A Perfectly Matched Layer Method for the Navier-Stokes
Equations
MASSACHUSETI
OF TECHN,
by
John P. Whitney
B.S., Physics (2004)
LIBRAF
Massachusetts Institute of Technology
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
41"0
Master of Science in Aeronautics and Astronautics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2006
©
Massachusetts Institute of Technology 2006. All rights reserved.
........................eronautics and Astronautics
January 6, 2006
Author ................
Departmertof
Certified by...
.................................
Profess
Accepted by.......
............
....
..
..............
Jaime Peraire
Astronautics
and
of Aeronautics
Thesis Supervisor
.
-.-----.
...
Jaime Peraire
Professor of Aeronautics and Astronautics
Chair, Committee on Graduate Students
........
.
...
. ..
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUL 10 2006
LIBRARIES
-ma
A Perfectly Matched Layer Method for the Navier-Stokes Equations
by
John P. Whitney
Submitted to the Department of Aeronautics and Astronautics
on January 6, 2006, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics and Astronautics
Abstract
The Perfectly Matched Layer Method (PML) has found widespread application as a highaccuracy, non-reflecting boundary treatment in many wave propagation simulations. However, in the area of computational fluid dynamics, its application has been mostly limited
to the linearized Euler equations. Attempts to apply PML to the nonlinear Euler equations
have found a tendency for the method to go unstable. Even so, in light of the method's
computational efficiency and high accuracy, finding a robust and stable implementation is
highly desirable. Here, the method is extended to the Navier-Stokes equations, and is implemented with a high-order discontinuous Galerkin finite element method (DGFEM). The
weaknesses and strengths of the method are investigated, and its performance is assessed
when applied to complex flows; in particular, a viscous cavity flow is investigated. Stabilizing adjustments to the method are made, and future work is indicated for increased utility
and flexibility of the method.
Thesis Supervisor: Jaime Peraire
Title: Professor of Aeronautics and Astronautics
3
4
Acknowledgments
First, I would like to thank Arthur and Linda Gelb for their generous support of my work
through a graduate student fellowship.
I would like to thank my wife, Janelle, and daughter, Grace, for putting up with my late
nights and long hours.
I am strongly indebted to all the members of the Project X team, whom without, this
work would never have been possible.
I am most appreciative to Per Olof-Persson and Professor Darmofal for their many helpful
suggestions, discussions, and diagnoses.
And last, but not least, I am greatful for my advisor, Professor Peraire: besides his interest, enthusiasim, and ingenuity-without his signature, I could never graduate!
Peter Whitney
January 3, 2006
5
6
Contents
1
Introduction
11
2
The Perfectly Matched Layer Method
13
2.1
2.2
3
4
13
PML for the Linearized Euler Equations . . . .
2.1.1
Split-P M L . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.2
Unsplit-PM L . . . . . . . . . . . . . . . . . . . . . .
14
PML for the Nonlinear Euler and Navier-Stokes Equations .
18
2.2.1
Choosing PML Parameters
. . . . . . . . . . . . . .
19
2.2.2
Determining the Reference State . . . . . . . . . . .
20
21
Discontinuous Galerkin Implementation
3.1
Navier-Stokes and PML Interior Terms . . . . . . . . . . . . . . . . . . . . .
21
3.2
Boundary and PML Interface Terms . . . . . . . . . . . . . . . . . . . . . .
24
3.3
Solver D etails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
27
Method Evaluation
4.1
4.2
Euler Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
. . . . . . . . . . . . . . .
28
. . . . . . . . . . . . . . . . . . . . . . . . . .
31
. . . . . . . . . . . . . . . . . . . . . . . .
33
Navier-Stokes Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.2.1
Viscous Cavity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.2.2
Vortex Shedding Flow . . . . . . . . .
. . . . . . . . . . .
39
4.1.1
Propagation of a Circular Acoustic Wave
4.1.2
Convective Instability
4.1.3
Non-Uniform Mean Flow
Conclusions and Future Work
45
A Flux Functions and Jacobians
47
5
7
8
List of Figures
.
.
. . .
. . . . . . . . . . . . . . . . .. ..
2-1
Restrictions on o-, and -y. .
2-2
Sample profile for ax and o-y.
3-1
Duplicated nodes on a DG element.
4-1
Basic uniform square mesh used, shown here with a two-cell wide PML buffer
...
...
.
16
..................
. . . . . . . . . . . . . . . . . . . . . .
all around . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-2
Characteristic-PML comparison at iteration n
momentum, (c) entropy
4-3
=
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
30
Relative performance of PML (solid) and characteristic (dotted) boundary
Differences between 6
=
31
0 and 6 > 0 cases for density at position (2,18). The
solid horizontal line is the maximum error for the 6 = 0 case.
4-5
22
425: (a) pressure, (b) y-
conditions for a circular acoustic pulse. . . . . . . . . . . . . . . . . . . . . .
4-4
15
. . . . . . . .
32
Comparison of PML methods without and with a convective correction term,
for an oscillating pressure source superimposed over a Mach 0.5 uniform
horizontal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-6
The entropy wake for an oscillating pressure source in a Mach 0.5 uniform
flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-7
33
34
(color) Comparison of PML methods without and with a convective correction term, for an oscillating pressure source superimposed over a horizontal
shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
. . . . . .
35
4-8
(color) Illustration of a catastrophic instability in a PML buffer.
4-9
(color) Maximum of the absolute value of qi, for the entire PML buffer, for
different values of 6, for the horizontal shear flow problem. . . . . . . . . . .
4-10 Cavity mesh and domain boundary conditions.
36
Dashed lines denote the
PML-interior domain boundaries . . . . . . . . . . . . . . . . . . . . . . . .
37
4-11 A comparison between cavity flows with characteristic boundary conditions
and with PML boundary conditions. . . . . . . . . . . . . . . . . . . . . . .
9
38
4-12 Diam ond m esh..................................
. 40
4-13 Close-up of vortices shed from the diamond body. . . . . . . . . . . . . . . .
40
4-14 Sound generated by the diamond body.
41
. . . . . . . . . . . . . . . . . . . .
4-15 A comparison between characteristic boundary conditions and PML boundary conditions for a vortex shedding diamond. . . . . . . . . . . . . . . . . .
10
43
Chapter 1
Introduction
One requirement for many time-accurate numerical simulations is an accurate and robust
treatment of the boundary conditions.
Frequently, boundary conditions are the largest
source of error in a simulation, making a proper treatment of the boundaries essential to
the simulation's success.
In many cases, confidence in knowing the solution outside the computational domain
is high enough to warrant an encoding of this knowledge into the boundary conditions.
The "far-field" or "asymptotic" class of boundary treatments take this approach. However,
reaching parts of the domain where confidence in the prescribed solution is high enough
may require a large domain, which can come at a high cost in computing resources.
Another approach is to view the boundary as an information gate. Information can
leave the computational domain, but cannot enter from the exterior. One example of this
approach is the "characteristic" boundary treatment for the Euler equations. This method
is attractive because it is easy to implement, local in nature, robust, and computationally
inexpensive. The method's weaknesses include low accuracy and its prescription for only
linear perturbations.
While the method can work well in a nonlinear simulation, flow
features with strong nonlinearity may cause excessively large errors at the boundary.
The limitations in our ability to control the flow of information at the boundaries lead
to methods that attempt to blur the boundary line.
These methods go by names like
"buffer zone", "buffer layer", "sponge layer", and others. The idea is to envision a simple
exterior solution, and then gradually encourage the interior solution to conform to this
exterior specification. The method of "encouragement" varies by method, but friction-type
damping through a source term of the form -o(u - uo), is one common approach. The
damping coefficient, o-, is gradually increased from zero-at the the interior-buffer interface,
to its largest value-at the boundary. Forcing approaches lead to unwanted reflections of
exiting disturbances when the disagreement between the local solution and the desired
11
exterior solution is large. This requires the damping region to be quite large-sometimes
even larger than the non-buffered domain, to ensure that the damping coefficient increases
gradually enough.
While these buffer methods may seem crude and inefficient, their robustness and ability
to handle nonlinear disturbances make them the state-of-the-art for nonlinear problems in
computational fluid dynamics [11, 16]. The typical boundary treatment for these problems
is gradual damping, combined with grid stretching, with the buffer region terminated by
either characteristic or far-field boundary conditions. While robust, these approaches are
expensive, since very large buffer regions are needed to compensate for their poor accuracy.
The Perfectly Matched Layer method (PML), introduced by Berenger [10], has become
an important boundary treatment in all manner of time-accurate numerical simulations.
It was originally developed for electromagnetic simulations, but the principle has been
extended to the simulation of many other wave propagation phenomena. It was applied
to the linearized Euler equations with a uniform, horizontal mean flow [20, 21], and has
recently been applied to linearized Euler problems with a non-uniform, horizontal mean
flow [22].
At first, the PML method appeared to be the "silver bullet" of boundary treatments.
Essentially, through numerical artifice, extra degrees of freedom in the governing equations
are introduced inside a small buffer layer, and these extra degrees of freedom allow for
the mathematical specification of zero numerical reflection for linear pressure, entropy, and
vorticity waves of any wavelength, impacting the buffer at any angle of incidence. However,
these extra degrees of freedom also allow extra ways for instability to creep into the solution.
This tendency toward instability has prevented widespread adoption of PML-type methods,
and much of the current research is aimed at addressing this issue [2,9,22].
In this work, application of the PML method to both the nonlinear Euler equations and
the full Navier-Stokes equations is investigated.
After introducing the general details of
the method, specifics of the discontinuous Galerkin (DG) implementation are presented. A
substantial portion of this paper will discuss the various types of instabilities encountered
and efforts to suppress them. In this work, the goal is not to make a detailed comparison
between PML and other methods. Previous research [3,20,21,23,29] has well documented
the gains in accuracy and computational efficiency from the PML method. Rather, this
research will focus on applying PML to more complicated nonlinear and viscous problems,
and look for ways to make the method more robust and stable.
12
Chapter 2
The Perfectly Matched Layer
Method
The Perfectly Matched Layer Method has been applied in the numerical simulation of many
different physical problems [15,17,19]. The common goal of each PML-type method is to
damp linear disturbances in an outer buffer region, without the creation of physical or
non-physical reflections that contaminate the interior solution. For the linearized Euler
equations, PML can be shown to guarantee, in the continuous case, zero reflection at the
interior-PML interface, for linear disturbances of any incident angle and any wavelength.
However, in a discrete implementation, this ideal is not always achieved. Different PML
methods, compared head-to-head, may not be equally effective, in spite of their equivalence
in the continuous case [3].
2.1
PML for the Linearized Euler Equations
First, consider the Perfectly Matched Layer Method applied to the Euler equations. The
two-dimensional Euler equations can be written in the so-called "quasi-linear" form
&u
at
+ A
ou
&u
+ B=u0,(2.1)
Oy
Ox
where u is the state vector, and A and B are the Jacobian flux matrices. The state vector
may be composed of primitive variables or conservative flux variables, with the Jacobian
matrices varying with that choice. For the linearized Euler equations, with a uniform mean
flow, A and B are constant, and depend only on the mean flow state. In the nonlinear
case, A and B will be local functions of the state, e.g. A = A(u(x, y)).
From here on,
an unprimed state, u, will denote a full variable, while a primed state, u', will denote
13
perturbations from a fixed, mean flow.
2.1.1
Split-PML
Berenger's original method was first adapted to the linearized Euler equations, with a
uniform mean flow, by Hu [20]. Hu's original formulation uses the same strategy of variable
splitting,
+ aou' + A
at
Here, u' has been split, such that u'
2+
a
=0
(2.2)
0.
(2.3)
ouU' + B-
ay
=
u'1
+ u'2 . The damping coefficients, ax and o, are
in general functions of x and y, and have restrictions on where they can be non-zero. Note
that when o-r and o- are zero, the Euler equations are recovered.
Figure 2-1 shows a typical arrangement of PML buffers intersecting at a corner. This
PML form is only indicated for a rectangular domain, although other formulations have
been developed for circular domains [15].
In general, PML buffers can be prescribed for any combination of the four boundaries
of a rectangular domain. Buffers attached to the left and right boundaries are called xlayers, and top and bottom buffers are called y-layers. For example, PML layers could be
prescribed for the left, right, and top boundaries. These layers would overlap to form two
corner-layers in the upper-left and upper-right corners.
In the x-layer, o-, must be zero, and a, must be only be a function of x, and cannot vary
in the y-direction. The opposite is true for the y-layer, which must have or, set to zero, and
og varying only in the y-direction. When x- and y-layers overlap, the overlapping region
is called a corner-layer, in which o-, and o- both continue their profiles into the corner.
Figure 2-2 helps illustrate this, and shows example profiles of ax and o-y.
2.1.2
Unsplit-PML
Instabilities have been observed when the split-PML equations are applied to a discrete
numerical simulation, for both acoustic and electromagnetic wave simulations [1, 20, 29].
Since the advent of the split-PML formulation, significant work has been done to put PML
methods on a firmer mathematical footing, and find forms that avoid instability. The socalled "unsplit" form has been formulated for both Maxwell's equations [1] and the linearized
Euler equations [21], with the intention of stabilizing the method.
Below, the derivation of Hu's unsplit-PML for linearized Euler is sketched, and the
14
"corner-layer"
"y-layer"
"x-layer"
physical
domain
Figure 2-1: Restrictions on crg and o- which satisfy the perfectly matched conditions, for
x-layer, y-layer, and corner-layer regions of the PML buffer. The thick line denotes the
boundary of the computational domain, and the dashed lines show the PML-PML and
PML-interior interfaces.
reader is referred to [21] for a full derivation and proof of perfectly matched behavior. The
derivation begins by writing the linearized Euler equations in the frequency domain, and
then applying the following complex change of variables
=
z) ,
1+
+
±
77=
) y,
(2.4)
which results in
-iwi'+
1
+
06'
. . A+
1
. B
+ w
oi6
=0,
(2.5)
where w is the frequency, and 6' is the frequency-domain state. After rearranging terms,
define
=
7
-,
-U,
w
(2.6)
and finally transform back to the time domain. The linearized Euler equations, now having
15
Figure 2-2: Here, o-, and -, are shown as smooth, increasing functions, but they may in
general take on any functional form in their respective directions. Additionally, there is no
continuity requirement (o- = 0) at the leading edges, even though that is the case in this
example.
undergone a complex change of spatial variables, have become
Du'
a
at
&u'
Bu'
+ Aa
+ B
+-y
ax
ay
Oq
B
A -- + o-xB+ (o- + O-y)u' +
Y
ojy
x
at
Oyq = 0,
= u'.
(2.7)
(2.8)
These equations are mathematically equivalent to the split-PML form of (2.2) and (2.3). If
you add (2.2) and (2.3) together and transform to the frequency domain, you can recover
(2.5), which is simply the frequency domain form of the unsplit-PML equations, (2.7) and
(2.8).
This unsplit-PML formulation has the same restrictions on a-x and o- as the split form.
The auxiliary variable, q, is again only defined inside the PML buffer region. The unsplit
form keeps the original linearized Euler equations intact, with PML terms simply added
on. The additional terms are (apparently) Cartesian flux terms in the auxiliary variable
(although with Jacobian flux matrices that depend on the mean flow), and friction-type
damping terms.
In PML formulations for electromagnetic wave simulations, it has been reported that
switching to an unsplit form has improved the stability of the method [1, 2, 9].
For the
Euler equations, it is not clear that an unsplit form provides any direct improvement.
Any improvement in stability would be a discrete effect, since the split and unsplit forms
are mathematically identical in the continuous case.
16
However, some instabilities in the
linearized Euler PML equations have been identified as convection-driven instabilities, which
are present in both forms.
It has been shown that the PML method is stable for waves that satisfy
w dw
dk> 0
k dk'
(2.9)
where w is the frequency and k is the wavenumber. This condition is satisfied if the group
velocity (dw/dk) and the phase velocity (w/k) of a wave have the same sign. If this relationship is not satisfied, which can be shown to happen for acoustic waves in x- and
corner-layers, then it is possible for certain unstable wave modes to grow exponentially in
the PML buffer.
To circumvent this problem, applying coordinate transformations similar to the Prandtl-Glauert
transformation have been proposed [21]. As the first step in deriving these new unsplit-PML
equations, the following transformation is applied to the linearized Euler equations
x=x,
=yoy,
(2.10)
t=t + ox,
which uses the definitions
M2
(2.11)
1),
1 - M2 UO
1 -M2,07
70
Yo
where MO and uO are the Mach number and horizontal velocity of the mean flow. While
similar to one chosen in [21], this transformation maintains the dimensionality of the equations. Next, the complex change of variables is applied and the auxiliary variable defined,
as before. Finally, apply the inverse of transformation (2.10), which yields the convectionstabilized unsplit-PML equations,
-+Au'_u
OU+A
58t
B
Dq
0x
+B
5y
+ o-A
YDx
+ uxB
Dy
+ (UX + oY)u'+
Uxoryq + oxoA(u' + oyq) = 0,
Dq
/
9 q = u.
(2.12)
( 2 .13 )
The only difference between (2.7) and (2.12) is the addition of the convective correction
term, o-/3oA(u' +
c-yq).
When the mean flow is not convective,
#0 is zero,
and the correction
drops out. Note also that the correction is zero for y-layers (since o-x = 0 inside them), which
do not exhibit convective instability. This instability and the correction are investigated
numerically in section 4.
17
2.2
PML for the Nonlinear Euler and Navier-Stokes Equations
Begin by writing the Navier-Stokes equations in strong, conservative form
U+ V -YFi(u)
at
- V - F (u, Vu) = 0,
(2.14)
I =PV
(2.15)
where the conservative state is given by
pE
in which p is the fluid density, u and v are the x- and y-components of velocity, and E is
the specific internal energy. The inviscid flux, Y, and the viscous flux, Fv, are given in the
Appendix. Note that the primes have been dropped from the state vector, indicating the
full variables are now being considered, and not their linearizations.
There is no difference in the nonlinear Euler and Navier-Stokes PML formulations. The
latter is simply the former, with the viscous terms tacked on. While it may be possible to
come up with viscous-specific PML terms, no attempt was made here.
For linearized Euler, the damping terms in (2.12) are of the form, ou'. In the nonlinear
case, those terms become, o(u - uo), where uo is a spatially varying (and perhaps time
dependent) reference solution. Choosing uO is discussed in section 2.2.2.
The PML flux terms in (2.12) are linear, and must be replaced with their nonlinear
equivalents. The proposed Navier-Stokes PML equations are
at
-+
V - FJ(u) - V - F (u, Vu)+
Navier-Stokes equations
UYA(u) a+
ax
oB(u)
ay,
+ (ox + oy)(u - uo) + o xcyq
L fPML
PML flux terms
damping terms
+ ox,3A(u)(u - uo + o-yq)
0,
(2.16)
PML convective correction
aq
=u - uo,
at
-LU
18
(2.17)
where 0 is now defined as
#X
(X )
(2.18)
1 r(X, y) 2 Uo (X, y)
The auxiliary variable, q, does not change in this new nonlinear form. The Jacobians are
now nonlinear functions of the local state, and the convective correction coefficient, /, is
now spatially varying, instead of being constant throughout the PML domain. The flux
Jacobians, A and B, are given in the Appendix. The restrictions on
o and uo are the same
as those in the linearized Euler case.
2.2.1
Choosing PML Parameters
While having many parameters in a method allows for increased customization and control,
it prevents the method from becoming a robust tool that non-experts can apply to general
problems. For PML, the user must choose the magnitude of the damping coefficients, their
spatial profiles, and the widths of the buffers.
To characterize the performance of a PML buffer for general problems, non-dimensional
relationships for the damping coefficients and buffer widths are desired. It was found that
appropriate non-dimensionalizations are
uh
c
(2.20)
-D
Ah'
D
where D is the buffer width,
,(2.19)
Ah is the element size, and c is the speed of sound.
In general, it is best to select the highest possible damping coefficients which will not
destabilize the calculation. When the mean flow is quiescent, the limiting values for 0,
and ay are the same. However, when the mean flow is horizontal convection, the critical
value for o is usually about 5-10 times lower than the critical a,. A look at equation (2.7)
reveals that the x-layer contains no x-direction PML flux terms, while the y-layer does.
This y-layer convection decreases the stability margin of oy. It was found that setting both
o
and
ory to about seventy percent of the critical value for ay was good practice. It is
certainly possible to set ax higher, but doing so only increases the tendency of the x-layer
to become unstable through convective instability.
For non-DG discretizations, it has been common to vary ox and
oa in a smooth fashion:
from zero at the interior-PML interface, to a maximum value at the boundary, with a
quadratic increase often used. It was noted in
19
[4]
that for DG discretizations, the damping
coefficients could be set constant throughout the PML buffer, with a discontinuous jump
from zero at the interior-PML interface, without any degradation in perfectly matched
behavior. That observation was confirmed in this work.
Equation (2.20) simply states that the damping is proportional to the number of elements in the buffer. For most purposes, it was found that 2-6 elements was usually a
sufficient buffer width to achieve high accuracy at the boundaries.
2.2.2
Determining the Reference State
In Perfectly Matched Layer methods, damping inside the buffer layers is applied relative to
a reference state, uO. When PML is applied to linearized equations, the reference state is
trivially the mean flow state. In this work, for all Euler simulations, where the behavior of
the flow was essentially linear, uo was fixed to be the mean flow, which is of course known
beforehand. However, for problems with strong nonlinear behavior, it is much harder to
determine uO a priori. For example, in the case of a wall-bounded compressible flow, the
solution near the boundaries may not be known beforehand. Consider two methods to set
the reference state when it is unknown:
1. Fix the reference state to an approximate mean flow.
2. Evolve the reference state as a time-average of the solution.
To apply method 1, a lower-fidelity model is used to construct the reference state. For a wallbounded compressible flow, a similarity solution could be used as the reference solution. In
unbounded flows with bodies, the far-field conditions could be found using a panel method.
However, many nonlinear problems are not simple enough to allow for an approximate
solution to be used as an accurate reference solution.
Method 2 is attractive because, in this case, the reference state is, by construction,
equal to the mean flow. However, there are two main flaws in this approach. First, if the
problem is impulsively started (e.g. an airfoil problem is initialized to a uniform mean flow),
then disturbances due to initial transients will be large, which may corrupt the average. In
practice, this transient corruption was observed to be a small effect. The second problem
is more serious. If the problem has sources of entropy, then disturbances with a non-zero
mean will convect into the outflow PML buffer. These non-zero mean disturbances can
"burn" through the layer, since they give positive feedback when the reference state uses a
cumulative average. To circumvent this problem, the reference state can be frozen after a
specified amount of time.
Unfortunately, there is no one-size-fits-all solution to this reference state issue, and the
best approach strongly depends on the problem at hand.
20
Chapter 3
Discontinuous Galerkin
Implementation
The discontinuous Galerkin method (DG) was introduced by Reed and Hill in 1973 [27].
They used DG to simulate neutron transport. Since then, DG has been applied to to general
hyperbolic and elliptic problems. It has also been applied to the Euler and Navier-Stokes
equations [5, 6, 13, 14, 18, 24, 25].
A comprehensive review of DG methods can be found
in [12].
3.1
Navier-Stokes and PML Interior Terms
The method of discretizing (2.16) and (2.17) using the discontinuous Galerkin finite element
method (DGFEM) is now summarized. For notational convinience, define the following:
(a, b), =
a - b.
(3.1)
Standard procedure is to multiply equation (2.16) by a test function, v, and integrate by
parts once. However, the basis functions are discontinuous, and are only defined inside each
element. Since the inter-element flux is not unique, it must be calculated at each interelement edge, as opposed to the continuous Galerkin case, where the inter-element fluxes
cancel, and fluxes need only be evaluated at the domain boundary. Figure 3-1 shows an
example element and illustrates that each edge node is duplicated, and adjoining elements
each "own" one of the nodes from each pair.
After integration by parts, the Euler term in (2.16) becomes
(V - Fi, v)p, = (Fi - n, v)qn, - (Vv, Fi),.
21
(3.2)
Figure 3-1: Illustration of an element, Qj, and one of its neighbors. As an example, nodes
for a 2"d-order Lagrange basis (P2) are shown, with the nodes shown slightly inside the
element edges, only to illustrate that each element has its own private edge nodes. The
labeled nodes, q- and q+, have the same position on the inter-element edge, but are "in"
different elements.
With shared nodes at the element edges, the ambiguous fluxes at shared nodes must be
resolved by solving the Riemann problem.
A well-studied problem in the finite-volume
community, simply replace the flux term, Fi - n, with a flux function,
(V - Fi, v)Q
= (jn,
v)
- (Vv,JFi)Q. .
(3.3)
For this implementation, the Roe flux function was used. Duplicating the edge nodes and
having to calculate inter-element fluxes might seem unnecessary, but it has the benefit of
stabilizing the calculation for convective flows. Additionally, since there is no continuity
requirement at the edges, high-order basis functions may be used with ease, and high-order
implementations do not require extended stencils.
In this implementation, the viscous terms of (2.16) were discretized using the second
method of Bassi and Rebay (BR2) [7,8]. The BR2 scheme has both optimal accuracy for
all order basis functions, and has a nearest-neighbor stencil. This last property is beneficial
if implicit time-stepping is used.
While the Euler equations can be cast in conservative form, the PML flux terms in
(2.16) are not conservative, and require a different approach. Begin as before by integrating
22
by parts over the entire domain,
(YA
Ox
+ oA-B
Oy
,vo
S((onrxA
=
q
(o,AV) +
),
+ a-xnyB)
.(Bv),
(3.4)
For each pair of edge nodes, q- is the auxiliary state native to the element under consideration, and q+ is native to the adjoining element. Notice that q has been replaced with a
flux function, q, to resolve the values at inter-element edges. Notice the more complicated
form of the area integrals; for the Euler equations, the derivatives could pass through the
Jacobians (a miracle of their "quasi-linear" nature), to leave only derivatives of the basis
vectors. Here, this is not possible, so the area integrals are simplified by integrating each
element by parts again, this time element-wise. This leads to
(UYA
a + o-zB
,o
=
+ (-yA
(oYnxA + o-xnyB) (-q- q-) , v
+ oB
,I v
(3.5)
Notice the similarity to the original form. The area term is simply an element-wise integration of the original equations, and the flux term is a correction for any disagreement in
q between node pairs on the inter-element edges. If all paired nodes agree, then the flux
terms are zero.
All that remains is to evaluate -q.First, define
(3.6)
A, - A(u)nxo-y + B(u)ny o-.
Then, resolve q with the Roe flux function,
q=
1
(q+ + q-) -
This leads to,
An(u) (- - q)
=
1
sign (An) (q+
1
(An 2
IAnI)
completing the discretization of the interior terms.
23
(q+
-
q-).
(3.7)
q),
(3.8)
3.2
Boundary and PML Interface Terms
To complete the discretization, specification of the flux functions at the boundaries and at
the interior-PML interface is required. To avoid confusion, the term boundary will hereafter
refer only to the outer boundary of the computational domain, and not to the interior-PML
interface.
The computational domain is composed of the interior, physical domain, and the PML
buffer region. The Navier-Stokes equations are solved over the entire domain. Thus, boundary conditions for their terms are set only at the boundary, and not at the interior-PML
interface. In a sense, a normal Navier-Stokes problem is solved, with the usual boundary
conditions (e.g. characteristic, wall, etc.), with a PML layer placed on top of the NavierStokes problem-just in a thin buffer region along the edges.
For the Navier-Stokes flux terms, the known interior state and an "exterior" state are
inputs to the flux function. For some boundary conditions, the exterior state is extrapolated
from interior states. For a "characteristic" boundary condition, the full exterior state is a
"known" state that the user specifies.
The PML flux terms in (2.16) appear to require specification at both the boundary and
the interior-PML interface. This is tricky because q is not defined outside the PML region.
Disturbances inside the PML buffer are damped, and q tends to zero as they approach
the boundary. In general, setting the boundary value of q to zero showed no change from
cases where the PML boundary fluxes were not calculated at all (i.e., the contributions to
the residual from the boundary q-fluxes were not included). However, at the interior-PML
interface, there is no good way to set the "interior" value of q.
Essentially, q at the
interior-PML interface assumes whatever value the incoming disturbance requires, in order
to produce perfectly matched behavior. Forcing it to zero causes non-smooth, and nonperfectly matched behavior. Therefore, the PML flux terms at the interior-PML interface
are not calculated. This is not completely satisfying. There are convection-like flux terms in
q, in equation (2.16), and some sort of upwind specification appears necessary to maintain
stability. It turns out that this can be a problem: consider a pulse train of density waves,
with a non-zero mean, that impinge on an outflow PML buffer. Since the mean is non-zero,
the time average of the RHS of equation (2.17) will be positive.
This allows q to grow
linearly. In practice, this has been observed to lead to catastrophic instabilities. A typical
example where this can occur is when the wake from a body impinges on the outflow PML
buffer.
24
To remedy this situation, the following change to equation (2.17) is proposed
oq+ j
=~
_u - uo,
0t
Ox
(3.9)
where 6 is a small percentage of the mean flow velocity. By introducing a small advection
in q, an upstream value for q must be specified when evaluating the advective flux across
the interior-PML interface at the outflow buffer. Adding this term breaks the perfectly
matched behavior of the equations, but for small 6 the impact is negligible. Even for very
small
6, this advection term prevents waves with a non-zero mean value from destabilizing
the calculation. A numerical example is discussed in section 4.1.3.
3.3
Solver Details
For this work,
3
rd-
and 4th-order Lagrange polynomials (P3 and P4) were used to represent
the solution. The nonlinear Euler-PML or Navier-Stokes-PML equations were integrated in
time using the standard 4th-order Runge-Kutta method. Additional details specific to this
implementation may be found in [18, 24].
25
26
Chapter 4
Method Evaluation
The proposed method was tested by simulating several types of acoustic and convective
flows, using both the Euler and Navier-Stokes equations. Each of the tests explored different
operating regimes, and were designed to uncover different weaknesses in the method. The
goal of this work was to find situations where the method failed, determine why it failed,
and then change the method appropriately.
4.1
Euler Simulations
For Euler simulations, the following non-dimensionalization was used
Xref
L,
mref = poL
(4.1)
tref = Lfy/c.,
where L is a characteristic length, p, and c, are the free-stream density and sound speed,
and
7 is the (constant) specific heat ratio. This choice implies that Uref
co/yi,
and
results in non-dimensionalized primitives,
P = p/po,'
P = p/po,
ft =
ufy-/c
W.
(4.2)
The nonlinear Euler equations were solved in the following simulations, although most of
the problems are largely linear in nature. Nonlinear behavior will be the focus of the NavierStokes simulations, while the Euler problems are similar to traditional benchmark problems
from the literature.
27
4.1.1
Propagation of a Circular Acoustic Wave
In this test, a regular, square domain, as shown in Figure 4-1 was considered. The PML
buffer is two "cells" wide, where one cell is a square region composed of two right-triangular
elements. For calculations in this test case, a 4th-order Lagrange basis (P4) was used. This
combination of a coarse mesh with high-order elements is typical of aeroacoustic calculations, and is much more efficient than using a fine mesh of low-order elements.
24
Figure 4-1: Basic uniform square mesh used, shown here with a two-cell wide PML buffer
all around.
The boundary states at all four domain boundaries are fully specified to a Mach 0.5
uniform flow. The solution is initialized with an isentropic pressure and density disturbance,
located at the center of the square domain, with a Gaussian footprint. Specifically, the initial
state is set to
P
Pu
Pv
pE
where the pressure, p, is given by
I,
I
Soof
Pu 00
(4.3)
0
+
±
_( (X-Po)2+
pu2
(Y-
9)2
(4.4)
For this specific simulation, E = 0.1, xo = yo = 10, and r = 1.5. A fixed timestep, dt = 0.075,
28
was used. This Gaussian "blip" creates a circular, isentropic, acoustic wave that radiates
outward from the center of the domain. The disturbance is initialized in the moving frame,
and the "center" of the circular wave travels downstream with the mean flow.
PML Performance
Figure 4-2 shows a comparison of characteristic and PML boundary conditions at iteration
n
=
425, at which point the wave should lie entirely outside the domain. For this simu-
lation, o.= o = 0.5, constant throughout the buffer. The full nonlinear Euler equations
were solved in this simulation. The total domain size is the same in both the PML and
characteristic cases. The shading scales for each pair in Figure 4-2 are the same. Note that
some of the features in the characteristic plots are saturated at that range.
In the characteristic plots, notice that the artificial reflection at inflow is especially
strong for the y-momentum. The PML solution is not problem-free, however. There is a
small entropy wave created when the sound wave hits the PML buffer. Essentially, the
buffer must damp the energy of the incoming wave, and this energy heats the local fluid,
creating an entropy disturbance that convects downstream. This phenomena is not unique
to the PML method-it is endemic to all friction-type damping methods. However, notice
from the greyscale range that the entropy wave is quite small. If the damping is done more
gradually, instead of as a step function, the resulting wave will be of a lower amplitude and
larger width.
Figure 4-3 compares PML and characteristic solutions against a "reference" solution.
The reference solution is solved on a larger domain, such that numerical reflections do
not contaminate the original domain, at least up to iteration n = 1000. The differences
are computed at the point (2,18), which is close to the upper left corner, a particularly
error-prone spot.
For this problem, the improvement realized by PML over characteristic is about two
orders of magnitude. After the wave passes by, one would expect the error to tend toward
zero. In Figure 4-3, the error tends toward ~ 10-6, which is approximately the error level
of the discretization.
Effects of q-convection on PML
In equation (3.9), the q-equation is modified to contain a small advection term. The strength
of the new advection term is controlled with the parameter 6. The circular acoustic pulse
case is now repeated for non-zero 6. The results are shown in Figure 4-4. The plotted error
is found by comparing the solution to the case where 6 = 0. The plot shows the additional
29
PML
CharacteristicP
(a)
(b)
(c)
p [0.999, 1.001]
p [0.999, 1.001]
pv [-5x10- 4 ,5x10-
4
pv [-5x10- 4,5x10-
]
4
]
s [-1.5x10- 5 ,1.5x10-5]
s [-1.5x10- 5 ,1.5x10-5]
Figure 4-2: Characteristic-PML comparison at iteration n = 425: (a) pressure, p, is plotted, with a white-to-black range of [0.999, 1.001]; (b) y-momentum, pv, is plotted from
[ -5x 10~ 4 , 5x 10~4 ]; (c) entropy, s, is plotted from [ -1.5x10- 5 ,1.5x10-5 ]. The greyscale
ranges are the same for each set of plots.
error incurred by having a non-zero 6. The horizontal solid line shows the maximum error
in the 6 = 0 case. It is clear from the plot that if 6 is one percent of the flow velocity or
less, then the additional error from the advection term will not be significant.
30
12
p-error at (2,18)
.
102
.,
pu-error at (2,18)
i101
10
:3
:3
108
200
600
400
800
600
400
200
1000
800
1000
800
1000
n
n
pE-error at (2,18)
pv-error at (2,18)
10 2
10 r
10
-4
10
200
600
400
800
10-8
1000
200
600
400
n
n
Figure 4-3: Relative performance of PML (solid) and characteristic (dotted) boundary
conditions for a circular acoustic pulse.
4.1.2
Convective Instability
In Sections 2.1.2 and 2.2, the possibility of instabilities driven by the convection of the mean
flow was discussed, and a coordinate change to solve the problem was described. Changing
coordinates introduced an extra term in the PML equations, which appears as the last term
in equation (2.16).
In the circular wave simulation, after the wave interacts with the boundaries, the solution
tends toward a quiescent state. This allows for an easy evaluation of the method, but a
single wave is not a very "abusive" problem. To incite instability, a problem that forces
the boundary to handle a continuous barrage of incoming waves is required. To that end,
a Mach 0.5 uniform flow will be considered as before, except now with a pressure forcing
term. At each instant, an oscillating pressure disturbance, Ap, is superimposed with the
31
p-error at (2,18) due to q-convection term
103
/\
I
10 -
5= 10% of u
/\
',
II
Ii
-
10-5
8 = 1% of u
.
CL
0.1% of u
I5=
*.
.......
*o'f"u
0
............ ...
10
8
0.01% of u
107
maximum error without q-convection
10-8
100
200
300
400
500
600
700
800
900
1000
Figure 4-4: Differences between 6 = 0 and 6 > 0 cases for density at position (2,18). The
solid horizontal line is the maximum error for the 6 = 0 case.
current local pressure. The disturbance is localized with a Gaussian footprint,
AP =
'eoe
2
/sin
For this simulation, E = 0.1, xo = yo = 10, ?7= 1.5, and T
(T
=
t).
(4.5)
4. For the PML buffer, o- = 5,
constant throughout the x-layers. A larger-than-usual damping coefficient is chosen because
over-damping is usually necessary to trigger the instability. If the damping coefficient is
low enough, depending on the properties of the problem, convection-driven instabilities
may never develop. As standard operating procedure, a-x and c, were set around seventy
percent of the critical value for o-y (in this case,
-y,crit ~ 0.7), and for moderately long
times the convective instability does not appear, unless the damping coefficients are raised
substantially.
Figure 4-5 contrasts the behavior of the PML x-layer with and without the convective
correction term given in (2.16).
The instability is clearly seen in the left image. Notice
that it develops in the upper-right corner first. This is because waves impacting here are
less resolved than waves traveling on the other diagonal. This resolution bias is a property
of the structured grid.
It seems likely that the instability is enhanced by errors in the
discretization.
32
Previous investigations of this instability used finite difference discretizations, which
are less stable than a DGFEM discretization.
Numerical filtering was applied by some
researchers to stabilize PML against the convective instability [20,22,29]. While in extreme
circumstances the instability is observed, under normal operating conditions, it does not
presented a problem. This is likely due to the stabilizing diffusion that DG introduces.
NO Convective Correction
WITH Convective Correction
p [0.97, 1.03]
p [0.97, 1.03]
Figure 4-5: Comparison of PML methods without and with a convective correction term,
for an oscillating pressure source superimposed over a Mach 0.5 uniform horizontal flow.
The static pressure, p, is plotted with range [0.97,1.03] at iteration n = 1500. The top and
bottom boundaries have characteristic boundary conditions and no PML buffer, which lead
to "eggcrate"-looking errors in the solution.
4.1.3
Non-Uniform Mean Flow
In this example, PML is applied to a non-uniform, horizontal Euler flow, and the long-term
stability of the method is tested. Consider a horizontal shear flow, with a velocity that
varies in the y-direction according to
u(y) =
[(ui + U2) + (ui - u 2 ) tanh
2(y
yo)),
(4.6)
where ui is the maximum velocity above the shear layer, u 2 is the velocity below the shear
layer, yo is the vertical position of the shear layer, and 6 is the shear layer thickness. The
temperature profile is given by the Crocco relation for compressible flow,
(y)
T(y)=T+
ui - u(y)
U2_-
U-U2
T2
U1-U2
33
+
7 - 1
2
(U1 - u(y)) (u(y) - U2 )-
(4.7)
For this simulation,
= 0.6, u 2 = 0.4, Ti = 1.0, T 2 = 0.8, yo = 10, 6
-i
=
4.
(4.8)
The density is determined from the ideal gas law. As in section 4.1.2, an oscillating pressure
source, given previously by equation (4.5), with the same parameters, is superimposed
over the flow at (10,10), the center of the domain. To illustrate the "wake" generated by
this pressure forcing, Figure 4-6 plots the entropy generated for the same pressure source,
superimposed on a Mach 0.5 uniform flow. Entropy is plotted for clarity, since the radiation
field is isentropic.
s [-8 x 10-3,8 x 10-3
Figure 4-6: The entropy wake for an oscillating pressure source in a Mach 0.5 uniform flow.
Notice how the wake is absorbed by the outflow PML buffer (o, = oy = 0.5).
A non-uniform mean flow is now being considered, so the convective correction term,
according to (2.16), is spatially varying. The PML parameters for this simulation are the
same as was used for the uniform flow investigation of the convective instability. Remember,
the damping coefficient, for the purposes of exciting instability, is raised from its normal
value of 0.5 to 5. As shown in Figure 4-7, the instability is clearly present at this high level
of damping, and applying the correction easily removes it. Again, the point is largely moot,
since for normal values of the damping coefficients, convective instabilities are almost never
observed, and diffusion from the discretization appears sufficient to suppress them.
The larger problem, however, is that for long times, the "wake" in this simulation will
overload the PML buffer. Figure 4-8 illustrates the severity of the problem.
In must be noted that this happens over a very long time. If the time for a wave to
34
NO Convective Correction
WITH Convective Correction
pu at n = 1500
pu at n = 1500
Figure 4-7: (color) Comparison of PML methods without and with a convective correction
term, for an oscillating pressure source superimposed over a horizontal shear flow. The
x-momentum, pu, is plotted at iteration n 1500, with equal color scales for each case.
M at n=38,500
M at n=10, 000
Figure 4-8: (color) Illustration of a catastrophic instability that can develop after a PML
buffer is exposed to non-zero mean disturbances from a wake, which develops over long
time. The Mach number is plotted in both images, with the color scales set equal.
travel across the domain is characterized by td = 20/coo, then the calculation goes unstable
around t = 170td, which is a quite long time. However, for problems with a stronger wake,
the instability can be counted on to appear sooner. It has been observed that the likelihood
of instability is linked to the growth of the absolute value of q, especially the q-"density",
35
qi. Figure 4-9 verifies that, in the absence of the small advection term, the components of
q that correspond to the non-zero mean components of u - uo will increase linearly. Thus,
qi will increase linearly without bound, which leads to a catastrophic instability after a
long time. However, Figure 4-9 also shows that a one percent advection will easily stabilize
this growth.
long time behavior of maximum value of
1q,I
18
16
14
12
10
CE
3.5
n
x 104
Figure 4-9: (color) Maximum of the absolute value of qi, for the entire PML buffer, for
different values of 6, for the horizontal shear flow problem. The iteration is n.
4.2
Navier-Stokes Simulations
For Navier-Stokes simulations, the following non-dimensionalization was used
Xref
L,
mref = poL3,
tref = L/uoo.
(4.9)
This choice results in non-dimensionalized primitives,
p=p/pou22,
P
= p/po,
i = u/Uoc.
(4.10)
While the previous test cases were all solved using the full nonlinear Euler equations, they
were predominantly linear. In the following simulation, PML is tested for the Navier-Stokes
36
equations, and a rectangular cavity flow and a vortex shedding case are investigated.
4.2.1
Viscous Cavity Flow
Under certain conditions, flow past a rectangular cavity can generate acoustic radiation,
caused by self-resonance. For high enough Mach and Reynolds numbers, the shear layer
across the top of the cavity is unstable, and its unsteady impingement on the downstream
corner causes strong acoustic radiation. This phenomenon is a model problem for noise
generated by wheel-wells, weapons bays, and other body cavities on aircraft.
The mesh and boundary conditions for our cavity simulation are shown in Figure 410. The flow conditions and cavity geometry for our simulation correspond to the L2 case
described in [28].
The inflow Mach number is 0.6, and the Reynolds number (based on
cavity depth) is 1500. A fixed timestep, dt = 0.003, was used. The solution was represented
with a 3rdorder Lagrange basis. To initialize the calculation, the solution near the wall
was set (even over the cavity mouth) to a Blasius solution, and the solution inside the
cavity was initialized quiescent. While far from correct, this rough guess leads to a smooth
solution much sooner than a uniform flow initialization. For the PML case, the reference
full state specification
uniform
inflow
-
-
parallel flow
-
constant
pressure
beginning of
isothermal wall
2
Figure 4-10: Cavity mesh and domain boundary conditions. Dashed lines denote the
PML-interior domain boundaries. The mesh contains 3571 elements, and the smallest
triangle edge has a length 0.065 times the cavity depth.
state was set to a cumulative average of the solution.
The damping coefficients in all
buffers were increased from zero to 4.0 linearly, as a hedge against any new instabilities
due to non-linearity or non-hyperbolicity (seen mostly in the bottom of the outflow buffer).
37
The maximum value of the damping coefficients, 4.0, again corresponds to approximately
seventy percent of the critical value for ory. The solution was allowed to develop without
PML, and then at n
=
10, 000 the PML buffers were turned on.
Temperature contours at n = 20,000 are plotted in Figure 4-11, with and without the use
of PML buffers. The shading scales for each case are equal, and were adjusted to emphasize
the radiated field-at the expense of saturating the plot near the cavity and boundary layer.
Clearly the radiated waves in the PML case are much cleaner, and qualitatively match the
solution behavior found in [28]. The characteristic case suffers from a variety of spurious
waves and disturbances.
The PML equations can only be expected to behave well when the flow can be adequately
represented by the linearized Euler equations. In this problem, that is surely the case in
Characteristic
PML
Figure 4-11: A comparison between cavity flows with characteristic boundary conditions
and with PML boundary conditions. The images are both taken at iteration n = 20,000
with inflow conditions: M = 0.6, and ReD = 1500. The temperature is plotted, with the
shading scales set equal for the two cases.
38
all three buffers-except in the thin boundary layer that cuts through the outflow buffer
(the maximum differential pressure of the radiated waves is about 0.7 percent of the mean).
While here at n = 20, 000 the solution does not suffer from instability, around n = 30, 000,
instabilities localized to the boundary layer begin to creep in, and eventually the solution
will go unstable.
Misbehavior of some sort is to be expected, since the PML equations
don't apply in the boundary layer region. Finding some workaround to this problem is
highly desirable, given the superior performance of PML. If a robust tool is immediately
desired, the outflow buffer could be eliminated, and the top and inflow buffers kept. Then,
a traditional method could be applied at the outflow: moving the boundary back some, and
perhaps adding a friction-type buffer to absorb any pockets of vorticity swept out of the
cavity.
4.2.2
Vortex Shedding Flow
This last Navier-Stokes test case illustrates the future work that must be done to develop
PML into a robust tool for handling the boundaries of tough Navier-Stokes problems. In
this test, a diamond shape, with a stream-wise length of 1 and a width of 1/2, was simulated
in a Mach 0.5 flow, at a Reynolds number of 4000, based on the diamond length. The mesh
is shown in Figure 4-12. With that level of mesh, 4000 was the highest Reynolds number
achievable (the solution would diverge at R = 5000).
At this Reynolds number, vortices are shed from the diamond chaotically. Figure 4-13
plots Mach number, and clearly shows these vortices. Note the doublet arrangement of the
bottom vortex pair.
In Figure 4-14, at a different instant, the temperature is plotted, with the shading set
to emphasize the radiated sound. A body in a flow is a dipole source for sound. Large
sound waves can be seen, which are generated as vortices are shed. Additionally, sound
can be seen to emanate from the doublets. In this simulation, all four outer boundaries
have characteristic boundary conditions that specify the full free-stream state. Note the
prominent wave in the bottom right-hand corner; this is a spurious wave caused by the
exiting of a vortex. The vortices are very nonlinear: the average vortex has an absolute
density amplitude approximately 50 to 60 percent of the free-stream density. With such a
high degree of nonlinearity, they render useless any PML buffer in their path, no matter
how gradual the damping. All attempts at outflow PML buffers ended in instability. This
cannot be scored as a failure for PML, however, since the method is not prescribed for such
nonlinear situations.
The issue remains: these exiting vortices need to be damped out before they can be
39
-1
0
1
2
4
3
5
6
7
8
Figure 4-12: This mesh was generated using the distmesh2d code [26]. The length (flowwise) is 1 and the width is 1/2.
Figure 4-13: Close-up of vortices shed from diamond. The Mach number is plotted (darker
is larger) with the scaling set to emphasize the vortices. The free-stream Mach number is
0.5, and the Reynolds number is 4000, based on the diamond length.
allowed to impact a domain boundary. This leaves traditional methods as the only option.
However, the flow should still have linear behavior at the top, bottom, and inflow buffers,
and it should still be possible to implement PML there.
40
This flow also illustrates the problem of determining the reference state. If the buffers
are placed at such a distance from the body that the mean flow is essentially equal to
the free-stream, then there are no issues, and the reference state can be fixed to the freestream state. However, this would require placing the buffers many body lengths away, and
would necessitate a much larger domain than is probably wanted. For the mesh in this
example, the buffers are not very far away from the diamond-the pressure disturbance of
the diamond reaches noticeably into the buffers. Therefore, some effort must be made to
determine the reference state.
The approach taken in this work was to use a cumulative average of the solution to build
up the reference state. In this test problem, the domain was initialized to the free-stream
values everywhere. After several thousand iterations, an accurate reference state is indeed
found, but before a stable average is reached, the "current" reference state is not useable
for PML. The reason is that before a stable average is reached, the current average and
the current state disagree by a non-trivial amount. Of course their disagreement decreases
as the average state is approached, but in the beginning their disagreement is large, which
leads to undefined (usually bad) behavior of the PML buffers. Therefore, in this problem,
as in the cavity flow, the PML buffers are not "turned on" until a stable average is reached.
For this problem, the PML buffers were turned on at n = 30, 000.
If the flow is subsonic, an alternative would be to use a panel method to find an approx-
Figure 4-14: The temperature is plotted, at n
sound waves in the radiated field.
41
15, 000, with the scale set to emphasize
imate mean flow, initialize the problem to that solution, and then use a cumulative average
as the reference solution. Ideally, the approximate solution would be close enough to the
actual mean flow to allow PML to operate from the beginning. This would cut down on the
start-up time for the problem. Improving the start-up behavior by finding a better initial
solution is beneficial in general, not just for PML.
Figure 4-15 shows a comparison between using characteristics and using PML. There
are small differences, but the impact is not as clearly seen as in the cavity case. The buffer
in this case would probably have been more effective if it were thicker. One difficulty in
making a visual comparison is that due to the chaotic nature of the flow, soon after the
instant shown, the shed time of the vortices, and the subsequent behavior of the solutions,
become very different. For long times, the average behavior should show differences, but
solving the problem for a long enough time to make a meaningful statistical comparison
was computationally prohibitive.
42
Characteristic
PML
Figure 4-15: A comparison between characteristic boundary conditions and PML boundary conditions for a vortex shedding diamond. The images are both taken at iteration
n = 35, 000. The pressure is plotted, with the shading scales set equal for the two cases.
43
Chapter 5
Conclusions and Future Work
In previous work, the Perfectly Matched Layer Method (PML) has been successfully applied
to linearized Euler simulations. While PML does not apply, and will not remain stable,
in regions of nonlinear or viscous flow, it surely can be applied to nonlinear and viscous
simulations-when the flow behavior at the buffers is essentially linear.
Herein, a PML method was formulated for the Navier-Stokes and nonlinear Euler
equations, and was implemented using the discontinuous Galerkin finite element method
(DGFEM). DGFEM-based PML showed less tendency toward instability than other discretizations, and no special filtering operations were necessary for stability of the method.
It was also found that under standard conditions, the convective instability was not present,
and under extreme conditions, when it did appear, it was easily controlled. Several test
cases were conducted successfully using the nonlinear Euler equations and PML.
To push the limits of this PML implementation, a viscous cavity flow problem and a
vortex shedding case were investigated. In the cavity problem, since the outflow buffer
contained a boundary layer, the calculation eventually became unstable. However, it was
shown qualitatively that PML performed much better than characteristics alone. For the
vortex shedding case, nonlinear vortices made the outflow buffer a nonstarter, and a nontrivial mean flow indicated areas for improvement in the determination of the reference
state.
If PML is to become a robust tool for viscous and nonlinear problems, several challenges
remain. First, additional experience with the method is needed if a universal method for
determining the reference state in complex flows is to be found. Second, work is needed to
allow PML to handle nonlinear disturbances and small regions of non-hyperbolicity (boundary layers). Perhaps there is a way to safely and locally "turn off" PML in the vicinity of
such events.
While many hurdles remain in the quest to deploy PML as a robust and flexible tool for
45
nonlinear and viscous problems, its high performance and low cost continue to motivate its
development.
46
Appendix A
Flux Functions and Jacobians
The nonlinear Euler equations written in the so-called "quasi-linear" form are
where u
=
ou
ou
Ou
at
OX
Oy
(p, pu, pV, pE) is the conservative state vector, where p is the density, u and v
are the x- and y-components of velocity, and E is the specific internal energy. The Jacobian
flux matrices, A and B, are given by
1
0
0
( 3 -- )u
-(y - 1)v
- - 1
V
u
0
0
+
2
AB=
-
2
uv
2
-yuE + (-y - 1)uq
-yE
0
-
B=
(-y-3)V
2
2
uv
+(-_1)u
+
(V2 + 3U2 )
-
- _1)uv
-yu
0
1
0
V
U
0
2
-(~-
2
-yvE + (y - 1)vq 2
1)u
-(y - 1)uv
(3 - y)v
yE - 2Z1(u
2
y
+ 3v 2 )
-1
'o
The compressible, two-dimensional Navier-Stokes equations in strong, conservation form
are
Ou
--
at
+V
-F(u)
-
V .F(u, Vu) = 0,
47
with conservative state, u, and inviscid flux vector, FY = (Fr, FY), given by
Pu
pv
pu2+p
Puv
puv
pv2 + P
pu H
pvH
where the pressure is p = (y - 1) [pE -
jp (U2 + v 2 )],
and the specific enthalpy is H
E + p/p. The viscous flux, F, = (F',F'), is given by
0
2(2
au
'9v
p2ax
an
-
p(2
FY
u-
=
ay)
a
+
)u+ p(2-+ )v+
0
au + v
8av
u
y(ay ax)
a)V +pA(LU+ a)U + K
22
p (2!Lv-
where p is the dynamic viscosity and n is the thermal conductivity.
48
V
V
=
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