Embed Size (px)
Citation preview
MARCH/APRIL 2007 THIS ARTICLE HAS BEEN PEER-REVIEWED. 21
Stochastic ComputationalFluid Mechanics
S T O C H A S T I CM O D E L I N G
Numerical solution of differentialequations with stochastic inputs hasdirect impact on simulations of flowsystems in at least three areas: uncer-
tainty quantification, stability of noisy systems, andcoarse-grained and multiscale representation. Inthe first category, the past few years have seen anincreasing interest in uncertainty quantification inlarge-scale numerical simulations. In simulations,just as in experiments, we often question ourresults’ accuracy and construct error bounds aposteriori, but the new objective is to model un-certainty from the simulation’s start, not to do itsimply as an afterthought. Researchers have usednumerical accuracy and error control in flow sim-ulations for some time now, at least in the moremodern discretizations, but there’s still an uncer-tainty component associated with the physicalproblem (specifically, with diverse factors such asconstitutive laws, boundary and initial conditions,transport coefficients, source and interaction terms,geometric irregularities, and so on).
We encounter noisy nonlinear systems in manyapplications, from the nanoscale (as in self-assem-bly processes) to the macroscale (such as large, sud-den, disturbances in flow past an aircraft).Bifurcations and chaotic transitions in stochasticdynamical systems can differ greatly from instabil-ities in deterministic dynamical systems.1 How ex-actly extrinsic stochasticity interacts with intrinsicstochasticity caused by the systems’ nonlinear in-teractions is an intriguing and not-well-understoodproblem. In turbulent boundary layers, in which awide range of small scales exists, the mean flowseems to be totally unaffected, even in the presenceof substantial background turbulence.2 In contrast,for other flow systems at a low Reynolds number,even small amounts of noise can have a significanteffect on the mean flow’s structure.3
The third area in the aforementioned list in-volves complex systems with an extremely largenumber of degrees of freedom—models of turbu-lent flows at a very high Reynolds number or atom-istic simulations of mesoscopic size systems.Frequently, we use a coarse-grained procedure toremove degrees of freedom making relatively smallcontributions to the overall system’s energy. Theremoved degrees of freedom are usually accountedfor by stochastic terms in the dimensionally re-duced “effective” or “upscaled” equations. In thedissipative particle dynamics method,4 for example,coarse-graining of the molecular dynamics method
Stochastic simulations in computational fluid dynamics let researchers model uncertaintiesbeyond numerical discretization errors. The authors present examples of stochasticsimulations of compressible and incompressible flows and provide analytical solutions forverifying these newly emerging methods for stochastic modeling.
GUANG LIN, XIAOLIANG WAN, CHAU-HSING SU,AND GEORGE EM KARNIADAKIS
Brown University
1521-9615/07/$25.00 © 2007 IEEE
Copublished by the IEEE CS and the AIP
22 COMPUTING IN SCIENCE & ENGINEERING
leads to a system of stochastic ordinary differentialequations (ODEs) that must be solved efficientlyfor a very large number of particles.
In this article, we present an efficient nonstatis-tical approach for modeling extrinsic stochasticityfor both compressible and incompressible flows.We consider two prototype cases—flow past awedge and flow past a bluff body—and quantify un-certainties associated with inflow disturbances, ran-dom motions, and random geometric roughness.This is a new field, so properly verifying the resultsis particularly important. To this end, we’ve derivedanalytical solutions for small disturbances based onstochastic perturbation analysis, and we employthese solutions in the verification procedure. Vali-dation of stochastic computational fluid dynamics(CFD) models is a much more difficult task becauseit requires very accurate characterization experi-mentally of both inputs and outputs, which is cur-rently lacking in existing data.
Polynomial Chaos and Its VariantsPolynomial chaos represents a stochastic processby a spectral expansion based on Hermite or-thogonal polynomials in terms of Gaussian ran-dom variables. Roger G. Ghanem and Pol D.Spanos pioneered its use in solving stochastic dif-ferential equations by employing a Galerkin pro-jection to derive an equivalent system ofdeterministic equations; this can, typically, besolved with standard numerical techniques.5
Dongbin Xiu and George E. Karniadakis devel-oped generalized polynomial chaos (gPC), whichuses a broader family of trial bases and is based onthe orthogonal polynomials from the Askeyscheme.6 As an example, gPC can express a gen-eral second-order random process T(�) as
, (1)
where � is the random event, and the family {�i} isan orthogonal basis in terms of the random vector�(�) with the following orthogonality relation
,
where �ij is the Kronecker delta, and �� denotes theensemble average with respect to the probabilitydensity function (PDF) of �. For a certain randomvector �, we can choose the gPC basis {�i} in such away that its weight function has the same form asthe PDF of �. Gaussian random variables, for ex-ample, are associated with Hermite polynomials,uniform random variables are associated withLegendre polynomials, and so forth.
Global gPC expansions work effectively formany physical applications, but increasing thepolynomial order might not be efficient for someproblems, such as those with random frequenciesor low regularity in the parametric space. Based ongPC, Xiaoliang Wan and Karniadakis developedthe multi-element extension (ME-gPC), which de-composes the random space into finite elements asin the deterministic finite element method.7 Fig-ure 1 illustrates this idea in the one-dimensionalcase. We assume that the range [a, b] of the 1Drandom variable � is decomposed into elements ek:= [ak, bk]. We then define a new random variable�k, k = 1, 2, …, N, in each random element, ek, as
, (2)
with a re-scaled PDF
, k = 1, …, N,
where N is the number of random elements, f (� ) isthe PDF of �, and
is the probability that � is located in random ele-ment ek. We first approximate the desired randomfield u(� ) locally via gPC, where the degree of per-turbation is effectively decreased by Equation 2from O(1) to O((bk – ak)/2)). Subsequently, wegather the information from all random elementsto obtain the statistics of u(� ).
Because the PDF of � is also decomposed to-gether with the random space, gPC’s orthogonal-ity in the entire random space will be, in general,lost in random elements. However, given an arbi-trary PDF, we can construct the following orthog-
Pr( ) ( )ξ ξ ξ∈ = ∫e f dk ek
ff
eb a
k kk
k
k k( )( ( ))
Pr( )ζ
ξ ζξ
=∈
−2
ξ ζ=−
++b a b ak k
kk k
2 2
φ φ φ δi j i ij, = 2
T Ti ii
( ) ˆ ( ( ))ω φ ξ ω==
∞
∑0
f(�)
a ak bk b –1 1
ek ek
fk (�k)
Figure 1. Schematic of decomposition and mapping in multi-element generalized polynomial chaos (ME-gPC). The procedure issimilar to the deterministic finite element method.
MARCH/APRIL 2007 23
onal system numerically on the fly, from the three-term recurrence relationship
�i+1(�k) = (�k – �i)�i(�k) – i�i–1(�k), i = 0, 1, ��� ,
�0(�k) = 1, �–1(�k) = 0,
where {�i(�k)} is a set of (monic) orthogonal poly-nomials,
�i (�k) = + lower-degree terms, i = 0, 1, … ,
and the coefficients �i and i are uniquely deter-mined by a positive (probability) measure fk(�k).The orthogonal system {�i(�k)} will then serve as alocal gPC basis. For problems related to low regu-larity in the parametric space, we use the ME-gPCmethod adaptively to achieve high efficiency.8
Galerkin versus Collocation ProjectionFor simplicity and clarity, we consider the stochas-tic equation L(x, t, �(�); u) = f (x, t; �(�)) with a gen-eral (nonlinear) differential operator L, where x ��d, d = 1, 2, 3, indicates the physical space and t thetime. In the Galerkin formulation, we first applythe gPC expansion to
and
.
Here P is the total number of basis modes. Then,performing the Galerkin projection on both sidesof the equation, we obtain
, (3)
where k = 0, 1, …, P – 1. In contrast, in the collo-cation formulation, we employ Delta functions �(� – �k) as test functions, k = 0, ..., M – 1, where{�k} is a proper set of grid (quadrature) points onthe range of �(�), and M is the number of gridpoints. By applying collocation projection on bothsides of the equation, we obtain
L(x, t, �k; u) = f(x, t; �k). (4)
Obviously, Equation 4 shares the same format asthe original equation, whereas Equation 3 is dif-ferent. In particular, if the operator L is nonlinear,Equation 3 is a system of ordinary or partial differ-
ential equations with the unknowns ui coupled to-gether, which introduces a real challenge to designan efficient deterministic solver. For complex fluiddynamical systems, such as compressible flows, it’seasier to use the collocation projection to obtainthe governing equations.
Note that the stochastic collocation method isequivalent to solving a deterministic problem ateach grid point. Once we obtain the numerical so-lutions at all collocation points, we can evaluate therandom solution’s statistics using the correspond-ing quadrature rule, for example,
,
where wk represents the integration weights, and �denotes the expectation.
High Dimensionality and Sparse GridsXiu and Jan S. Hesthaven9 constructed a stochas-tic collocation method based on sparse grids byusing the Smolyak algorithm.10 This algorithm isa linear combination of tensor product formulas,and the resulting grid set has a significantlysmaller number of grids compared to the full ten-sor product rule. Sparse grids depend weakly onthe random space’s dimensionality, and hence aremore suitable for applications with large dimen-sional random inputs. Recently, sparse grids forhigh-dimensional integrals have also received a lotof attention.11–13
Representation of Stochastic InputsWe represent the stochastic inputs with theKarhunen-Loeve (KL) decomposition14 given by
, (5)
where v(x; �) denotes the random process, denotes the corresponding mean, {�i(�)} is a set ofuncorrelated random variables with zero mean andunit variance, and x is the spatial or temporal co-ordinate. Also, i(x) and �i are the eigenfunctionsand eigenvalues of the covariance kernel Rhh(x, y),respectively—that is,
�DRhh(x, y)i(y)dy = �ii (x). (6)
Specifically, for a time-dependent stochasticprocess, we usually consider the exponential co-variance kernel
v( )x
v v ii
i i( ; ) ( ) ( ) ( )x x xω λ ψ ξ ω= +=
∞
∑1
�[ ]( , ) ( , , ) ( ) ( , , )u t u t f d u t wk kk
Mx x x= ≈
=
−
∑ξ ξ ξ ξ0
1
∫∫
⟨⎛
⎝⎜
⎞
⎠⎟ ⟩ = ⟨ ⟩
=
−
∑L t u fi ii
P
k kx, , ( ); , ,ξ ω φ φ φ0
1
f t fiP
i i( , ; ( ))x ξ ω φ= =−Σ 0
1
u t uiP
i i( , ; ( ))x ξ ω φ= =−Σ 0
1
24 COMPUTING IN SCIENCE & ENGINEERING
, (7)
where A is the correlation length. This correspondsto a first-order Markov chain in time, which in dis-crete form is
v0 = �0, vi+1 = cvi + q�i+1, i � 0, (8)
where , , and �t is the time step.Discrete forms of stationary spatial-dependent
random processes usually differ from time-depen-dent ones. This is due to the time series’ unilateralnature, which only depends on past values as op-posed to the spatial process’s dependence in all di-rections. By solving the following stochasticHelmholtz equation, we can obtain the correspond-ing spatial covariance kernel Rhh(v(x1, �), v(x2, �)),15
�v – k2v = f(x), (9)
where the random forcing term f (x) is a white-noise process, a function of the spatial vector xthat satisfies
�[f(x1)f(x2)] = �(x1 – x2). (10)
Equation 9 corresponds to a discrete model fora second-order autoregressive process in space,where the value of v at x depends on its neighbors.Here’s a simple 1D example:
. (11)
The parameter c is the constant correlation coeffi-cient, a gives the measure of the stochastic forcing’sstrength, and �i are independent random variableswith zero mean and unit variance.
Stochastic SimulationsLet’s first look at some results for compressible in-viscid steady flows and then for incompressible un-steady flows.
Stochastic Compressible FlowsFor supersonic flows, random inflow disturbancescan substantially modify the flow field, thus ren-dering traditional predictive CFD tools invalid.We study the shock dynamics in two-dimensionalsupersonic flows past a wedge, assuming two dif-ferent random disturbances. These correspond torandom inflow velocity or random oscillations ofthe wedge around its apex, and they can be steadyin time or time-dependent (the latter could be amodel for aeroelastic motions, such as flutter os-cillations). Figure 2 shows a schematic. We con-sider small perturbations and assume that theshock slope’s perturbation is small. We approxi-mate the flow between the shock and the wedgeas isentropic.
We denote the wedge angle by �0, the shock an-gle by 0, and the incoming flow velocity W1 withits normal component u1 = W1sin 0. The stream-lines behind the shock are parallel to the wedgesurface, and we denote the velocity by W2 and itsnormal component to the shock by u2 = W2sin( 0 –�0) = W1cos 0tan( 0 – �0).
First, we assume that the wedge angle is fixed,but the shock is subject to random inflow pertur-bation that we describe as a uniform random vari-able. We’ve obtained an analytical solution for theperturbed shock path
, (12)
where s = tan( 0 – �0), �M1 is the perturbed partof the inflow Mach number, and H( 0) = dM1/d .The mean and variance of the perturbed shockpath are then
�z(x; �)� = 0,
z x x sM
H( ; ) ( )
( )ξ
χ= +
Δ1 2 1
0
vc
v v ai i i i= + ++ −2 1 1( ) ξ
q c= −1 2c et
A=−Δ
R v t v t
v t v t e
hh ( ( , ), ( , ))
[ ( , ) ( , )]
1 2
1 2
ω ω
ω ω= =�
−− −| |t tA1 2
y
x
�2
p2
c2
�1
p1
c1
W1
W2
→→
�0
u1
u2
0
0
vu
v2
v1
Random perturbed shock
Unperturbed shock
M1 > 1
d
0 – �0
Figure 2. Sketch of supersonic flow past a wedge and definition of acoordinate system. We assume two different random disturbancescorresponding to random inflow velocity or random oscillations ofthe wedge around its apex. We’re interested in investigating howsuch random disturbances affect the shock path.
MARCH/APRIL 2007 25
(13)
Next, we assume that the wedge inflow is deter-ministic and describe the wedge oscillations as auniform random variable. In this case, we obtain
. (14)
The mean and variance for this case are similar tothe previous case, consistent with physical intuition.
Numerically, we solve the 2D Euler equationsfollowing the polynomial chaos approach for ran-dom inflow and random oscillations. In the lattercase, we use a transformation based on a boundary-fitted coordinate system approach, so that we solvethe Euler equations in a stationary domain:
ut + f(u)x + g(u)y = T(u), (15)
where u = [�, m, n, E]T, f = [�u, um + p, un, u(p + E)]T,and g = [�v, vm, vn + p, v(p + E)]T; for random inflow,T = 0, whereas for random oscillations, T = [0,–�cos�0(uw)t, �sin�0(uw)t, �(uw)t(nsin�0 – mcos�0)]T.Here, uw represents the wedge’s random motion.
In these examples, we consider the case in whichthe inflow velocity is perturbed by random fluctu-ation, which we describe as a random variable withamplitude � = 0.01 and � = 0.18. Figure 3 presentsthe mean of density �(x, y; �) corresponding to in-flow perturbation described as a random variablewith amplitude � = 0.18. We note that the mean re-sembles a fan expansion. Figure 4 plots the per-turbed shock path’s variance as a function of thedistance from the wedge apex x on the wedge sur-face for two amplitude values: � = 0.01 and � = 0.18.From Equation 13, we know that the perturbedshock’s variance is proportional to x2. Indeed, foramplitude � = 0.01, we can verify that the pertur-bation solution from Equation 13 and the numer-ical solution for amplitude � = 0.01 match exactly.However, for amplitude � = 0.18, we see in Figure4 that the numerical solution deviates from theanalytical solution in Equation 13 because thatequation doesn’t hold for large amplitudes.
Stochastic Wedge RoughnessNext, let’s consider the perturbation of an obliqueattached shock in supersonic flow past a wedge dueto random roughness on the wedge’s surface.16 Fig-ure 2 defines the problem, and we use the same as-
z x x sM
H( ; ) ( )
( )ξ
εξχ
= − +1 2 1
0
Var z x x sM
H
x s
( ( ; )) ( )( )
(
ξε ξχ
= +⟨ ⟩
= +
2 2 2 12 2 2
20
2
1
1 22 2 12 2
203
)( )
.M
H
εχ
Deterministic
2.2
2.1
2.0
1.9
1.8
1.7
1.6
Figure 3. Mean of �(x, y; �) induced by random inflow perturbationdescribed as a random variable with amplitude, � = 0.18. Note thatthe mean resembles a fan expansion.
x
<Z2 >/
�2
0 1 2 3 4 50
5
10
15 Perturbation solutionME-gPC: � = 0.01ME-gPC: � = 0.18
Figure 4. Variance of perturbed shock paths as afunction of x induced by random inflowperturbation, which is described as a randomvariable with amplitude � = 0.01 and � = 0.18.The perturbed shock path’s variance obtainedfrom the numerical simulations for amplitude �= 0.01 is proportional to x2, which matches withthe perturbation solution. However, foramplitude � = 0.18, the numerical solutiondeviates from the analytical solution becausethe perturbation analysis doesn’t hold for largeamplitude.
26 COMPUTING IN SCIENCE & ENGINEERING
sumptions as in the previous case.Using stochastic perturbation analysis, we obtain
the perturbed shock path z(x; �) as
, (16)
where �h(x) is the perturbation on the wedge and
,
,
� = 1 – s/m, and
.
Both � and are less than 1; we obtain M2, F( 0) =d�/d , and
from normal shock relations.We assume the length of the wedge is finite (x �
[0, L]) and describe the perturbation at the roughwedge y(x; �) = �h(x; �) as a space-dependent ran-dom process, described as a second-order auto-regressive process between [0, L] with zero meanand exponential covariance:15
(17)
where A is the correlation length in space. We setthe rough perturbation at x = 0 and L to be zero(h0 = hN = 0).
Numerically, we solve the 2D Euler equations inEquation 15 with T = 0, following the collocationapproach. Because it’s a steady-state problem, wesolve Equation 15 by marching in time until theflow reaches a steady state. Figure 5 shows the con-tours for the normalized standard deviation of theperturbed pressure � (p)/e. Figure 6 presents thevariance of the perturbed shock as a function of x.We describe the random roughness as a randomprocess with correlation length A/L = 1; for ampli-tude � = 0.0003, we can verify that the analytical so-lution from Equation 16 and the numericalsolution match exactly. However, for the larger am-plitude � = 0.03, we see in Figure 6 that the nu-merical solution loses its symmetry and deviatessubstantially from the analytical solution obtainedfrom Equation 16.
Noisy Flow Past aStationary Circular CylinderLet’s consider a model problem that appears inmany engineering applications: noisy flow past a
hb
h h a i N
b e
i i i i
x
= + + = −
=
− +
− Δ2
1 2 11 1( ) , , , ...,ξ22
2
2
2232 20 5A
xAa x e, . ( ) ,= − Δ
⎧
⎨⎪⎪
⎩⎪⎪
− Δ
GM
M P
dPd
( )χγ χ0
22
22
2
21=
−
β = −+
m sm s
rF GF G
( )( ) ( )( ) ( )
χχ χχ χ0
0 0
0 0=
−+
qF G
( )( ) ( )
χχ χ0
0 0
2=+
z x q s rh xn
n
nn
( ; ) ( ) ( )( ; )ξ ε αβ ξαβ
= + −=
∞
∑1 2
0
Roughregion
Figure 5. Contours of the normalized standard deviation of theperturbed pressure �(p)/e. Red denotes high values, and bluedenotes lower values.
<Z2 >/
�2
x0 1 2 3 4 5 6
0
5
10
15
20Analytical solution� = 0.0003� = 0.03
Figure 6. Variance of the perturbed shock path asa function of x. The random roughness isdescribed as a random process with correlationlength A/L = 1.
MARCH/APRIL 2007 27
stationary circular cylinder. The inflow velocity u =U(t) is random, which Figure 7 indicates with rightarrows that have rippled ends. The 2D incom-pressible Navier-Stokes equations govern the flow:
, i = 1, 2
(summation over j).
We normalize all length scales with the cylinder di-ameter D, all velocity components with the meaninflow velocity �[U], and the pressure with ��2[U](where � is the fluid density). We define theReynolds number as Re = UD/�, where � is thefluid’s kinematic viscosity.
For a deterministic uniform flow past a station-ary circular cylinder, we can model the lift force FLon the cylinder as harmonic in time at the sheddingfrequency fs:
,
where is the maximum amplitude of the in-stantaneous deterministic lift coefficient. We wantto extend such a harmonic model to the stochasticcase. If the inflow velocity u = U(t; �) is random, thecorresponding lift force and vortex shedding fre-quency will also be random. We can express the in-stantaneous stochastic harmonic model of CL as
CL(t) = AL(�)cos(2�fs(�)t),
and rewrite fs(�) as
,
where is the standard deviation of fs and X � [a,b] is a random variable with unit variance. We thenobtain the following statistics of CL:
�[CL] 0, as t ,
, as t ,
where f(x) is the PDF of X. We note that the sec-ond-order moment approaches a constant value foran arbitrary random variable � in contrast to the os-cillatory behavior for the deterministic case.
Let’s consider the following boundary conditionat the inflow:
u = 1 + ��, v = 0,
where � is a random variable with zero mean andunit variance, and � is a constant indicating the de-gree of perturbation. We let �[Re] = 100. For 10percent symmetric noise (� = 0.1), the range of theReynolds number is Re � [90, 110]. Using an em-pirical relation between the vortex-shedding fre-quency and the Reynolds number,17 we find thatthe corresponding vortex-shedding frequency israndom: fs � [0.1592, 0.1697]. We’ve shown thatthe random frequency can severely weaken gPC’sefficiency in long-term integration.18 gPC’s accu-racy with polynomial order p = 15, for example, willreach an error of O(1) at t�[U]/D � 40 for this case.To simulate 10 percent symmetric noise (uniform
�[ ] ( ( )) ( )C A x f x dxL La
b2 12
→ ∫ ξ
σ f s
f f Xx s f s( ) [ ]ξ σ= +�
CLd
F t U DC f tL Ld
s( ) cos( )= 12
22ρ π
∂
∂=
u
xj
j0
∂∂
+∂∂
= − ∂∂
+∂∂
ut
uux
px
u
xi
ji
j i
i
j
1 2
2�[Re]
x–10 0 10 20
–5
0
5
= 0∂n∂u
u = U(t)
y
Figure 7. Schematic of the domain and mesh for noisy flow past acircular cylinder. The size of the domain is [–15D, 25D] � [–9D, 9D],and the cylinder is at the origin with diameter D = 1. The meshconsists of 412 triangular elements.
0 50 100 1500
0.05
0.10
0.15
0.20
0.25
0.30
0.35
tE[U]/D
RMS
of C
L Uniform distributionBeta distributionDeterministic
Figure 8. Instantaneous root-mean-square (RMS) of lift coefficients.A uniform distribution in [ ] and a beta distribution Be(2, 2)in [ ] are considered with � = 0.1.− 7 7,
− 3 3,
28 COMPUTING IN SCIENCE & ENGINEERING
and beta PDFs), we use ME-gPC with N = 20 andp = 8, which can reach an accuracy of O(10–3) in therange t�[U]/D 150, corresponding to roughly 20shedding periods. We can see in Figure 8 that theroot-mean-square (RMS) of lift coefficient ap-proaches a constant value quickly, in agreementwith the stochastic analytic model of CL. The RMSapproaches approximately the same value, 0.258,for both uniform and beta noise, which is 20 per-cent bigger than the time-averaged deterministicRMS, or 0.215, without noise.
Subsequently, we study the random amplitudeAL by incorporating a linear model, AL(�) = A0 +c1�, and a quadratic model, AL(� ) = A0 + c2� + c3�
2,where A0 = 0.3372 is the deterministic amplitudeat Re = 100 given by direct numerical simulations,and ci, i = 1, 2, 3 are undetermined constants. Us-ing the first- and second-order moments, we canresolve ci from the ME-gPC results. For the uni-form noise, we obtain ci = 0.1402, 0.1100, and0.0109; for the beta noise, ci = 0.1414, 0.1101, and0.0112. These two sets of numbers agree well be-cause the linear and quadratic models are inde-pendent of the PDF of �. We observe from thelinear model that the response degree is c1/A0 � 42percent in contrast to the 10 percent noise at theinflow, which implies that the lift coefficient isvery sensitive to upstream noise. Due to the qua-dratic term, the RMS of CL is bigger than the de-terministic one at Re = 100 as Figure 8 illustrates.Figure 9 shows the PDFs of AL. We see thatthey’re well described by the quadratic model,which is consistent with the deterministic modelof lift coefficient.17 In contrast to symmetricnoise, the PDF of AL has a clear bias toward lowerReynolds numbers.
We’re at the dawning of the sto-chastic modeling age in fluidmechanics, which means there’smuch more work to be done.
The computational complexity of a typical sto-chastic simulation is more than an order of mag-nitude greater than its deterministic counterpart,but parallel algorithms are rather straightforwardto implement and can significantly reduce the re-quired simulation time. Establishing “error bars”in CFD that reflect not only numerical uncer-tainty but also uncertainty in physical modelingand geometry provides great credibility to simu-lation and will make it an indispensable tool in de-signing complex flow systems. It’s an importantstep in establishing simulation-based certificatesof fidelity in flow design. In addition, stochastic-simulated responses can serve as a form of sensi-tivity analysis that could potentially guideexperimental work and dynamic instrumentationand make the simulation–experiment interactionmore meaningful.
Several outstanding issues related to long-timeintegration, stochastic discontinuities, adaptivity,and high dimensionality still exist. All of them arekey elements to establishing multi-element gPC(with Galerkin or collocation projections) as a“mainstream” stochastic simulation approach anda powerful alternative to Monte Carlo simulation.These methods are more suitable for unsteady sim-ulations and several orders of magnitude more ef-ficient than MC for the stochastic correlated inputstypically encountered in CFD applications.
AcknowledgmentsThis work was supported by the US Air Force Office of
(a) (b)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
AL
0.1 0.2 0.3 0.4 0.5 0.6AL
Linear modelQuadratic model
ME-gPC: N = 20, p = 8Linear modelQuadratic model
ME-gPC: N = 20, p = 8
Figure 9. Probability density functions (PDFs) of AL given by different models: (a) uniform noise and (b) beta noise. Therelation between � and AL is well described with a quadratic model.
MARCH/APRIL 2007 29
Scientific Research’s computational mathematics pro-gram and the US National Science Foundation’s AMC-SSprogram.
References1. E. Simiu, Chaotic Transition in Deterministic and Stochastic Systems,
Princeton Univ. Press, 2002.
2. P.K. Maciejewski and R.J. Moffat, “Heat Transfer with Very HighFree-Stream Turbulence, Part I: Experimental Data,” J. Heat Trans-fer, vol. 114, no. 44, 1992, pp. 827–833.
3. D. Lucor and G.E. Karniadakis, “Noisy Inflows Cause a Shedding-Mode Switching in Flow Past an Oscillating Cylinder,” PhysicalRev. Letters, vol. 92, no. 15, 2004, pp. 154501(1–4).
4. P. Espanol and P. Warren, “Statistical Mechanics of DissipativeParticle Dynamics,” Europhysics Letters, vol. 30, no. 4, 1995, pp.191–196.
5. R.G. Ghanem and P. Spanos, Stochastic Finite Elements: A SpectralApproach, Springer-Verlag, 1991.
6. D. Xiu and G.E. Karniadakis, “The Wiener-Askey PolynomialChaos for Stochastic Differential Equations,” SIAM J. ScientificComputing, vol. 24, no. 2, 2002, pp. 619–644.
7. X. Wan and G.E. Karniadakis, “Multi-Element Generalized Poly-nomial Chaos for Arbitrary Probability Measures,” SIAM J. Scien-tific Computing, vol. 28, no. 3, 2006, pp. 901–928.
8. X. Wan and G.E. Karniadakis, “An Adaptive Multi-Element Gener-alized Polynomial Chaos Method for Stochastic Differential Equa-tions,” J. Computational Physics, vol. 209, no. 2, 2005, pp. 617–642.
9. D. Xiu and J.S. Hesthaven, “High Order Collocation Methods forDifferential Equations with Random Inputs,” SIAM J. ScientificComputing, vol. 27, no. 3, 2005, pp. 111–113.
10. S. Smolyak, “Quadrature and Interpolation Formulas for TensorProducts of Certain Classes of Functions,” Soviet MathematicsDoklady, vol. 4, 1963, pp. 240–243.
11. E. Novak and K. Ritter, “High Dimensional Integration of SmoothFunctions over Cubes,” Numerische Mathematick, vol. 75, no. 11,1996, pp. 79–97.
12. G. Wasilkovski and H. Wozniakowski, “Explicit Cost Bounds of Al-gorithms for Multivariate Tensor Product Problems,” J. Complex-ity, vol. 11, no. 1, 1995, pp. 1–56.
13. T. Gerstner and M. Griebel, “Numerical Integration Using SparseGrids,” Numerical Algorithms, vol. 18, nos. 3 and 4, 1998, pp.209–232.
14. M. Loeve, Probability Theory, 4th ed., Springer-Verlag, 1977.
15. C.H. Su and D. Lucor, “Covariance Kernel Representations ofMultidimensional Second-Order Stochastic Processes,” J. Com-putational Physics, vol. 217, no. 1, 2006, pp. 82–99.
16. G. Lin, C.H. Su, and G.E. Karniadakis, “Modeling RandomRoughness in Supersonic Flow past a Wedge,” Proc. AIAA Aero-space Sciences Meeting and Exhibit, AIAA Press, 2006, pp.2006–0124.
17. R. D. Blevins, Flow-Induced Vibration, Van Nostrand ReinholdCompany, 1990.
18. X. Wan and G.E. Karniadakis, “Long-Term Behavior of Polyno-mial Chaos in Stochastic Flow Simulations,” Computational Meth-ods in Applied Mathematics & Eng., vol. 195, nos. 41–43, 2006,pp. 5582–5596.
Guang Lin is a PhD candidate in the Division of AppliedMathematics at Brown University. His research interestsinclude scientific computation, uncertainty quantification,and MHD flow simulations. Lin has an MSc in appliedmathematics from Brown University. Contact him at [email protected].
Xiaoliang Wan is a PhD candidate in the Division of Ap-plied Mathematics at Brown University. His research in-terests include stochastic modeling and high-ordernumerical methods for partial differential equations. Con-tact him at [email protected].
Chau-Hsing Su is a professor of applied mathematics atBrown University. His research interests include the studyof spatial random processes and shock wave propagation.Su has a PhD in aeronautical engineering from PrincetonUniversity. Contact him at [email protected].
George Em Karniadakis is a professor of applied math-ematics at Brown University. His research interests in-clude spectral methods on unstructured grids, parallelsimulations of turbulence in complex geometries, andmicrofluidics simulations. Karniadakis has a PhD inmechanical engineering from MIT. Contact him at [email protected].
Writers
For detailed information on submittingarticles, write to [email protected] orvisit www.computer.org/cise/author.htm.
Letters to the Editors
Send letters to Jenny Stout, Lead Editor, [email protected]. Providean email address or daytime phone number with your letter.
On the Web
Access www.computer.org/cise/ or http://cise.aip.org for information.
Subscribe
Visit https://www.aip.org/forms/journal_catalog/order_form_fs.htmlor www.computer.org/subscribe/.
Subscription Change of Address (IEEE/CS)
Send change-of-address requests for magazine subscriptions [email protected]. Be sure to specify CiSE.
Subscription Change of Address (AIP)
Send general subscription and refund inquiries to [email protected].
Missing or Damaged Copies
Contact [email protected]. For AIP subscribers, [email protected].
Reprints of Articles
For price information or to order reprints, send email [email protected] or fax +1 714 821 4010.
Reprint Permission
Contact William Hagen, IEEE Copyrights and Trademarks Manager,at [email protected].
www.computer.org/cise/
How toReach CiSE
