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Computational Fluid Dynamics
Computational Fluid Dynamics
http://www.nd.edu/~gtryggva/CFD-Course/
Grétar Tryggvason
Lecture 21April 5, 2017
Computational Fluid Dynamics
We can expect that the development of numerical methods for
computing fluid flow will continue and we will see methods that
are
• More accurate• More robust• More versatile
than current methods.
We are also seeing new trends and a shift in emphasize to topics
that in the past were on the sidelines.
Computational Fluid Dynamics
New Directions for Computational Physics:
Computational predictivity
Multiscale/multiphysics Integrated simulations of complex
systems
Computational Fluid Dynamics
Section 605 of the America COMPETES Reauthorization Act of 2010
[2] is titled Promoting use of high-end computing simulation and
modeling by small and medium sized manufactures, and states that
Congress finds that:
(1) the utilization of high-end computing simulation and
modeling by large-scale government contractors and Federal research
entities has resulted in substantial improvements in the
development of advanced manufacturing technologies; and
(2) such simulation and modeling would also benefit small- and
medium-sized manufacturers in the United States if such
manufacturers were to deploy such simulation and modeling
throughout their manufacturing chains.
Ensuring American Leadership in Advanced Manufacturing” by the
President’s Council of Advisors on Science and Technology (PCAST)
and the President’s Innovation and Technology Advisory Committee
(PITAC) states that “powerful computational tools and resources for
modeling and simulation could allow many U.S. manufacturing firms
to improve their processes, design, and fabrication.”
H. R. 5116
One Hundred Eleventh Congress of the
United States of America AT THE SECOND SESSION
Begun and held at the City of Washington on Tuesday, the fifth
day of January, two thousand and ten
An Act To invest in innovation through research and development,
to improve the competi-
tiveness of the United States, and for other purposes.
SECTION 1. SHORT TITLE; TABLE OF CONTENTS.
(a) SHORT TITLE.—this Act may be cited as the ‘‘America
COM-PETES Reauthorization Act of 2010’’ or the ‘‘America Creating
Opportunities to Meaningfully Promote Excellence in Technology,
Education, and Science Reauthorization Act of 2010’’.
(b) TABLE OF CONTENTS.—The table of contents for this Act is as
follows:
Sec. 1. Short title; table of contents. Sec. 2. Definitions.
Sec. 3. Budgetary impact statement.
TITLE I—OFFICE OF SCIENCE AND TECHNOLOGY POLICY
Sec. 101. Coordination of Federal STEM education. Sec. 102.
Coordination of advanced manufacturing research and development.
Sec. 103. Interagency public access committee. Sec. 104. Federal
scientific collections. Sec. 105. Prize competitions.
TITLE II—NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Sec. 201. NASA’s contribution to innovation and competitiveness.
Sec. 202. NASA’s contribution to education. Sec. 203. Assessment of
impediments to space science and engineering workforce
development for minority and under-represented groups at NASA.
Sec. 204. International Space Station’s contribution to national
competitiveness en-
hancement. Sec. 205. Study of potential commercial orbital
platform program impact on
Science, Technology, Engineering, and Mathematics. Sec. 206.
Definitions.
TITLE III—NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
Sec. 301. Oceanic and atmospheric research and development
program. Sec. 302. Oceanic and atmospheric science education
programs. Sec. 303. Workforce study.
TITLE IV—NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY
Sec. 401. Short title. Sec. 402. Authorization of
appropriations. Sec. 403. Under Secretary of Commerce for Standards
and Technology. Sec. 404. Manufacturing Extension Partnership. Sec.
405. Emergency communication and tracking technologies research
initiative. Sec. 406. Broadening participation. Sec. 407. NIST
Fellowships. Sec. 408. Green manufacturing and construction. Sec.
409. Definitions.
Computational Fluid Dynamics
February 2012
Cyberinfrastructure for 21st Century Science and Engineering
Advanced Computing Infrastructure Vision and Strategic Plan
1 iii
Reports on the future of simulations
Computational Fluid Dynamics
https://www.youtube.com/watch?v=TGSRvV9u32M
HPC Video
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Computational Fluid Dynamics
Validation & Verification
Uncertainty Quantification
Computational Fluid Dynamics
Demonstrating that a numerical solution is correct is a standard
expectation of ALL numerical predictions. At the most elementary
level this is done by grid refinement
The general expectation is that the grid is changed by a factor
of 2 and the solution shown for three grids. If this is
impractical, a grid refinement can be done on a smaller problem
with comparable physical parameters
A new code is generally expected to go through more extensive
studies, ideally converging to an exact solution at the expected
rate
Computational Fluid Dynamics
In early work, a comparison of numerical results with
experimental results was used to demonstrate the correctness of a
code (this is still done by inexperienced investigators)
It has, however, been understood for some time that the issue is
more involved and we need to distinguish between the mathematical
model of a physical process and the numerical solution of the
mathematical model.
More precisely………
Computational Fluid Dynamics
To accurately predict the behavior of a system computationally,
we need:
A correct and accurate code. The code correctness is
demonstrated by Verification
An model that accurately describes the process under study: The
accuracy of the model is established by Validation
A set of input parameters whose accuracy and reliability is
known. Assessing the accuracy of the results is done by Uncertainty
Quantification
Computational Fluid Dynamics
From: Schlesinger, S. Terminology for Model Credibility,
Simulation, Vol. 32, No. 3, 1979; 103-104. Cited in W. L. Oberkampf
and T.G. Trucano. Verification and Validation in Computational
Fluid Dynamics. SANDIA REPORT SAND2002-0529 (2002)
Computational Fluid Dynamics
Verification
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Computational Fluid Dynamics
Verification: Show that the code solves the equations that it is
intended to solve with the expected accuracy. Verification consists
of Code Verification (which can be done once and for all) and
Solution Verification which must be done for all new problems.
The key tool to verify a code and solution correctness is grid
refinement.
Computational Fluid Dynamics
Code Verification is independent of any physical reality. The
goal is simply to show that the code correctly solves the equations
that it is intended to solve. If an exact solution is available,
then it can be used. If not, the Method of Manufactured Solutions
(MMS) can be used.
Computational Fluid Dynamics
From:AIAA.GuidefortheVerificationandValidationofComputationalFluidDynamics
Simulations, American Institute of Aeronautics and Astronautics,
AIAA-G-077-1998, Reston, VA, 1998. Cited in W. L. Oberkampf and
T.G. Trucano. Verification and Validation in Computational Fluid
Dynamics. SANDIA REPORT SAND2002-0529 (2002)
Computational Fluid Dynamics
Method of Manufactured Solutions
Computational Fluid Dynamics
The basic idea is to add a source term to the equations that
force the solution to take a given value. Suppose we have:
L(u) = 0Taking
u = qobviously does not satisfy this equation. However if we add
a source to the RHS, given by
g = L(q)
L(u) = gThen is a solution to u = q
Computational Fluid Dynamics
Or, said a little differently, we write
L(q)− L(q) = 0Which obviously is true for any q. The solution
to
u = qis obviously
g = L(q) L(u) = g
Or, is the exact solution to
L(u) = L(q)
u = q
where
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Computational Fluid Dynamics
Thus, the Method of Manufactured Solutions consists of
1. Picking a function q2. Computing the source term3. Solving
the original equation with the new source
term
Notice that there is no requirement that the function q
satisfies the equations or that it has any physical meaning.
Computational Fluid Dynamics
Trivial Example:
d 2udx2
= 0; 0 < x < 1; u(0) = 1; u(1) = 0;
q = 1− x2;
dqdx
= −2x;d 2qdx2
= −2; ⇒ g = −2
The solution is given by: u = 1− x
Take:
Solve:
d 2udx2
= −2;dudx
= −2x + C1; u = −x2 + C1x + C2
C1 = 0; C2 = 1; ⇒ u = 1− x2 As we intended!
Computational Fluid Dynamics
For a more “real” example consider
∂f∂t+ f
∂f∂x
= D∂2 f∂x2
q = A+ sin(x + Ct)
With the manufactured solution:
First we rewrite our equation as:
L( f ) =
∂f∂t+ f
∂f∂x
− D∂2 f∂x2
= 0
So our source is
g =
∂q∂t+ q
∂q∂x
− D∂2q∂x2
Computational Fluid Dynamics
q = A+ sin(x + Ct)
∂q∂t
= C cos(x + Ct);∂q∂x
= cos(x + Ct);∂2q∂x2
= − sin(x + Ct);
Our manufactured solution is
We have
So the source term is
g = C cos(x + Ct) + A+ sin(x + Ct)( )cos(x + Ct) + sin(x +
Ct);
From: P. J. Roache. Fundamental of Verification and Validation.
Hermosa, 2009
Computational Fluid Dynamics
f = A+ sin(x + Ct)Therefore,
Is an EXACT solution to
∂f∂t+ f
∂f∂x
= D∂2 f∂x2
+ g
g = C cos(x + Ct) + A+ sin(x + Ct)( )cos(x + Ct) + Dsin(x +
Ct);
Computational Fluid Dynamics
% Method of Manufactured Solution for the 1D Burgers
equation%------------------------------------------------------------n=61;
nstep=2000; length=2*pi;h=length/(n-1);diff=0.05;dt=1.0/nstep
f=zeros(n,1); y=zeros(n,1); ex=zeros(n,1); time=0.0; for i=1:n;
x(i)=h*(i-1);endA=1.0; C=1.0; for i=1:n, f(i)=A+sin(x(i)); end;
%initial conditions
for m=1:nstep+1,m for i=1:n, ex(i)=A+sin(x(i)+C*time); end;
%exact solution hold off;plot(f,'linewidt',6); axis([1 n -1.0,
3.0]); % plot solution hold on;plot(ex,'r','linewidt',2); %pause; %
plot exact solution err=0.0;for i=1:n, err=err+h*(ex(i)-f(i))^2;
end;err=sqrt(err) y=f; % store the solution for i=2:n-1,
g=C*cos(x(i)+C*time)+(A+sin(x(i)+C*time))*cos(x(i)+C*time)+diff*sin(x(i)+C*time);
f(i)=y(i)-0.5*(dt/h)*y(i)*(y(i+1)-y(i-1))+...
diff*(dt/h^2)*(y(i+1)-2*y(i)+y(i-1))+dt*g; % advect by centered
differences end;
g=C*cos(x(n)+C*time)+(A+sin(x(n)+C*time))*cos(x(n)+C*time)+diff*sin(x(n)+C*time);
f(n)=y(n)-0.5*(dt/h)*y(n)*(y(2)-y(n-1))+diff*(dt/h^2)*(y(2)-2*y(n)+y(n-1))+dt*g;
% do endpoints f(1)=f(n); % for periodic time=time+dt %
boundariesend;
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Computational Fluid Dynamics
2
1Solution at time 1
The error versus grid spacing
Computational Fluid Dynamics
Error in boundary conditions destroys second order
convergence!
The manufactured solution is used in code verification exactly
as we would use an exact solution to the original equations. If we
do not, for example, get the expected convergence rate, then there
must be an error somewhere
Correct second order convergence 2
1
h
Computational Fluid Dynamics
∂f∂t+ f
∂f∂x
= D∂2 f∂x2
q = ex sin t( )
g = ex cos t( ) + ex sin t( )( )2 − Dex sin t( )
The manufactured solution can be selected in many different
ways. We could, for example, pick:
From: P. J. Roache. Fundamental of Verification and Validation.
Hermosa, 2009
Which gives us the source
For the Burger’s equation
Computational Fluid Dynamics
1. Manufactured solutions should be smooth analytic functions
like polynomials, trigonometric, or exponential functions so that
the solution is conveniently computed.
2. The solution should exercise every term in the
equations3. The solution should have sufficient number of
derivatives4. The derivatives should be bounded by small
constants5. The solution should not prevent the code from running
to
completion6. The solution should be defined on a connected
subset of two-
or three-dimensional space7. The solution should be constructed
in a manner such that the
differential operators in the PDE’s make sense.
Adopted from: K. Salari and P. Knupp. Code Verification by the
Method of Manufactured Solutions. SAND2000 – 1444 (2000)
Guidelines for the MMS
Computational Fluid Dynamics
Generating the source terms can be complicated for complex
operators. This can, however, easily be done using symbolic
manipulation software such as MAPLE or Mathematics
Computational Fluid Dynamics
Verification involves two different steps:
Code verification Generally done once to ensure that the code is
correct, using for example the method of manufactured solutions
Solution verificationDone every time the code is used to produce
a solution to ensure that the solution errors are acceptable (that
the solution is converged)
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Computational Fluid Dynamics
Egrid1 = Chp
Egrid 2 = Chr
!
"#$
%&
p
Egrid1Egrid 2
=Chp
Chp!
"#$
%&r p = r p
p = log
Egrid1Egrid 2
!
"#
$
%& / log r( )
Suppose we have a supposedly pth-order solution on two grids
where h2=h1/r. The error is then
The ratio of the errors is
Which allows us to compute the actual order:
Computational Fluid Dynamics
Theoretical convergence rate
h
The actual order p, for the Burger’s equation example, computed
as on the previous slide
Computational Fluid Dynamics
From: C.J. Roy, C.C. Nelson, T.M. Smith, C.C. Ober, Verification
of Euler/Navier–Stokes codes using the method of
manufacturedsolutions, Int. J. Numer. Meth. Fluids 44 (6) (2004)
599–620. Cited in: C.J. Roy. Review of code and solution
verification procedures for computational simulation. Journal of
Computational Physics 205 (2005) 131–156
Euler Equations for 2D FlowRoy, Nelson, and Smith tested two
codes using:
Computational Fluid Dynamics
The method of manufactured solutions does not address all issues
of code verification, such as domain size and boundary conditions.
Nevertheless, it is emerging as one of the major tool in ensuring
that a given set of equations is correctly solved.
Computational Fluid Dynamics
Solution Verification (Correct code, wrong solution)
While code verification is usually done once, solution
verification needs in principle to be done every time the code is
used to generate a solution. In practice an experienced user will
have a good idea about the necessary resolution.
A correct code but insufficient resolution or other numerical
parameters (iteration errors, for example) can lead to inaccurate
and even wrong solutions. For new problems the accuracy must be
verified.
Computational Fluid Dynamics
Richardson Extrapolation
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Computational Fluid Dynamics
Here, f(0) is the (unknown) exact solution and C is a constant
determining the magnitude of the error. Given f(h) and f(2h),
estimate C. Once C is known, we can use the above formula to find a
better estimate for the solution. This procedure is called
Richardson Extrapolation and is widely used in practice.
�
f h( ) = f 0( ) + Ch2 + HOT
The approximate solution can be written as a Taylor series
around the exact solution f(0):
Richardson ExtrapolationComputational Fluid Dynamics
We have:similarly, the solution on twice as coarse grid is:
subtracting to eliminate the h2 term:
Solving for the exact solution:
Since the Higher Order Terms (HOT) are at least O(h3), f(0) is a
better estimate than either f(h) or f(h2). Similar formulas can be
derived for schemes of different orders.
�
f h( ) = f 0( ) + Ch2 + HOT
�
f 2h( ) = f 0( ) + C4h2 + HOT
�
4 f h( ) − f 2h( ) = 4 f 0( ) − f 0( ) + HOT
�
f 0( ) = 4 f h( ) − f 2h( )3
+ HOT
Richardson Extrapolation
Computational Fluid Dynamics
For an p-th order scheme we have:
similarly, the solution on a finer grid is:
subtracting to eliminate the hn term:
Solving for the exact solution:
f h( ) = f 0( ) + Chp + HOT
f h / r( ) = f 0( ) + C h / r( )p + HOT
f h( ) − f h / r( )r p = f 0( ) 1− r p( ) + HOT
f 0( ) =
f h( ) − f 2h( )r p1− r p
+ HOT
Richardson ExtrapolationComputational Fluid Dynamics
Validation
Computational Fluid Dynamics
Validating a theory consists of comparing its predictions with
experimental results. It is thus at the core of science and as such
not a computational issue.
However, scientific computing has greatly increased our
abilities to solve complex models and this is leading to more and
more complex models, with new issues and challenges for
validation.
Computational Fluid Dynamics
From:AIAA.GuidefortheVerificationandValidationofComputationalFluidDynamics
Simulations, American Institute of Aeronautics and Astronautics,
AIAA-G-077-1998, Reston, VA, 1998. Cited in W. L. Oberkampf and
T.G. Trucano. Verification and Validation in Computational Fluid
Dynamics. SANDIA REPORT SAND2002-0529 (2002)
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Computational Fluid Dynamics
From:Oberkampf, W. L., and Trucano, T. G. Validation Methodology
in Computational Fluid Dynamics, AIAA 2000-2549, Fluids 2000
Conference, Denver, CO, 2000. Cited in W. L. Oberkampf and T.G.
Trucano. Verification and Validation in Computational Fluid
Dynamics. SANDIA REPORT SAND2002-0529 (2002)
Validation Hierarchy for a Hypersonic Cruise Missile
Computational Fluid Dynamics
At the present time there is no real theory covering how to
validate a computational model, except compare the predictions with
experiments or observations for selected cases. For complex models,
build by assembling sub-models, the sub-models are usually
validated independently.
C.J. Roy. Review of code and solution verification procedures
for computational simulation. Journal of Computational Physics 205
(2005) 131–156
AIAA. 1998. AIAA Guide for the Verification and Validation of
Computational Fluid Dynamics Simulations. American Institute of
Aeronautics and Astronautics.
Standard for Verification and Validation in Computational Fluid
Dynamics and Heat TransferV V 20 – 2009
Computational Fluid Dynamics
Volume 217, Issue 1, Pages 1-276 (1 September 2006) Special
Issue: Uncertainty Quantification in Simulation ScienceEdited by
George Em Karniadakis and James Glimm
Computational Fluid Dynamics
Uncertainty Quantification
Computational Fluid Dynamics
In most cases uncertainties do not only come from the numerical
solution but also from the problem specification. Those
uncertainties include:
• Material properties (density, viscosity, etc)• Domain
geometry• Boundary conditions• Model assumption
In principle these uncertainties can be treated in the same way
as experimental uncertainties.
Computational Fluid Dynamics
In practice, the quantification of uncertainties requires us to
assume that the error follows a particular distribution and it is
easiest to deal with uncorrelated errors.
The identification of the uncertainty in all model and input
parameters is challenging and there are considerable questions
whether standard uncertainty quantification will ever be able to
deal with “one-off events” or “Black Swans.” The role of incorrect
use of uncertainty models (Black-Scholes, etc.) in the recent
financial crisis suggests caution!
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Computational Fluid Dynamics
Definitions
Epistemic Uncertainty: Imperfect knowledge and models
(turbulence modeling)
Aleatoric Uncertainty: Physical variability. Generally cannot be
reduced. Described using probabilistic models.
For the most part, uncertainty quantification assumes that we
can use probabilistic approaches.
Computational Fluid Dynamics
Uncertainty Quantification
Data Assimilation: Determine the uncertainty in the input data
(material properties, geometry, etc.)
Propagation of Uncertainty: Map the uncertainty in the inputs
into uncertainty in the predicted values
Certification: develop reliability or confidence metrics
Computational Fluid Dynamics
Propagating uncertainty: Uncertainty in input parameters is
mapped into a distribution of possible outcomes
Uncertainty in input
Unc
erta
inty
in p
redi
ctio
n
Mapping by the computer code
Computational Fluid Dynamics
What are we looking for? Statistical description of the
prediction
Probability distribution for the predicted variable f
E f⎡⎣ ⎤⎦= zp z( )−∞∞
∫ dz
p z( )
z
Var f⎡⎣ ⎤⎦= z − E f⎡⎣ ⎤⎦( )2p z( )−∞
∞
∫ dz = E f 2⎡⎣ ⎤⎦− E f⎡⎣ ⎤⎦2
Expected value
Variance
Computational Fluid Dynamics
Two main approaches
Non-Intrusive Methods: Usually based on sampling. We run our
model for several values of the input parameters and construct a
distribution of possible outputs from the samples. Choosing the
sample points is the main task
Intrusive Methods: Derive expressions for the evolution of the
probability distribution and evolve those along with the solutions.
This requires us to change the code or write a new one
Computational Fluid Dynamics
Sampling Methods
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Computational Fluid Dynamics
Monte Carlo: Pick points at random. Can lead to “holes” and
“clusters” and usually converges slowly
Latin Hypercube: divide the domain into regions with equal
probability and pick at random within each interval
Collocations: select the sampling points based on assumptions
about the distribution
Computational Fluid Dynamics
Selecting the “best” sampling procedure is extremely important,
particularly for high dimensional problems.
Monte Carlo methods will give the right answer but generally
converge very slowly. Thus, significant effort has gone into
finding methods that minimize the number of sampling points needed.
In addition to sampling methods, significant effort has gone into
efforts to evaluate the integrals using properly selected
quadrature points
Computational Fluid Dynamics
Monte Carlo SamplingWoodward and Colella forward facing step
problem. Top: Velocity divergence field. Nominal (deterministic)
solution corresponding to Mach = 2.75. Bottom: Mean velocity
divergence field, corresponding to an uncertain input Mach number
[2.5 : 3.0]. Monte Carlo sampling based on 1000 realizations.
From: G. Iaccarino, P. Pettersson, J. Nordstrom and J.
Witteveen. NUMERICAL METHODS FOR UNCERTAINTY PROPAGATION IN HIGH
SPEED FLOWS. V European Conference on Computational Fluid Dynamics
ECCOMAS CFD 2010 J. C. F. Pereira and A. Sequeira (Eds) Lisbon,
Portugal,14-17 June 2010
G. Iaccarino, P. Pettersson, J. Nordström, J. Witteveen
2 UNCERTAINTY IN SHOCK-DOMINATED FLOW
Consider the supersonic flow over a forward step in a channel,
the classical Woodward-Colella test [8]. The objective is to
characterize the position of the shock waves at aparticular instant
of time after the impulsive start of the channel. In Fig. 1 the
nominalsolution obtained for an inflow Mach number of 2.75, at time
t = 2 on a grid consisting of≈ 16, 000 elements is reported. The
numerical solution is obtained using a second-orderspatial
discretization and an explicit Runge-Kutta integration [9]. The
velocity divergenceis used as a scalar field indicator to expose
the shock location.
We assume that the inflow Mach number is uncertain, specified as
a uniform randomvariable defined over the interval [2.5, 3.0]. In
the UQ analysis, we will consider the outputof interest to be the
statistical average (mean) of the velocity divergence.
A straightforward application of Monte Carlo (MC) method leads
to the results shownin Fig.2, an ensemble averaging of 1000
solutions obtained for randomly chosen Machnumbers in the above
mentioned interval.
Figure 2: Woodward and Colella [8] forward facing step problem.
Mean velocity divergence field. corre-sponding to an uncertain
input Mach number [2.5 : 3.0]. Monte Carlo sampling based on 1000
realizations.
It is obvious from the comparison of Fig. 1 and 2 that the mean
effect of the uncertaintyin the inflow conditions is to smear the
shock wave, an observation reported previouslyin the literature
[7].
In order to simplify the description of the spectral expansion
methods, we will discussnow a simplified shock dynamics problem
governed by the Burgers equations.
3 A MODEL PROBLEM
We consider the one-dimensional viscous Burgers equation:
∂u
∂t+ u
∂u
∂x= ν
∂2u
∂x2(1)
The viscosity is assumed to be small (ν = 0.02) and we are
interested in the time-dependent behavior that includes the
formation of shocks (or, more precisely, very sharpgradients). The
initial conditions are uncertain and assumed to be:
3
G. Iaccarino, P. Pettersson, J. Nordström, J. Witteveen
random variables, and the objective of the UQ analysis is to
determine the probability dis-tributions (or some statistical
moments) of the output. From a mathematical perspectivethe original
(deterministic) mathematical problem is cast in a stochastic
framework.
Sampling methods such as Monte Carlo have been traditionally
applied to solve suchstochastic problems; realizations are drawn
from input probabilistic distributions and theensemble of the
corresponding solutions is interpreted as an empirical distribution
of therandom solution allowing to generate statistical outputs [1].
The flexibility and simplicityof this approach has led to its wide
use, but its slow convergence limits its applicabilityfor
large-scale problems, motivating research in other
methodologies.
Figure 1: Woodward and Colella [8] forward facing step problem.
Velocity divergence field. Nominal(deterministic) solution
corresponding to Mach = 2.75.
Recently several approaches based on stochastic expansions of
the solution in the spacespanned by the uncertain variables have
been introduced [6]. Polynomial chaos methodsare particularly
attractive because they allow for a reformulation of the stochastic
prob-lem as a set of (coupled) deterministic problems that are
amenable to analysis [10, 11].On the other hand, this approach
requires modification of the existing computationaltool: is an
intrusive approach. An alternative class of methods, namely
stochastic collo-cation approaches, share a similar mathematical
structure but allow for reuse of existingdeterministic codes
[5]
Stochastic expansion methods exhibit exponential convergence
and, therefore, can beextremely effective when compared to
Monte-Carlo-type approaches. This advantage isa direct consequence
of the expected smoothness of the system response with respect
tovariability of the input quantities [10, 5]. As mentioned before
it is clear that in problemscharacterized by highly non-linear
responses and sharp transitions, stochastic expansionsmethods can
experience difficulties. In this paper, we explore the application
of suchmethods to unsteady high-speed compressible flow problems
governed by the Burgers andEuler equations. We illustrate how the
presence of shock waves hinders the convergence ofthe approach, and
discuss why both intrusive and non-intrusive formulations suffer
fromthis problem.
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Computational Fluid Dynamics
Stochastic Galerkin Methods
orPolynomial Chaos
Computational Fluid Dynamics
The alternative to sampling methods is to derive equations for
the mapping of the probability distribution.
To propagate the uncertainty, we expand the probability
distribution function as a series of orthonormal basis
functions.
The basis functions depend on the error distribution. For
Gaussian distribution we use Hermite polynomials. For a uniform
distribution Legendre polynomials are used.
f x,t,ξ( ) = fk x,t( )k=0
∞
∑ Hk ξ( )
Computational Fluid Dynamics
Gaussian distribution of the initial uncertainty is very
common.The Hermite polynomials are defined by:
http://www.efunda.com/math/Hermite/Hermite.cfm
H0 =1H1 = x
H2 = x2 −1
H3 = x2 − 3x
H4 = x4 − 6x2 + 3
HiH j = Hi ξ( )H j ξ( ) p ξ( )dξ = i!δ ij−∞∞
∫ p ξ( ) =12πe−ξ
2 /2
Inner product
where
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Computational Fluid Dynamics
In practice, we truncate the expansion and follow only a few
moments
f x,t,ξ( ) = fk x,t( )k=0
∞
∑ Hk ξ( ) ≈ fk x,t( )k=0
M
∑ Hk ξ( )
E f⎡⎣ ⎤⎦ = E fkk=0
M
∑ Hk⎡
⎣⎢
⎤
⎦⎥ = f0E H0⎡⎣ ⎤⎦ + fk
k=1
M
∑ Hk = f0
Var f⎡⎣ ⎤⎦ = E f − E f⎡⎣ ⎤⎦( )2⎡⎣⎢⎤⎦⎥= E fk
k=0
M
∑ Hk⎛⎝⎜
⎞⎠⎟− f
⎛
⎝⎜⎞
⎠⎟
2⎡
⎣
⎢⎢
⎤
⎦
⎥⎥=
E fkk=0
M
∑ Hk⎛⎝⎜
⎞⎠⎟
2⎡
⎣⎢⎢
⎤
⎦⎥⎥= fk
2
k=1
M
∑ E Hk2⎡⎣ ⎤⎦ = fk2k=1
M
∑ k!( )2
The expected value is given by
And the variance is:
Computational Fluid Dynamics
ExampleFollowing:www.dtic.mil/get-tr-doc/pdf?AD=ADA568130
See alsoPer Pettersson, Gianluca Iaccarino, Jan Nordström,
Numerical analysis of the Burgers’ equation in the presence of
uncertainty, Journal of Computational Physics, Volume 228, Issue
22, 1 December 2009, Pages 8394-8412, ISSN 0021-9991,
http://dx.doi.org/10.1016/j.jcp.2009.08.012.http://www.sciencedirect.com/science/article/pii/S0021999109004471
Computational Fluid Dynamics
Apply to inviscid Burgers Equation
∂fi∂ti=0
M
∑ Hi ξ( )+ f jH j ξ( )j=0
M
∑⎛
⎝⎜⎞
⎠⎟∂f j∂xi=0
M
∑ Hi ξ( )⎛
⎝⎜
⎞
⎠⎟ = 0
∂f∂t
+ f ∂f∂x
= 0
f x,t,ξ( ) = fi x,t( )i=0
M
∑ Hi ξ( )Substitute a finite series:
giving
This is still a partial differential equation and to convert it
into a set of discrete approximations we can use several
methods
Computational Fluid Dynamics
Gives:
HiH j = δ iji!
HiH jHk =0 if i + j + k is odd ormax(i, j,k) > s
i! j!k!s − i( )! s − j( )! s − k( )!
⎧
⎨⎪⎪
⎩⎪⎪
where s = (i + j + k) / 2
∂fi∂ti=0
M
∑ HiHk + fi∂f j∂xj=0
M
∑ HiH jHki=0
M
∑ = 0 k = 0,1,…,M
To generate a discrete set of equations we use the Galerkin
method where we multiply by the weight functions and integrate.
Using that for Hermite polynomials:
Computational Fluid Dynamics
In practice, the quantification of uncertainties requires us to
assume that the error is Gaussian
∂f∂t
+ f ∂f∂x
= 0
∂f0∂t
+ f0∂f0∂x
+ f1∂f1∂x
= 0
∂f1∂t
+ f1∂f0∂x
+ f0∂f1∂x
= 0
M =1; f x,t,ξ( ) = f0 +ξ f1
M = 2; f x,t,ξ( ) = f0 +ξ f1 + 2 ξ 2 −1( ) f2Resulting in three
coupled equations
Computational Fluid Dynamics
From: Per Pettersson, Gianluca Iaccarino, Jan Nordström,
Numerical analysis of the Burgers’ equation in the presence of
uncertainty, Journal of Computational Physics, Volume 228, Issue
22, 1 December 2009, Pages 8394-8412,
-
Computational Fluid Dynamics
Even the linear advection equation
can have uncertainty in
Initial conditions
Transport velocity
∂f∂t+U ∂f
∂x= 0
Computational Fluid DynamicsANRV365-FL41-03 ARI 12 November 2008
14:36
y, v
x, uFigure 3Horizontal and vertical velocity profiles at select
stations in the differentially heated cavity, with superposed6σ
uncertainty error bars. Figure reproduced with permission from the
Journal of Computational Physics.
Le Maı̂tre et al. (2004a) also studied Rayleigh-Bénard flow in
the Boussinesq limit using PC UQ.In this context, they considered a
cavity with a heated bottom wall. Above a critical Rayleigh
num-ber, the system transitions from a conductive to a convective
heat-transfer mode, as the instabilityof the flow leads to
convective motion. Uncertainty is prescribed in the bottom wall
temperature.This study explored the performance of a global
Wiener-Legendre GPC construction versus alocal Wiener-Haar scheme
employing a Haar wavelet basis. The results demonstrated the
supe-rior performance of the local construction when the parametric
uncertainty spans the bifurcationcorresponding to the critical
Rayleigh number. The failure of the global spectral expansion
torepresent a bifurcation in stochastic space is not surprising.
The local construction dealt with thebifurcation effectively.
Asokan & Zabaras (2005) reached similar conclusions in this
system usingGPC UQ in a stabilized, variational multiscale FEM.
Wan & Karniadakis (2006b) used ME-GPC for UQ in
incompressible flow and heat transfer ina 2D channel over an open
cavity with a spectral element solver. At high Reynolds number,
largestochastic perturbations were evident, and the local ME-GPC
construction was more efficientthan the global GPC.
5.4. Reacting FlowReacting flow presents serious challenges to
PC UQ, through the high dimensionality associ-ated with many
uncertain parameters and the strong nonlinearity of chemical
reactions. Phenixet al. (1998) first used PC UQ in isothermal
chemical ignition in their deterministic equivalentmodeling–method
approach, focusing on supercritical water oxidation. With this
chemical model,Reagan et al. (2003) employed nonintrusive WH PC
with LHS in ignition and 1D flames in isother-mal supercritical
water oxidation. They also later computed uncertain sensitivity
coefficients fromthe PC results (Reagan et al. 2005).
46 Najm
Ann
u. R
ev. F
luid
Mec
h. 2
009.
41:3
5-52
. Dow
nloa
ded
from
ww
w.a
nnua
lrevi
ews.o
rgby
Uni
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ame
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2/05
/13.
For
per
sona
l use
onl
y.
ANRV365-FL41-03 ARI 12 November 2008 14:36
00
1
1
−0.497
−0.355
−0.213
−0.071
0.0708
0.213
0.355
0.496
Nondimensionaltemperature
Figure 2Mean temperature field in a differentially heated
cavity. The scale on the right indicates the range of low-
tohigh-nondimensional temperature. The right wall temperature is
cold, with prescribed uncertainty, whereasthe left wall is hot and
deterministic. The top and bottom walls are adiabatic. Figure
reproduced withpermission from the Journal of Computational
Physics.
5.3. Thermofluid FlowLe Maı̂tre et al. (2001) studied uncertain
thermofluid flows using intrusive WH PC, in the con-text of
incompressible channel flow with temperature-dependent viscosity.
They later applied thisconstruction in modeling natural convection
in a differentially heated cavity with adiabatic topand bottom
walls and cold/hot sidewalls in the Boussinesq limit (Le Maı̂tre et
al. 2002). Nom-inal conditions corresponded to a steady laminar
recirculating flow regime. They presumed anuncertain cold wall
temperature and modeled it as a random process with a specified
correlationlength, which they represented using a KL approach. The
mean temperature field exhibits twolayers parallel to the vertical
walls and horizontal stratification in the vertical direction
(Figure 2).The temperature standard-deviation field has a similar
topology, with a maximum on the (right)cold wall, at which the
uncertain temperature is imposed, and a zero minimum at the (left)
hotwall, at which a deterministic high temperature is imposed.
Figure 3 shows the mean horizontaland vertical velocity profiles at
a number of stations in the cavity, with superimposed 6σ
uncer-tainty error bars. The mean velocity field highlights the
bulk average circulation of the flow inthe clockwise direction.
Uncertainty grows in both the temperature and the velocity fields
as fluidmoves downward along the right wall. This growth is driven
by the uncertainty in the temper-ature on that wall, and
uncertainty is convected along with the circulating mean velocity
field.Le Maı̂tre et al. (2004c) extended this study to the
non-Boussinesq limit, implementing the fullvariable-density
low-Mach-number equations, again using intrusive KL-PC. Numerical
stabilityrequired discrete global mass conservation in the
stochastic equations to ensure the solvability ofthe elliptic
equations for the pressure modes.
www.annualreviews.org • Uncertainty Quantification 45
Ann
u. R
ev. F
luid
Mec
h. 2
009.
41:3
5-52
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from
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From: Le Maˆıtre O, Reagan M, Najm H, Ghanem R, Knio O. 2002. A
stochastic projection method for fluid flow II. Random process. J.
Comput. Phys. 181:9–44
Natural convection in a differentially heated cavity with
adiabatic top and bottom walls and cold/hot sidewalls in the
Boussinesq limit. Uncertain cold wall temperature modeled as a
random process with a specified correlation length
Another Example
Computational Fluid Dynamics
While stochastic Galerkin methods can produce accurate results,
they require either very extensive rewrite of existing codes or
possibly new codes
In higher dimensions and for a large number of uncertainties the
Galerkin method becomes very complex
Computational Fluid Dynamics
In addition to the Galerkin Polynomial Chaos method, a number of
other approaches are under development such as the Probabilistic
Collocation method, Non-Intrusive Polynomial Chaos method and the
Stochastic Collocation method.
Some of them, such as the Stochastic Collocation seem to share
the simplicity of MC methods but produce results like PC
Computational Fluid Dynamics
Packages
Computational Fluid Dynamics
The Dakota Project
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The DAKOTA (Design Analysis Kit for Optimization and Terascale
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From: http://dakota.sandia.gov/about.html
-
Computational Fluid Dynamics
Overview
Slidehttp://dakota.sandia.gov/papers/DAKOTA_oneslide.pdf
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Computational Fluid DynamicsToolkit for Large-Scale
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Computational Fluid Dynamics
Uncertainty quantification is a challenging problem due to
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