
differential
In 1908, Blasius
these
these
equation
found
conditions,
partial
differential
for which
a solution
the governing
equations
solutions
to the parallel
equations
f = f(_?) only
conditions
= U(:c)f'O]
transibrm
f(O)
To date,
an analytic
f' _
are
1 as 'q _
correct
solution
is a simple
value
root
analysis
finding
to six significant
Deterministic
for the
freestream
8.2.
input
Reynolds
solution.
numbers
Stochastic
problem
with
can be obtained
(8.9)
that
which,
(Re -
(see [45]).
Under
J
The kinematic
a coefficient
= 1
although
reduces
series solutions
to finding
in our calculations,
fine 5000
of variation
and
the
large
with
value
value.
was solved
a finite
of f"(0)
Finding
by the
radius
such
that
the correct
Secant
method.
viscosity,
of 2%.
••
grid
point
velocity
in Figure
distributed
on 77 C [0, 10] was
x = 1 meter
for three
different
8.1.
u, was taken
The
mesh
profiles
sensitivity
as a normally
derivative
distributed
required
stochastic
for the
FOSM
from
au _ ou o,? _
Or'
0_10u
derivative
a constant
U(x).
to
_7 = 10 is a sufficiently
functions
u,_) are shown
M
,f'(oo)
= 0.469600.
A relatively
(8.8)
second
distributions,
is (see [45])
Similarity
Problem.
= 0
is not known
shows
digits
Problem.
numerical
parameter
method
U(x)
as an
_](z, y) = y v/U2ua'"
the problem
f"(0)
8.1.
The
= if(O)
Numerically,
(8.7)
used
velocity
variable
= 0
where
)
to this equation
established.
oc. An asymptotic
value of f'(O)
with
of a sinfilarity
to
(8.6)
of convergence
freestrealn
form reduce
+ f f"
in terms
and
u(x,y)
The boundary
for specific
flow over a flat plate
f'"
(8.5)
The
exist
in self-sinfilar
(8.4)
where
can be re-cast
(for the
SOSM
method)
Og-u _
Ou 2
u)? f,,(,l).
21/
can be found
U,1 (3f'('q)+
4u 2
39
by further
'l.f'"('l))
application
of the chain
rule to be
/iol
0.9
f(q)
-_
.......
_"
_
_
/
0.8
_-
///i
ii//
f'(q)
1.5
.........
,,
o7-
._o
2
1
/' /
0.6
o.5
r_
Re = 5,000
//1111
i1/
//
0.4
0.3
0.5
,'
/
//
Re=
10,000
Re=20,000
.........
i /
_//i /
0.2
,1'//I
1
0 0
2
3
00,
4
_
'
'
' 0.025
I
_
_
r
0.4
0.0008
0.3
0.0007
/
I/
0.1
'_
'_X\
///
0.2
I
d2u/dv
I ...\
/"
\
I
0.075
....
0.I t
MC- 1o00
--Q--
FOSM
0.0006
2
duldv
I
....
(meters)
\
@
.__
._>
_ 0.05
I
y
q
\
13
0.0005
"_
0.0004
....
-0.1
[] ....
SOSM
0.0003
g
-0.2
0.0002
-0.3
0.0001
r
-0.4
_
I
2
I
4
t
I
I
T
05
0i
I
6
::::::_:::bd_
y/8
q
YI(;.
bo u.H,d_
8.2.
vii
,Vo,-d.ml.c.,.._.,m.n.ol
Plots
of _l_e derivat.ives,
For
this
problem:
are
negligibly
where
t,he
Mean
are
shown
int, ervals
Carlo
dc'rh_ot/rc._
(/c.fl)
small
and
Mont, e Carlo
velocity
be
estimate
of the
parent,
l:_opulation,
at, a: =
8.3.
An
,',.d
mean.
distribut,
smaller
However.
a more
1000
enlarged
and
dL_l.ribu.l.m._
direct
were
since
ions
and
of
lh.c
=
t_r:.lm./l.!l
._l,'nd,'nt
viscosit.y.
t,o the
dcr'mlio,
on
th'm,flh
lh_
plot
of' roughly
for
30
(_
met.hod,_:
t,o Monte
Carlo
intervals.
1()
fi'om
so that
confidence
clarify.
seen
on
1000
the
in
Figure
and
right,
the
Monte
single
intervals
It is worth
V 1_0--0-0) and
only
shown
es of t,he mean
8.2.
variance
of Figure
8.2
for comparison.
ol)t, ained
t,he right
7v. are
estimat,
as
shown
SOSM
n_oment-based
comparison
also
10. 000
is display,:ed
in the
mean
terms
indist, inguishable
are
and
t.he
ive
are
Re
FOSM
omirted
a factor
U and
derivat,
simulations
view
The
by
by
second
1 meter
observed.
int, ervals
are
their
result, s after
readily
int, ervals
of the
hence
profiles
in Figure
can
non-dimensionalized
t.he contribut.ion
confidence
confidence
._c.,..,../l.i,;./t:q
l,/l_"v.
should
estimat,
i:s afforded
e the
1:6; the
Carlo
predict,
coincide
noting
be
standard
single
ion
wit, h
that,
used
realizations
the
for giviitg
deviat,
prediction
confidence
the
_Xionte
95%
the
ion
mean
l_est
of the
confidence
Mean
95% CI
Mean
95% CI
,.. _ =
0.75
0.5
/
/ /
0.25
/
f
0,5
0.75
i
t
t
f
/
f
0.25
/
I
J
07S0r
• a ....
1
I
0.75
y/5
y/6
0.001
MC
i
FOSM
G)
¢..t
"o
-8
0.0008
¢..
o
0.0006
•
,
I
_
I
250
Number
FIC;.
The
convergence
of simulations
are
also
low frequency
standard
needs
Note
a moderately
9.
deviation,
Summary.
to fundamental
for random
a source
in Figure
the
The
of the
accurate
then
estimate
relatively
problems
models
a non-linear
and
standard
and
of the
mean
are presented.
Sources
equation,
by the Monte
estimates
of the
Monte
Carlo
method
versus
the
number
lay on top of one another)
method
very
to accept
acceptable
of uncertainty
supersonic
Carlo
as can be seen by the
accurate
will be unacceptable.
is willing
presented
'm,cl,(-:'l'.s.
(which
If one wants
and probabilistic
Applications
1000
_l,l, y =0.17
SOSM
may yield
deterministic
in fluid mechanics.
sim'_l, lo.lio'n,_
high cost
and
I
750
deviation.
hence
few simulations
reviews
Burgers
Carlo
slow convergence
velocity
I
500
Simulations
predicted
FOSM
slow convergence
This paper
variable
term,
this
of Bqml, te
deviation
8.4.
relatively
meandering
deviation,
standard
of the standard
is shown
shown.
Coilvc.'rfl_m,c_:
8.4.
of
estimates
However,
a fairly
rough
of the
if one only
estimate
of the
results.
uncertainty
and
include
flow over wedges,
analysis
a discussion
a linear
methods
of selected
convection
expansion
corners,
applied
methods
problem
and
with
a thin
supersonic
airfoilaswellasincompressible
boundarylayerflow.
Themethodsdiscussed
andimplemented
are:IntervalAnalysis,Propagation
of errorusingsensitivity
derivatives,
MonteCarlosimulation,MomentmethodsandPolynonfialChaos.Althougheasyto implement,intervalanalysisoftenresultsin maximalerrorboundsthat arequitelarge.Thebasicprocedure
for
implementing
MonteCarlois presented
next. Althoughcomputationally
intensive,MonteCarlosolutions
arefrequentlyusedasa baseline
for comparison
with othermethods
sincetheyareknownto converge
to
theexactstochastic
solutionin thelimit of infinitesamplesize.First-andsecond-order
momentmethods.
popularbecause
of therelativelylowcostandutility in a designem:ironment
arecovered.
Thesemethods
generally
yieldgoodapproximations
whentheoutputprobabilitydensityfunctionis a Gaussian
distribution
or relativelycloseto Gaussian.
Next.Hermitepolynonfialchaos
is described
forsolvingstochastic
problems
involvingrandomvariables.
Themostsophisticated
ofthemethods
reviewed,
polynomial
chaosisbasedona
spectralrepresentation
oftheuncertaintywhichis subsequently
decomposed
intodeterministic
andrandom
components.
Often,highlyaccurate
resultsareobtainablefromthis approach
at lowercostthanXionte
Carlosimulations.
All methodswereimplemented
fora non-linear
formof thegeneralized
Burgersequationforwhichwe
obtainedanexactstochastic
solution.Tomimicthe behavior
of CFDcodes,weusedsecond
orderspatial
differencing
andix-nplement,
edNewton'smethodto soh,ethenon-linear
problem.In allcases,
approximately
teniterationswererequiredto achieve
machine
precision
results.Intervalanalysis
errorboundswereunacceptablylarge,evenwhenperformed
asa singlefunctionevaluation
out,
sidetheiterationloop.Bothfirstandsecond-order
momentmethods
produced
reasonable
estimates
ofthemeanandvariance
but thesecond
orderestimates
weresubstantially
better.Polynomial
chaos
solutions
ofvariousordersweregenerated.
The
first-orderchaossolutionswerecomparable
to the secondmomentsolutions.The third andfourth-order
solutions
wereveryaccurate
andmatchedtheexactPDFof thesolutioncloselyat all point,
sin thedomain.
\\:ealsoshowed
thatthetreatmentofboundaryconditions
andthequalityof thegridhasanimpacton the
error
convergence
Oblique
number
results
shocks
were
expansion
as a function
theory
to simulate
among
Finally,
variable
input
viscosity
folded
random
For this
among
curve
little
in which
We used
the
the
Carlo
flow turning
2 - D CFD
and first-order
the variables
demonstrate
Supersonic
angle
is treated
and
(:ode FLOW.f
moment
the
input
_iach
as well as shock
methods.
Parametric
the need to have knowledge
flow over a thin cosine
of the airfoil
shaped
as a random
variable
airfoil
about.
was studied
and the
impact
is examined.
steady
flow over a flat plate,
The equations
the similarity
example,
variables.
The thickness
was uncertain.
chaos.
considered
variables.
incompressible,
into
of the
flows by the Monte
on the lift-to-drag
we studied
gets
performed.
these
of correlation
of angles-of-attack.
of this uncertainty
were
to be random
the impact
the relationship
kinematic
expansions
considered
showing
at a variety
and
of the order
were solved
variable.
difference
Both
moment
was observed
the celebrated
in self-similar
methods
between
Blasius
variables
and Monte
the first-
flow. in which
for which
Carlo
the random
simulations
and second-moment
the
were
methods.
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