Modeling Uncertainty in Flow Simulations via Polynomial Chaos

Dongbin Xiu and George Em Karniadakis �

Division of Applied Mathematics

Brown University

Providence, RI 02912

Submitted to Journal of Computational Physics

February 8, 2001

Abstract

We present a new algorithm based on Polynomial Chaos in order to model the input uncertainty and its

propagation in incompressible ow simulations. Speci�cally, the stochastic input is represented spectrally by

employing the Wiener-Hermite polynomials. Randomness is thus expressed as a new continuous variable, in

addition to the space-time domain. A standard Galerkin projection is performed and the resulting set of coupled

equations is then solved to obtain the solution for each random mode. We implement the method in the context

of the spectral/hp element method, which maintains control of the numerical error while its spectral trial basis is

from the same Askey family of polynomials as the Polynomial Chaos. The algorithm is applied to micro-channel

ows with random wall boundary conditions, and to external ows with random freestream. Both Gaussian and

non-Gaussian inputs are considered, and the results are in good agreement with exact solutions and Monte Carlo

simulations. The Polynomial Chaos based stochastic simulation, although more expensive than the corresponding

deterministic simulation, it generates all the solution statistics in a single run. Compared with the Monte Carlo

simulation, which requires at least thousands of runs to obtain accurate statistics, the Polynomial Chaos expansion

promises a substantial speed-up (at least one thousand in the current simulations), and thus it is a very e�ective

approach in modeling uncertainty propagation in ow simulations.

�Corresponding author, gk@cfm.brown.edu

1

1 Introduction

There has been recently an intense interest in veri�cation and validation of large-scale simulations and in modeling

uncertainty [1, 2, 3]. In simulations, just like in the experiments, we often question the accuracy of the results and we

construct a posteriori error bounds, but the new objective is to model uncertainty from the beginning of the simulation

and not simply as an afterthought. Numerical accuracy and error control have been employed in simulations for

some time now, at least for the modern discretizations, e.g. [4, 5]. However, there is still an uncertainty component

associated with the physical problem, and speci�cally with such diverse factors as constitutive laws, boundary and

initial conditions, transport coeÆcients, source and interaction terms, geometric irregularities (e.g. roughness), etc.

.

Most of the research e�ort in CFD research so far has been in developing eÆcient algorithms for di�erent ap-

plications, assuming an ideal input with precisely de�ned computational domains. With the �eld reaching now

some degree of maturity, we naturally pose the more general question of how to model uncertainty and stochastic

input, and how to formulate algorithms in order for the simulation output to re ect accurately the propagation of

uncertainty. To this end, the Monte Carlo approach can be employed but it is computationally expensive and it is

only used as the last resort. The sensitivity method is an alternative more economical approach, based on moments

of samples, but it is less robust and it depends strongly on the modeling assumptions [6]. There are other more

suitable methods for physical applications, and there has already been good progress in other �elds, most notably in

seismology and structural mechanics. A number of papers and books have been devoted to this subject, e.g. [7],[8],

[9],[10], [11],[12],[13],[14].

The most popular technique for modeling stochastic engineering systems is the perturbation method where all

stochastic quantities are expanded around their mean via a Taylor series. This approach, however, is limited to small

perturbations and does not readily provide information on high-order statistics of the response. Another approach

is based on expanding the inverse of the stochastic operator in a Neumann series, but this too is limited to small

uctuations, and even combinations with the Monte Carlo method seem to result in computationally prohibitive

algorithms for complex systems [15]. A more e�ective approach pioneered by Ghanem and Spanos [10] in the context

of �nite elements for solid mechanics is based on a spectral representation of the uncertainty. This allows high-order

representation, not just �rst-order as in most perturbation-based methods, at high computational eÆciency. It is

based on the original ideas of Wiener (1938) on homogeneous chaos [16, 17] and the Hermite polynomials. The

classical Grad's method of moments for solutions of Boltzmann equation is based on Hermite polynomial expansions

2

as well [18]. The e�ectiveness of Hermite expansions was also recognized by Chorin in his early work [19], who

employed Wiener-Hermite series to substantially improve both accuracy and computational eÆciency of Monte Carlo

algorithms. However, the limitation of prematurely truncated Wiener-Hermite expansions, especially in applications

of turbulence [20], has also been recognized.

The spectral representation of the uncertainty is based on a trial basis (�) where � denotes the random event.

For example, the vorticity (! in two-dimensions) has the following �nite-dimensional representation

!(x; t; �) =

PXi=0

!i(x; t)i(�(�)):

Here !i(x; t) represents the deteministic part and can be interpreted as a scale (i) of vorticity uctuation. The

random trial basis is expressed in terms of multi-dimensional Hermite polynomials in powers of the random variable

�(�), which de�nes a speci�c probability distribution; for example, �(�) may be a Gaussian function. The polynomial

trial basis constructed in this way has been termed Polynomial Chaos (PC) by Wiener. It is a functional, as it is a

function of � which is a function of the random parameter � 2 [0; 1]. The theory of orthogonal functionals plays a

key role in the algorithms that we develop in this work. Note that the Monte Carlo algorithm is a subcase of the

above representation corresponding to the collocation procedure where the test basis is

i(�) = Æ(� � �i);

with Æ the Kronecker delta function, and �i referring to an isolated random event. For convergence, the Monte

Carlo method requires a great number of such events as it ignores interactions between the various scales unlike the

Galerkin representation.

In this work we propose a Polynomial Chaos (PC) expansion procedure to represent the uncertainty of the input in

ow simulations. The algorithms we develop are general but in this paper we concentrate on modeling the uncertainty

associated with boundary conditions. This situation is encountered, for example, in micro-channel ows [21] but also

in classical ows such, e.g. the freestream of ow past blu� bodies. We assume that the velocity at the boundaries

is described by a mean value and a random perturbation with certain propability distribution function (PDF). This

distribution can be assumed to have a known analytical form, e.g. Gaussian or lognormal distribution, or it can

be a measured distribution provided in tabulated form. The PC procedure can handle both Gaussian and non-

Gaussian representations, although for certain distributions there may exist a better representation than the original

Wiener-Hermite polynomial basis. To this end, we can replace the Wiener-Hermite expansions by the best spectral

polynomials of the Askey family [22] that match speci�c distribution functions. For example, the appropriate set for

3

Poisson distributions is the Charlier polynomials, for Gamma distributions the Laugerre polynomials, for binomial

distributions the Krawtchouk polynomials, for the beta kernel the Jacobi polynomials, etc. This can be appreciated

by the fact that the orthogonality weights of these polynomials match the probability measure of the corresponding

polynomials, and the same relationship exists of course for the Hermite polynomials and the Gaussian distribution.

The work we present in this paper is based on the pioneering work in solid mechanics of Ghanem & Spanos [10]

but applied here to incompressible Navier-Stokes equations. In addition, we employ the spectral/hp element method

in order to obtain a better control of the numerical error [23]. Our approach leads to homogeneous representations as

the mixed inner products required in the random and spatial disretizations are all spectral polynomials. In particular,

for certain distributions, e.g. beta kernels the corresponding Askey polynomial for the Wiener expansion is the Jacobi

polynomial, which is exactly the same as the trial basis employed in the spectral/hp element method [23].

In the next section we review the PC expansion, and in section 3 we address its implementation details as applied

to Navier-Stokes equations. In section 4 we present computational results for Gaussian and lognormal distributions,

and we compare with analytical stochastic solutions of Navier-Stokes we have obtained as well as with corresponding

Monte Carlo simulations. We conclude the paper with a discussion on possible extensions and applications of the

stochastic spectral/hp element method. We also include two appendices that explain how to generate partially

correlated Gaussian �elds and how to approximate the lognormal distribution that may be useful to CFD readers.

2 Representation of Random Processes

In this section we brie y review the PC expansion along with the Karhunen-Loeve (KL) expansion, another classical

technique for representating random processes. The KL expansion will be used to represent the known stochastic

�elds, e.g. the input. In the following analysis, we will use the symbol � to denote the standard Gaussian random

variable, i.e., Gaussian random variable with zero mean and unit variance.

2.1 Polynomial Chaos Expansion

The PC expansion was �rst proposed by Wiener [16]. According to the theorem by Cameron & Martin [24], it can

be used to approximate any functional in L2(C) and converges to the functional in the L2(C) sense. Therefore, PC

provides a means for expanding second-order random processes in terms of orthogonal polynomials. Second-order

random processes are processes with �nite variance, and this applies to all viscous ow processes. Thus, a second-

order random process X(�), viewed as a function of the independent random variable �, can be represented in the

4

form

X(�) = a0H0

+

1Xi1=1

ai1H1(�i1(�))

+

1Xi1=1

i1Xi2=1

ai1i2H2(�i1 (�); �i2 (�))

+

1Xi1=1

i1Xi2=1

i2Xi3=1

ai1i2i3H3(�i1 (�); �i2 (�); �i3 (�))

+ � � � ; (1)

where Hn(�i1 ; : : : ; �in) denotes the polynomial chaos of order n in the variables (�i1 ; : : : ; �in). The above equation

is the discrete version of the original Wiener-Hermite expansion, where the continuous integrals are replaced by

summations. The general expression of the polynomials is given by

Hn(�i1 ; : : : ; �in) = e12�T �(�1)n @n

@�i1 � � � @�ine�

12�T �; (2)

where � denotes the vector consisting of n random variables (�i1 ; : : : ; �in). For notational convenience, equation (1)

can be rewritten as

X(�) =

1Xj=0

ajj(�); (3)

where there is a one-to-one correspondence between the functions Hn(�i1 ; : : : ; �in) and j(�). The Polynomial Chaos

forms a complete orthogonal basis in the L2 space of random variables, i.e.,

< ij >=< 2

i > Æij ; (4)

where Æij is the Kronecker delta and < �; � > denotes the ensemble average. This is the inner product in the Hilbert

space of random variables

< f(�)g(�) >=

Zf(�)g(�)W (�)d�: (5)

and it is justi�ed by the Cameron & Martin theorem [24]. The weighting function is

W (�) =1p(2�)n

e�12�T �: (6)

What distinguishes the Wiener-Hermite expansion from many other possible complete sets of expansions is that

the polynomials here are orthononormal with respect to the weighting function W (�) that has the form of a n-

dimensional independent Gaussian probability distribution with unit variance. For example, the one-dimensional

5

(n = 1) polynomials are:

0 = 1; 1 = �; 2 = �2 � 1; 3 = �3 � 3�; : : : (7)

The orthogonality condition takes the speci�c form

< ij >=< 2

i > Æij = (i!)Æij : (8)

2.2 Karhunen-Loeve Expansion

The Karhunen-Loeve (KL) expansion [25] is another way of representing a random process. It is based on the spectral

expansion of the covariance function of the process. Let us denote the process by h(x; �) and its covariance function

by Rhh(x;y), where x and y are the spatial or temporal coordinates. By de�nition, the covariance function is real,

symmetric and positive de�nite. All eigenfunctions are mutually orthogonal and form a complete set spanning the

function space to which h(x; �) belongs. The KL expansion then takes the following form:

h(x; �) = �h(x) +

1Xi=1

p�i�i(x)�i(�); (9)

where �h(x) denotes the mean of the random process, and �i(�) forms a set of orthogonal random variables. Also,

�i(x) and �i are the set of eigenfunctions and eigenvalues of the covariance function, respectively, i.e.,

ZRhh(x;y)�i(y)dy = �i�i(x): (10)

If the random process itself h(x; �) is a Gaussian process, then the random variables �i form an orthonormal Gaussian

vector.

Among many possible decompositions of a random process, the KL expansion is optimal in the sense that the

mean-square error resulting from a �nite representation of the process is minimized. Its use, however, is limited as the

covariance function of the solution process is often not known a priori. Nevertheless, the KL expansion still provides

an e�ective means of representation of the input random processes when the covariance structure is known. In this

paper we will employ the KL procedure to represent the stochastic boundary conditions in the case of Gaussian

distributions. For non-Gaussian distributions we will employ Polynomial Chaos representations directly.

3 Polynomial Chaos Expansion for Navier-Stokes Equations

In this section we discuss the solution procedure for solving the stochastic Navier-Stokes equations by PC expansion.

The randomness in the solution can be introduced through boundary conditions, initial conditions, forcing, etc.

6

3.1 Governing Equations

We employ the incompressible Navier-Stokes equations

r � u = 0; (11)

@u

@t+ (u � r)u = �r�+ Re�1r2

u; (12)

where � is the pressure and Re the Reynolds number. All ow quantities, i.e., velocity and pressure are considered

stochastic processes. A random dimension, denoted by the parameter �, is introduced in addition to the spatial-

temporal dimensions (x; t), thus

u = u(x; t; �); � = �(x; t; �) (13)

We then apply the Polynomial Chaos expansion (3) to these quantities to obtain

u(x; t; �) =

PXi=0

ui(x; t)i(�); �(x; t; �) =

PXi=0

�i(x; t)i(�); (14)

where we have replaced the in�nite summation in (3) by a �nite term summation. The total number of expansion

terms, (P + 1), depends on the number of random dimensions (n) and the highest order of the polynomials chaos

(p) [10]:

P = 1 +

pXs=1

1

s!

s�1Yr=0

(n+ r): (15)

The most important aspect of the above expansion is that all random processes have been decomposed into a set

of deterministic functions in the spatial-temporal variables multiplying random coeÆcients that are independent of

these variables.

Substituting (14) into Navier-Stokes equations ((11) and (12)) and noting that the partial derivatives are taken

in physical space and thus they commute with the operations in random space, we obtain the following equations

PXi=0

r � ui(x; t)i = 0; (16)

PXi=0

@ui(x; t)

@ti +

PXi=0

PXj=0

[(ui � r)uj)]ij = �PXi=0

r�i(x; t)i + �

PXi=0

r2uii: (17)

We then project the above equations onto the random space spanned by the basis polynomials fig by taking the

inner product of above equation with each basis. By taking < �;k > and utilizing the orthogonality condition (4),

we obtain the following set of equations:

7

For each k = 0; : : : P ,

r � uk = 0; (18)

@uk

@t+

1

< 2

k >

PXi=0

PXj=0

eijk [(ui � r)uj)] = �r�k + �r2uk; (19)

where eijk =< ijk >. Together with < 2

i >, these coeÆcients can be determined analytically from the

de�nition of i and equation (2). This set of equations consists of (P + 1) system of Navier-Stokes equations for

each random mode coupled through the convection term. The solution associated with k = 0 term represents the

mean of the random solution. The solution corresponding to the �rst-order polynomial chaos, i.e., the k = 1; : : : ; n

terms where n is the dimension of random space, represents the Gaussian part of the solution. The rest of the

solution terms represent the nonlinear interactions between the mean and each Gaussian part of the solution and are

non-Gaussian.

3.2 Numerical Formulation

3.2.1 Temporal Discretization

We employ the semi-implicit high-order fractional step method, which for the standard deterministic Navier-Stokes

equations ((11) and (12)) has the form [26]:

u�PJq=0

�qun�q

�t= �

JXq=0

�q [(u � r)u]n�q ; (20)

^u� u

�t= �r�n+1; (21)

0un+1 � ^u

�t= Re�1r2

un+1; (22)

where J is order of accuracy in time and �; � and are the coeÆcients of the integration weights. A pressure Poisson

equation is obtained by enforcing the discrete divergence-free condition r � un+1 = 0

r2�n+1 =1

�tr � u; (23)

with the appropriate pressure boundary condition given as

@�

@n= �n �

�u+Re�1r�!n+1

�; (24)

where n is the outward unit normal vector and ! = r� u is the vorticity. The method is sti�y-stable and achieves

third-order accuracy in time; the coeÆcients for the integration weights can be found in [23].

In order to discretize the stochastic Navier-Stokes equations, we apply the same approach to the coupled set of

equations (18) and (19):

8

For each k = 0; : : : ; P ,

uk �PJ

q=0�qu

n�qk

�t= � 1

< 2

k >

JXq=0

�q

24 PXi=0

PXj=0

eijk(ui � r)uj

35n�q

; (25)

^uk � uk

�t= �r�n+1

k ; (26)

0un+1

k � ^uk

�t= Re�1r2

un+1

k : (27)

The discrete divergence-free condition for each mode r �un+1

k = 0 results in a set of consistent Poisson equations for

each pressure mode

r2�n+1

k =1

�tr � uk; k = 0; : : : ; P; (28)

with appropriate pressure boundary condition derived similarly as in [26]

@�k

@n= �n �

�uk +Re�1r�!n+1

k

�; k = 0; : : : ; P; (29)

where n is the outward unit normal vector along the boundary, and !k = r� uk is the vorticity for each random

mode.

3.2.2 Spatial Discretization

Spatial discretization can be carried out by any method, but here we employ the spectral/hp element method in

order to have better control of the numerical error [23]. In addition, the all-spectral discretization in space and

along the random direction leads to homogeneous inner products, which in turn result in more eÆcient ways of

inverting the algebraic systems. In particular, the spatial discretization is based on Jacobi polynomials on triangles

or quadrilaterals in two-dimensions, and tetrahedra, hexahedra or prisms in three-dimensions.

3.3 Post-Processing

The coeÆcients in the expansion of the solution process (14) are obtained after solving equations (25) to (29). We

then obtain the analytical form (in random space) of the solution process. It is possible to preform a number of

analytical operations on the stochastic solution in order to carry out a sensitivity analysis. Speci�cally, the mean

solution is contained in the expansion term with index of zero. The second-moment, i.e., the covariance function is

given by

Ruu(x1; t1;x2; t2) = < u(x1; t1)� u(x1; t1);u(x2; t2)� u(x2; t2) >

=

PXi=1

�ui(x1; t1)ui(x2; t2) < 2

i >�: (30)

9

Note that the summation starts from index (i = 1) instead of 0 to exclude the mean, and that the orthogonality

of the PC basis fig has been used in deriving the above equation. The variance of the solution, also called the

`mean-square' value, is obtained as

V ar(u(x; t)) =<�u(x; t)� u(x; t)

�2>=

PXi=1

�u2

i (x; t) < 2

i >�; (31)

and the root-mean-square (rms) is simply the square root of the variance. Similar expressions can be obtained for

the pressure �eld.

4 Stochastic Flow Simulations

In this section we present numerical results of the Polynomial Chaos solution to the Navier-Stokes equations. We �rst

consider a micro-channel ow where there is uncertainty associated with the wall boundary conditions. Subsequently,

we simulate a laminar ow past a circular cylinder with uncertain freestream. We model the uncertainty in the

boundary conditions both as Gaussian and as lognormal distributions in order to evaluate the convergence of the PC

method. In both cases we employ a two-dimensional Polynomial Chaos expansion (n = 2) with polynomial order up

to p = 4; these polynomial coeÆcients are listed in table 1.

Index p, Order of kth Polynomial Chaos

k Polynomial Chaos k

0 p = 0 1

1 p = 1 �12 �23 p = 2 �2

1� 1

4 �1�25 �2

2� 1

6 p = 3 �31� 3�1

7 �21�2 � �2

8 �1�2

2� �1

9 �32� 3�2

10 p = 4 �41� 6�2

1+ 3

11 �31�2 � 3�1�2

12 �21�22� �2

1� �2

2+ 1

13 �1�3

2� 3�1�2

14 �42� 6�2

2+ 3

Table 1: Polynomial terms for two-dimensional Polynomial Chaos.

4.1 Micro-channel Flow: Gaussian Input

We consider a micro-channel ow as shown in �gure 1. In micro- ows the no-slip boundary condition is not always

valid and appropriate slip boundary conditions should be used [21]. For gas ows Maxwell's boundary condition on

10

velocity slip or its high-order variants are appropriate at least in the slip ow regime. However, for liquids there

is no existing models for the molecule layering observed in molecular dynamics simulations [27]. Also, the e�ect of

roughness complicates further the validity or not of the no-slip boundary condition. Here we model such uncertainty

at the boundary by assuming that

u = 0 + r

where r is a random variable.

F=2ν

u=u1

u=u2

x

y

y=-1

y=1

Figure 1: Schematic of the domain for micro-channel pressure-driven ow.

The domain (see �gure 1) has dimensions such that y 2 [�1; 1] and x 2 [�5; 5]. The pressure gradient, acting

like a driving force, equals to twice the kinematic viscosity, and thus for a no-slip condition a parabolic pro�le is a

solution with centerline velocity equals to unity. Assuming that the two walls are moving with constant velocities

u1 and u2, then the exact solution is

u(x; y) = (1� y2) +1� y

2u1 +

1 + y

2u2 (32)

so it is a superposition of the Poiseuille and two Couette type ows.

4.1.1 Fully-Correlated Random Boundary Conditions

Let us now assume that the boundary conditions are random and spatially uniform, i.e.,

u1 = ��1 and u2 = �2�2;

where �1 and �2 are two independent standard Gaussian random variables with zero mean and unit variance and

�1, �2 are the standard deviations of u1 and u2, respectively. In this case, the exact solution of equation (32) still

11

applies and thus

u(x; y) = (1� y2) +1� y

2�1�1 +

1 + y

2�2�2 (33)

It can be seen from table 1 that the solution consists of the parabolic mean u0 and only the �rst two random modes

u1 and u2, linearly distributed across the channel. We will use this solution to validate the stochastic Navier-Stokes

solver.

y

u 0

u 1,u

2

-1 -0.5 0 0.5 10

0.25

0.5

0.75

1

0

0.005

0.01

0.015

0.02

u0

u1u2

Figure 2: Solution pro�le across the channel with uniform Gaussian boundary conditions.

Figure 2 shows the solution pro�le across the channel. The solution is obtained by setting �1 = 0:02 and

�2 = 0:01. A two-dimensional PC expansion is used (n = 2) with Polynomial Chaos of order p = 1. The numerical

results correctly obtain the distribution pro�les of the mean, the �rst and the second random modes. If p > 1 is

employed, all higher-order random modes are identically zero.

4.1.2 Partially-Correlated Random Boundary Conditions

Next we consider the case of non-uniform random boundary conditions, i.e. the boundary points are only partially-

correlated and thus the random contribution to the boundary condition varies along the walls. The boundary

conditions are assumed to be Gaussian random processes with correlation function in the form

C(x1; x2) = e�jx1�x2j

b ; (34)

12

where b is the correlation length. This correlation function has been employed extensively to model processes in a

variety of �elds and its use can be justi�ed on theorectical grounds [28].

Given the exponential form of the correlation function, the Karhunen-Loeve expansion of equation (9) can be

used to decompose the input random process. The corresponding eigenvalue problem of equation (10) can be solved

analytically. The eigenvalues are:

�n =2b

1 + b2!2n

; n = 1; 2; : : : (35)

and the eigenfunctions are:

fn(x) =

8><>:

cos(!nx)qa+

sin(2!na)

2!n

if n is odd,

sin(!nx)qa� sin(2!na)

2!n

if n is even,(36)

where [�a; a] is the size of the domain, and !n is determined by�1

b� !n tan(!na) = 0 if n is odd,

!n +1

btan(!na) = 0 if n is even.

The details of the derivations can be found in [10]. If the correlation function is not in the exponential form as in

equation (34) and the eigenvalue problem cannot be solved analytically, the correlation function can be constructed

numerically and a standard eigenvalue solver can be employed.

13

4

5

1

2

56

2

34

3

4

55

5

6

7

7

8

5

4

5

4

-5 0 5-1

0

1

8 0.00037 0.0002193066 0.000104885 1.23655E-054 -1.12158E-053 -7.85285E-052 -0.0002642041 -0.0005

5

5

6

7

7

8

2

345

1

2

34

5

6

5

3

3

4

-5 0 5-1

0

1

8 0.0002714297 0.0001940896 8.48499E-055 1.42857E-054 -1.42857E-053 -4.69169E-052 -9.14149E-051 -0.000133509

Figure 3: Deviation of mean solution from a parabolic pro�le in micro-channel ow with partially-correlated random

boundary conditions at the lower wall; Upper: u-velocity, Lower:v-velocity.

Here a = 5 and Re = 100; also the correlation length in (34) is set to b = 100. The eigenvalues from equation

(35) are

�1 = 9:675354; �2 = 0:1946362; �3 = 0:05014117; : : :

Due to the fast decay of the eigenvalues, we use the �rst two terms in the Karhunen-Loeve expansion (9). A fourth-

order Polynomial Chaos is used (p = 4) and �fteen expansion terms (P = 14) are needed. We also assume that

13

only the lower wall is considered to have the non-uniform random boundary condition. The upper wall is set to be

stationary and deterministic. The variance of Gaussian process of the lower wall is �2 = 0:12. A mesh with 10� 2

elemnts is employed and nineth-order Jacobi polynomial is used as the basis polynomial in each element. A parabolic

(no-slip) pro�le is used at the inlet and fully-developed conditions are asummed at the outlet.

1

1

4

1

4

1

3

2

1

1

2

3

1

-5 0 5-1

0

1

4 0.0716433 0.03187672 0.02331221 0.0105502

23

4

5

5

6

6

25

6 4

3

4

2

12

34

5

5 2 22 1

-5 0 5-1

0

1

6 0.0157875 0.006169684 0.001776943 0.001432742 0.0007764671 0.000312162

Figure 4: Contours of rms of u velocity (upper) and v velocity (lower).

Figure 3 shows the velocity contour plot of the deviation of the mean solution at steady-state from a parabolic

pro�le. The mean of u-veclocity remains close to the parabolic shape and the mean of v-velocity, although small in

magnitude, is non-zero, in contrast to the case where boundary conditions are uniformly random or deterministic.

Figure 4 shows steady-state solutions of the rms (root-mean-square) of u and v-velocity.

It is clear that theere is a developing region in the form of a boundary layer emanating at the inlet. Unlike

the fully-correlatred case, here all the higher-order expansion terms are non-zero, which implies that although the

random input is a Gaussian process, the solution output is not Gaussian. Since there is no analytic solution for

this ow with non-uniform random boundary conditions, we employ Monte Carlo (MC) simulation to validate the

Polynomial Chaos solution. The deterministic solver of the Navier-Stokes equations is used to compute the ow with

each realization of the Gaussian process from the exponential correlation function (34) as the boundary condition at

the lower wall. The MC solution is then obtained by gathering the statistics from the large ensemble of the solutions

from such realizations. A brief discussion on the generation of spatially correlated Gaussian �elds can be found in

Appendix A. For the comparison, we plot the non-dimensionalized variance of velocities, ums and vms, along the

centerline of the channel. It is de�ned as V ar(u)=�2 where V ar(u) is de�ned in equation (31) and �2 is the variance

of the input Gaussian �eld.

Figure 5 shows the solution of ums and vms along the centerline from MC simulation with 100, 500, 1000 and

14

x

Um

s

-4 -2 0 2 4

0

0.02

0.04

0.06

0.08

PCMC 100MC 500MC 1000MC 2000

x

Vm

s

-4 -2 0 2 4

0

0.01

0.02

0.03

0.04

PCMC 100MC 500MC 1000MC 2000

Figure 5: Variance of velocities along the centerline of the channel; Left: u-velocity, Right: v-velocity.

2000 realizations, together with the Polynomial Chaos (PC) solution. It is seen that the MC result converges as

the number of realizations increases and the solution converges to the PC solution. The advantage of Polynomial

Chaos expansion is evident here since the PC solution, with enough terms included for accurate approximation of

the randomness, is equivalent to the MC solution with in�nite number of realizations. While each single run of

PC solution is more expensive than a single deterministic run, in this case about 15 times more expensive as there

are 15 terms in the expansion, it is substantially faster than the Monte Carlo solution which requires thousands of

realizations in order to obtain solution statistics with comparable accuracy as the PC solutions. Note that here we

employ the standard Monte Carlo simulation without any acceleration algorithms, such as variance reduction [29].

4.2 Micro-channel Flow: Non-Gaussian Input

In this section we revisit the pressure-driven micro-channel ow discussed above but with non-Gaussian random

boundary conditions. Non-Gaussian distributions are not uncommon in micro- uidics where reactive ion etching

techniques cause anisotropies and preferential directions. The Polynomial Chaos expansion procedure can also be

applied to non-Gaussian processes. The particular non-Gaussian input we consider here corresponds to a lognormal

distribution. This process has been considered by Ghanem in [7] and [8]. It is chosen because its projection onto the

Polynomial Chaos basis can be obtained analytically. If this is not possible, as for most non-Gaussian processes, a

numerical projection procedure can instead be applied.

The lognormal process l is de�ned as

l(x) = eg(x); (37)

where g is a Gaussian process. We �rst apply the Karhunen-Loeve expansion of equation (9) to decompose the

15

Gaussian process g(x) as follows

g(x) = �g(x) +

nXi=1

gi(x)�i; (38)

where gi(x) =p�i�i(x) according to equation (9). We then expand the lognormal process l(x) by the Polynomial

Chaos

l(x) =

PXi=0

li(x)i(�): (39)

The expansion coeÆcients can be obtained by projecting the lognormal process onto the PC basis

li(x) =< l(x)i >

< 2

i >=

< eg(x)i >

< 2

i >: (40)

By using the Karhunen-Loeve expansion of g(x) (38), the above projection can be obtained analytically:

li(x) =< �i >

< 2

i >exp

0@�g(x) + 1

2

nXj=1

g2j

1A ; (41)

where the terms < �i > are listed below,

< �i >=

8>><>>:

gi when i = �i;gigj when i = �i�j � Æij ;gigjgk when i = �i�j�k � �iÆjk � �jÆki � �kÆij ;: : : : : : :

In case the process g is reduced to a random variable, the expansion of the lognormal variable can be simpli�ed to

l = exp

�g +

�2g2

!PXj=0

�jgj!j ; (42)

where �2g is the variance of the corresponding Gaussian variable g. A more detailed discussion about the lognormal

distribution and its Polynomial Chaos approximations can be found in Appendix B.

4.2.1 Fully-Correlated Random Boundary Conditions

We �rst consider the case where the boundary conditions at the walls are random but spatially uniform, i.e. all

grid points at the boundary are fully-correlated. The exact solution of equation (32) then applies. By substituting

the expansion of lognormal random variable (42) into the formula, we �nd that the exact solution consists of the

mean u0 and the random modes corresponding to the inputs from the two walls �1 and �2, linearly distributed across

the channel. There is no interaction between the two random walls so the cross terms, i.e., terms associated with

�i1�j2(i 6= j), in the expansion remain zero.

Figure 6 shows the velocity modes across the channel. The standard deviations of the underlying Gaussian

variables at lower and upper walls are �1 = 0:2 and �2 = 0:1, respectively. Fourth-order Polynomial Chaos is

employed (p = 4) and the total number of expansion terms is 15 (P = 14). Only the nonzero terms, i.e., terms

16

y

u 1,u

3

u 2,u

5

-1 0 10

0.07

0.14

0.21

0

0.04

0.08

0.12

u1

u2

u3

u5

y

u 6,u

10

u 9,u

14

-1 0 10

0.0005

0.001

0.0015

0

6E-05

0.00012

0.00018

u6

u9

u10

u14

Figure 6: Distribution of velocity random modes across the channel; Left: �rst- and second-order (non-zero) random

modes, Right: third- and fourth-order (non-zero) random modes.

associated with �1 or �2 only, are plotted (see table 1 for the indices of these terms). We see the linear distribution

of the di�erent random modes across the channel. All the cross terms, i.e., terms with the form �i1�j2(i 6= j), are zero

as expected. The highest order terms, u10 and u14, which are associated with the terms �41and �4

2, respectively, are

of the order of 10�6. This implies that the fourth-order PC expansion is adequate in this case.

4.2.2 Partially-Correlated Random Boundary Conditions

Next we consider spatially non-uniform input. The wall boundary condition is considered as a lognormal random

process as in equation (37). The underlying Gaussian process g(x) is exactly the same as the one in section 4.1.2 and

the same two-term Karhunen-Loeve expansion is applied to decompose g(x). Monte Carlo simulation is conducted

since no exact formula is known. Again only the lower wall is considered to be random and the standard deviation

of the underlying Gaussian process is � = 0:5. The non-dimensionalized variance of velocities, de�ned similarly as

in section 4.1.2, is plotted along the centerline of the channel in �gure 7.

Monte Carlo solutions corresponding to 100, 500 and 1,000 realizations are plotted, together with the solution

from the Polynomial Chaos expansion. It can be seen that as the number of realizations increases, the MC solution

converges to PC solution. The convergence rate is relatively slow due to the large variance of the random input.

4.3 Flow Past a Circulat Cylinder

In this section we simulate two-dimensional incompressible ow past a circular cylinder with random correlated

uctuations superimposed to the freestream. More speci�cally, the in ow takes the form

uin = �u+ g;

17

x

Um

s

-4 -2 0 2 40

0.05

0.1

PCMC 100MC 500MC 1000

x

Vm

s

-4 -2 0 2 4

0

0.01

0.02

0.03

PCMC 100MC 500MC 1000

Figure 7: Variance of velocities along the centerline of the channel; Left: u-velocity, Right: v-velocity.

where g is a Gaussian random variable or process. The size of the computational domain is [�15; 25]� [�9; 9] and

the cylinder is at the origin with diameter of D = 1. The Reynolds number Re = 100 based on the mean value of

the in ow velocity �u. The domain consists of 412 triangular elements and the Jacobi polynomial in each element for

spatial discretization is sixth-order.

4.3.1 Fully-Correlated Gaussian Freestream

In this case uin = �u + g and g = �� is a Gaussian random variable, where � = 0:01 is its standard deviation.

This results in a one-dimensional PC expansion and we employ fourth-order PC in the simulation. Figure 8 shows

the history of pressure signal at the rear stagnation point on the cylinder surface. The signal of the corresponding

deterministic simulation is also plotted and it is denoted as PD in dotted line, for reference purposes. It is seen that

the amplitude of the mean pressure signal is smaller compared to the deterministic signal due to di�usion induced

by the randomness. The higher modes, which describe the stochastic component of the solution, all start from zero

then develop gradually from P1 to P4 over time.

A stationary periodic state is reached after t = 350 in convective time units. Because only the mean and �rst

random modes (Gaussian part) have non-zero boundary inputs, the development to non-zero value of the higher

random modes is due to the interactions of the random modes through the nonlinear convective terms in equation

(19). Figure 9 shows the instantaneous vorticity distribution from the deterministic simulation and the mean solution

from the stochastic simulation at the same time t = 370; both simulations started with identical initial conditions

corresponding to a converged periodic state at Re = 100. Both plots use exactly the same contour levels so that the

randomness-induced di�usion in the stochastic solution can be identi�ed in the wake of the cylinder. In �gure 10 a

18

snapshot of rms of vorticity for the stochastic simulation is shown at the same time t = 370.

200 250 300 350

-0.16

-0.12

-0.08

-0.04

0 PD

P0

200 250 300 350

-0.04

0

0.04

0.08 P1

P2

P3

P4

Figure 8: Pressure signal of cylinder ow with uniform random in ow: � = 0:01. Upper: High modes. Lower: Zero

mode (mean).

It should be noted that this problem, although still relatively simple, is computationally much more complex

than the channel problem, and the approach of Monte Carlo simulation is prohibitively expensive.

4.3.2 Partially-Correlated Gaussian Freestream

We consider now the more realistic case of partially-correlated freestream velocity. We describe this partial correlation

by the exponential covariance kernel of equation (34) with variance �2 = 0:022. A relatively large correlation length

is chosen (b = 100) such that the �rst two eigenmodes are adequate to represent the process by Karhunen-Loeve

expansion (9). Thus, we employ a two-dimensional PC expansion and third-order polynomials (p = 3) (see table 1

for relevant terms).

Figure 11 shows the pressure signal, together with the deterministic signal for reference (denoted as PD in dotted

line). We see that the stochastic mean pressure signal has a smaller amplitude and it is out of phase with respect

to the deterministic signal. Although initially, the stochastic response follows the deterministic reponse, eventually

there is a change in the Strouhal frequency as shown in �gure 12. Speci�cally, the Strouhal frequency of the mean

stochastic solution is lower than the deterministic one due to e�ective lowering of the Reynolds number induced by

randomness. We now examine the response of the higher random modes. The �rst random mode �1 is dominant and

19

x

y

0 10 20-6

-4

-2

0

2

4

6

x

y

0 10 20-6

-4

-2

0

2

4

6

Figure 9: Instantaneous vorticity contours: Upper - Deterministic solution with uniform in ow; Lower - Mean

solution with uniform Gaussian random in ow.

11 1

4 2

1

2

1

4 5

3 4

3

1

3

1 1

x

y

0 10 20-6

-4

-2

0

2

4

67 1.422276 1.296765 1.014294 0.6071433 0.4975912 0.3454891 0.227166

Figure 10: Contours of rms of vorticity �eld with uniform Gaussian random in ow.

20

�2 is subdominant. Correspondingly, in the higher modes, only the terms associated with �1, i.e., terms P3 � �21and

P6 � �31are active (see table 1). These high modes are also plotted in �gure 11 (upper plot).

In �gure 13 we present velocity pro�les along the centerline for the deterministic and the mean stochastic solution

at the same time instant. We see that signi�cant quantitative di�erences emerge even with a relatively small 2%

uncertainty in the freestream. In �gure 14 we plot instantaneous vorticity contours for the mean of the vorticity and

compared it with the corresponding plot from the deterministic simulation. There is clearly a strong di�usive e�ect

induced by the randomness. In �gure 15 we plot contours of the corresponding rms of vorticity. It shows that the

uncertainty in uences the most interesting region of the ow, i.e., the shear layers and the vortex street and not the

far �eld.

Time

Pre

ssur

eS

igna

l

200 250 300 350 400-0.3

-0.25

-0.2

-0.15

-0.1

-0.05 PD

P0

Time

Pre

ssur

eS

igna

l

200 250 300 350 400

-0.04

0

0.04

0.08P1

P2

P3

P6

Figure 11: Pressure signal of cylinder ow with non-uniform Gaussian random in ow. Upper: High modes. Lower:

Zero mode (mean).

5 Summary and Discussion

We have developed a stochastic spectral method to model the uncertainty associated with boundary conditions in

simulations of incompressible ows. This method can also be applied to model uncertainty in the boundary domain,

e.g. a rough surface, and also to model the uncertainty associated with transport coeÆcients, e.g. the eddy viscosity

in large eddy simulations or other transport models. It sets the foundation for a composite error bar in CFD [30]

that includes, in addition to the numerical error, contributions due to imprecise input to the simulation. Clearly, this

uncertainty is propagated nonlinearly and the new method quanti�es statistically the uncertainty in the solution.

More speci�cally, we have employed Wiener-Hermite polynomials to represent the stochastic solution and in

21

0.05 0.1 0.15 0.2 0.25 0.30

0.5

1

1.5

2

2.5

3

3.5

4x 10

−4

Frequency

Spe

ctru

m

Deterministic signal Mean random signal: σ=0.02

Figure 12: Frequency spectrum for the deterministic (high peak) and stochastic simulation (low peak).

x0 5 10 15 20

-0.5

-0.25

0

0.25

0.5

0.75

1

UDeterministic

UMean

VDeterministic

VMean

Figure 13: Instantaneous pro�les of the two velocity components along the centerline for the deterministic and the

mean stochastic solution.

22

x

y

0 10 20

-5

0

5

x

y

0 10 20

-5

0

5

Figure 14: Instantaneous vorticity �eld : Upper - Deterministic solution with uniform in ow; Lower - Mean solution

with non-uniform Gaussian random in ow.

22

122

2

12

1

2

224

2

1

x

y

0 10 20

-5

0

5

7 1.546386 1.307815 1.14 0.8949323 0.62 0.2761361 0.124748

Figure 15: Instantaneous contours of rms of vorticity �eld with non-uniform Gaussian random in ow.

23

particular to discretize the random dimension. We have veri�ed in [31] that exponential convergence is obtained

for stochastic ordinary di�erential equations, for which exact solutions are available. On the other hand, it has

been reported in the literature that Wiener-Hermite expansions may be converging slower for certain distribution

functions, e.g. for Poisson processes [32]. This, however, can be recti�ed by employing a more suitable spectral

basis instead of the Hermite polynomials. For example in [32] multivariate Charlier polynomials were found to best

represent Poisson processes. This can be extended to many di�erent known distributions by employing a basis from

the general family of Askey polynomials [22], which includes polynomials both for continuous as well as discrete

processes. For example,

� Laguerre polynomials are associated with the Gamma distribution,

� Meixner polynomials with the Pascal distribution,

� Krawtchouk polynomials with the Binomial distribution,

� Jacobi polynomials with the Beta kernel, and

� Hahn polynomials with the Hypergeometric distribution.

For an arbitrary distribution any Wiener-Askey expansion would converge, in accord with the Cameron & Martin

theorem [24], however certain representations converge faster than others similarly to deterministic spatial expansions.

As regards eÆciency, a single Polynomial Chaos based simulation, albeit computationally more expensive than the

deterministic Navier-Stokes solver, is able to generate the solution statistics equivalent to the Monte Carlo simulation.

In the latter, thousands of realizations are required for converged statistics, which is prohibitively expensive for most

CFD problems in practice. For example, to obtain converged statistics using the standard Monte Carlo approach

for the two-dimensional ow past a cylinder that we presented in section 4.3 it would require currently more than a

year of computation on a standard workstation compared to about six days for the Polynomial Chaos simulation. Of

course a reduced variance version could be employed and also the Monte Carlo simulation is embarassingly parallel,

and thus it can be accelerated greatly. However, the Polynomial Chaos based stochastic simuilation can also be done

in parallel. One possible strategy is to distribute one random mode per processor, as the bulk of the work is parallel

due to linearity, and transpose the data to perform the nonlinear products in a separate step. Even with a modest

parallel eÆciency, the aforementioned computation would be completed in less than a day on a �fteen-processor

computer.

24

A Generation of Correlated Gaussian Random Fields

While numerous routines are available for generating independent Gaussian random variables, it is often needed to

generate a correlated Gaussian �eld (vector) with given correlation structure. In general, this can be performed by

means of: (1) Spectral representation; (2) ARMA (Auto-Regressive Moving Average) modeling; and (3) Covariance

matrix decomposition procedures. There are two approaches for covariance matrix decomposition methods: The

Cholesky decomposition method and the modal decomposition method. In this section we brie y review the Cholesky

decomposition method [33] and the �rst-order ARMA method.

A.1 The Cholesky Decomposition Method

Without loss of generality, we consider the normal random vector x = (x1; x2; � � � ; xn) with zero mean, i.e., each

xi(i = 1; : : : ; n) is a Gaussian random variable with zero mean. Let us denote its covariance matrix as

C =

264

�11 � � � �1n...

. . ....

�n1 � � � �nn

375 : (43)

We then call x the normal random vector of N(0;C).

index

Ran

dom

field

0 25 50 75 100

-3

-2

-1

0

1

2

3

Independent Gaussian fieldCorrelation length = 1Correlation length = 10

Figure 16: Correlated Gaussian random �eld with exponential correlation structure (correlation length is non-

dimensionalized by the size of the domain)

Let y be distributed N(0; In) where In is the unit matrix of size n, i.e., y is an independent (uncorrelated)

25

Gaussian random �eld. Let also x = Ay, then x is distributed N(0;AAT ) where AT is the transpose of A. This

result is a special case of a more general theorem by Anderson [34].

The problem of generating Gaussian random �eld with given correlation structure C is then transformed to the

problem of �nding matrix A such that AAT = C. Since matrix C is real and symmetric, this can be readily done

by the Cholesky's decomposition, where A takes the form of a lower-triangular matrix:

ai1 = �i1=p�11; 1 � i � n;

aii =

q�ii �

Pi�1k=1

a2ik; 1 < i � n;

aij =h�ij �

Pj�1k=1

aikajk

i=ajj ; 1 < j < i � n;

aij = 0; i < j � n:

(44)

Once the aij 's have been determined according to the given �ij 's, we need to generate an n-dimensional independent

Gaussian vector y, and subsequently perform the transformation x = Ay. The resulting Gaussian random vector x

will have the desired correlation structure C = f�ijg.

Figure 16 shows the correlated Gaussian �eld of the exponential correlation function de�ned in (34) with unit

variance. The correlation length is non-dimensionalized by the size of the domain. The two results with correlation

length of 1 and 10 are both obtained by applying the above Cholesky decomposition method to the same independent

Gaussian �eld also shown in the �gure. We see that the original �eld is uncorrelated and by increasing the correlation

length the correlation structure of the �eld becomes stronger.

A.2 First-Order ARMA Process Method

Another way of generating correlated Gaussian �eld is based on the idea of employing the �rst-order Auto-regressive

(AR(1)) process, also called the Markov process. This method works speci�cally for the exponential covariance

function (34)

C(x1; x2) = ejx1�x2j

b ; (45)

where b is the correlation length. The stationary AR(1) process is de�ned as,

xi = �xi�1 + �yyi; j�j < 1; (46)

where fyig is the set of independent Gaussian variables with zero mean and unit variance. It can be shown that the

stationary AR(1) process has the following properties [35]:

E(xi) = 0; (47)

V ar(xi) =�2y

1� �2; (48)

C(xi; xi�k) = �jkj; k = 0;�1;�2; : : : (49)

26

It is then straightforward to show that by choosing the parameters

� = e1b ; �y =

p1� �2; (50)

the resulting random vector x = fxig is a Gaussian random �eld with the exponential covariance kernel (equation

(45)).

B Lognormal Distribution and its Approximations

While the normal distribution arises from the sum of many small e�ects according to the Central Limit Theorem, the

lognormal distribution arises as the result of a multiplicative mechanism. Let X denote a Gaussian random variable,

the lognormal random variable is obtained by taking the exponential of X ,

Y = eX ; X = lnY: (51)

A random variable Y whose logarithms are normally distributed is said to have the lognormal distribution.

The PDF (Probability Density Function) of the Gaussian variable X is:

fX(x) =1

�Xp2�

exp

"�1

2

�x�mX

�X

�2#; �1 � x �1; (52)

where mX and �2X are the mean and variance of the Gaussian random variable X , respectively.

Equation (51) de�nes a one-to-one monotonic transformation. It is well known that when such transformation in

the form of Y = g(X) exists, the PDF of the new random variable Y is:

fY (y) =

����dg�1dy(y)

���� fX(g�1(y)): (53)

Therefore, upon substitution the PDF of the lognormal distribution is:

fY (y) =1

y�Xp2�

exp

"�1

2

�ln y �mX

�X

�2#; y � 0: (54)

The moments of lognormal distribution can be calculated as follows [36]:

< Y r >= ermXe12r2�2

X ; r = 1; 2; : : : (55)

The Polynomial Chaos expansion to the lognormal random variable is given in equation (42).

Y = exp

�mX +

�2X2

� PXj=0

�jXj!

j ; (56)

27

Upon substituting the one-dimensional Polynomial Chaos (equation (7)) we obtain the �rst few terms of the approx-

imation:

Y = c

�1 + �X� +

�2X2!

(�2 � 1) +�3X3!

(�3 � 3�) + � � ��; c = emX+

�2X

2 : (57)

The Taylor expansion-like formula approximates the lognormal distribution with a Gaussian distribution and higher-

order non-Gaussian correction terms. The PDF of the Polynomial Chaos approximation can be obtained via several

common techniques such as the method of Cumulative Distribution Function (CDF) [37]:

1. First-order approximation:

Y1 = c(1 + �X�); (58)

f1(y) =1

c�Xf(�); � =

�yc� 1�: (59)

2. Second-order approximation:

Y2 = c

�1 + �X� +

�2X2!

(�2 � 1)

�; (60)

f2(y) =1

c�X

1

r[f(�1) + f(�2)] ; (61)

where

�1;2 =1

�X(�1� r); r =

q�1 + �2X + 2y=c:

3. Third-order approximation:

Y3 = c

�1 +

�2X2!

(�2 � 1) +�3X3!

(�3 � 3�)

�; (62)

f3(y) =1

c�X

�4Xb2

�1 +

�4X (1� �2X )

b2

��1 +

1� 3y=c

r

�f(�); (63)

where

r =q2� 6y=c+ 9(y=c)2 � 3�2X + 3�4X � �6X ; b = �2X (1� 3y=c+ r)

1=3;

and

� = � 1

�X+�X � �3X

b� b

�3X:

In the above equations, the function f(x) = 1p2�e�x

2=2 is the PDF for the standard Gaussian variable � which has

zero mean and unit variance. Figure 17 shows the PDFs of the �rst-, second- and third-order PC approximations

to the lognormal distribution, together with the exact lognormal PDF. The results are obtained with mX = 0 and

�X = 0:3. It can be seen that the �rst-order approximation (58) is symmetric because it only contains the Gaussian

term. As more high-order terms are added, the PDFs become asymmetric and the third-order expansion gives a very

good approximation.

28

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

exact 1st−order2nd−order3rd−order

Figure 17: Probability density functions (PDF) of the lognormal distribution and its 1st-, 2nd- and 3rd-order

Polynomial Chaos approximations.

Acknowledgements

We would like to thank Dr. M. Jardak, D. Lucor, Prof. C.-H. Su and Prof. R. Ghanem for useful discussions. This

work was supported by ONR and computations were performed at Brown's TCASCV and NCSA's (University of

Illinois) facilities.

29

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