Embed Size (px)
Citation preview
Modelling of a ground
impedance boundary in a
time-domain CAA method
E. van der Pol
Master Thesis in Applied Mathematics
August 2012
Modelling of a ground impedanceboundary in a time-domain CAAmethod
Summary
In this thesis the modelling of a ground impedance boundary condition in a time-domain com-putational aeroacoustic method is investigated. A ground impedance boundary is a boundarythat absorbs part of the incoming acoustic wave and reflects the rest. In the literature severalmodels are proposed to model the quantity of absorption, the impedance, for a range of groundtypes. All of the proposed models are defined in the frequency domain, but the computationalaeroacoustic method calculates in the time domain. Unfortunately, the transformation to thetime domain is not straightforward, and therefore approximations are proposed. One ap-proximation is a broadband approach and is described by poles, another is a single-frequencyapproximation.There are several ways to discretize the pole approximation. In the literature a recursiveimplementation is proposed. However, in this thesis a new implementation is proposed. Theimplementation is given as a system of ordinary differential equations. This system canbe solved with the same fourth-order time-integration method, as is used in the compu-tational aeroacoustic method, which makes the entire system fourth-order accurate. Sometwo-dimensional test cases show the influence of the ground impedance boundary conditionon a sound wave.
Master Thesis in Applied MathematicsAuthor: E. van der PolSupervisor(s): A.E.P. VeldmanSecond supervisor: A.C.D. van EnterExternal supervisor: J.C. Kok and W. RozemaDate: August 2012
Institute of Mathematics and Computing ScienceP.O. Box 4079700 AK GroningenThe Netherlands
UNCLASSIFIEDNationaal Lucht- en Ruimtevaartlaboratorium
National Aerospace Laboratory NLR
Executive summary
Modelling of a ground impedance boundary in atime-domain CAA method
Report no.NLR-TR-2012-349
Author(s)E. van der Pol
Classification reportUnclassified
DateAugust 2012
Knowledge area(s)Computational Physics &Theoretical AerodynamicsAeroacoustics & ExperimentalAerodynamics
Descriptor(s)Ground impedance boundaryconditionComputational aeroacoustics
Problem areaAircrafts make noise during take-off and landing. This can be quiteannoying to people living nearbyairports, but also a danger to the en-vironment. Sound walls and otherobjects have been placed nearbynoisy areas such as airports or high-ways to block part of the sound,but these objects only reflect thesound. This holds for most man-made surfaces that are basicallysolid and they are called acousti-cally hard. Naturally most groundsare not acoustically hard, but sur-faces covered with grass or othervegetation. These grounds absorbsome of the incoming sound andtherefore reflect only part of it.
NLR uses a computational aeroa-coustic (CAA) method to accuratelypredict the propagation of sound ina fluid. So far the method can onlyhandle acoustically hard bound-aries. As said before, most surfacesare not acoustically hard, so an im-plementation of acoustically softboundaries is wanted. Acousticallysoft boundaries can be modelled asimpedance boundaries where theimpedance describes the quantity ofthe absorption.
Description of workThis investigation concers themodelling of impedance bound-aries and the implementation ofthe impedance boundaries in thecomputational aeroacoustic methodof NLR. The impedance describeshow an impedance boundary re-flects and absorbs sound wavesfor a certain ground type and fre-quency. Therefore the impedanceis defined in the frequency domain.For all acoustically soft surfacesan impedance can be prescribed,so models are necessary to deter-mine what the impedance is fordifferent ground types. Since 1949models can be found in the liter-ature that describe the impedancefor different ground types and fre-quencies. Some of these models arebased on analytical derivations andothers are based on experimentalvalues. For further investigationthe experimentally-derived andphysically-valid Miki model waschosen.
The computational method of NLRpredicts propagation of sound in thetime domain and the impedance isdefined in the frequency domain.Therefore the impedance needsto be transformed to the time do-
UNCLASSIFIED
UNCLASSIFIED Modelling of a ground impedance boundary in a time-domain CAAmethod
main with a Fourier transform.Most of the models from the lit-erature are not easily or not at alltransformable to the time domain.Therefore approximations have tobe made. There are different waysof approximation the impedance,including a single-frequency and abroadband approximation. In thiscase the broadband approximationthat is described by poles is chosenfor further investigation.
Once a choice for a model andapproximation are made theimpedance is discretized to im-plement into NLR’s computationalmethod. In the literature some dis-cretizations can be found, but a newone based on ordinary differentialequations is derived. This method isthen applied to a one-dimensionalfinite-volume method in Matlab.With this method the accuracyand stability of the system with animpedance boundary condition are
investigated.
Results and conclusionsIt is established that the new dis-cretization of the impedanceboundary keeps the system fourth-order accurate and stable for small(enough) time steps. After expend-ing the discretization to higherorder, the impedance boundaryis applied into NLR’s computa-tional method. With this methodtwo-dimensional test cases weresimulated. The first test case simu-lates a point source and investigatesthe sound pressure level. Anothertest case initializes an acousticpulse and simulates its propagationover a large domain.
ApplicabilityNow that the impedance boundaryis implemented in NLR’s compu-tation method, calculations withacoustically soft surfaces can becarried out.
UNCLASSIFIED
Nationaal Lucht- en Ruimtevaartlaboratorium, National Aerospace Laboratory NLR
Anthony Fokkerweg 2, 1059 CM Amsterdam,P.O. Box 90502, 1006 BM Amsterdam, The NetherlandsTelephone +31 20 511 31 13, Fax +31 20 511 32 10, Web site: www.nlr.nl
Nationaal Lucht- en Ruimtevaartlaboratorium
National Aerospace Laboratory NLR
NLR-TR-2012-349
Modelling of a ground impedance boundary in atime-domain CAA method
E. van der Pol
No part of this report may be reproduced and/or disclosed, in any form or by any means, without the prior
written permission of the owner.
Customer National Aerospace Laboratory NLR
Contract number ----
Owner National Aerospace Laboratory NLR
Division Aerospace Vehicles
Distribution Limited
Classification of title Unclassified
August 2012Approved by:
Author Reviewer Managing department
NLR-TR-2012-349
Contents
1 Introduction 5
2 Modelling of aeroacoustics 6
2.1 Linearized Euler Equations 6
2.2 Boundary conditions 7
2.2.1 Solid wall 8
2.2.2 Inflow and outflow boundaries 8
2.2.3 Partially absorbing surfaces 9
3 Impedance ground surfaces 9
3.1 Frequency vs. time domain 9
3.2 Material dependence 10
3.3 Hard-backed layered ground 11
3.4 Physical validity 11
3.5 Impedance models 12
3.5.1 Zwikker-Kosten 12
3.5.2 Delany-Bazley/Miki 15
3.5.3 Other models 18
3.6 Approximations 19
3.6.1 Poles 19
3.6.2 MSD+ 20
3.7 Conclusions 22
4 Numerical method 23
4.1 Computational problem 23
4.2 Time integration 25
4.3 Spatial schemes and artificial diffusion 26
4.3.1 Artificial diffusion 29
4.4 Discretized boundary conditions 29
4.4.1 Solid wall 30
4.4.2 Symmetry boundary 31
4.4.3 Impedance boundary with recursive convolution 31
4.4.4 Impedance boundary with a system of ODEs 33
3
NLR-TR-2012-349
4.5 One-dimensional test case 34
4.5.1 Accuracy 37
4.5.2 Stability 39
5 Two-dimensional problems 43
5.1 Single frequency 45
5.2 Gaussian pulse 48
6 Conclusions and recommendations 51
References 53
Appendix A ENSOLV routines 55
A.1 pol.h 55
A.2 bcpole.F 55
A.3 dim.h 61
A.4 bcond.F 61
A.5 rdbcd.F 61
A.6 updbcd.F 63
Appendix B Discretization methods for the MSD+ approximation 64
B.1 Integration discretization 64
B.2 Discretization with auxiliary parameters 65
B.3 System of ODEs 66
4
NLR-TR-2012-349
1 Introduction
An aircraft emits a lot of noise. This is not a problem for the environment when an aircraft is fly-
ing at 30.000 feet, but during takeoff and landing an aircraft can make so much noise that it may
be annoying to people and also an environmental hazard. All kinds of objects and effects can
strengthen or damp the amount of noise that is heard. Think of a sound wall placed nearby busy
highways to damp the noise to residential neighbourhoods, or when the wind is directed towards
you, the noise seems to strengthen. When there is a solid ground like a runway, the noise reflects
upon this runway. However looking around an airport like Schiphol, there is a lot of grass and
vegetation. One of the properties of ground covered with grass or vegetation is that part of the
noise is absorbed by the ground. This property is even stronger in a layer of snow covering the
ground. Therefore it is very interesting to model the influence this property has on the propaga-
tion of sound.
NLR uses a computational aeroacoustic method to model the propagation of sound, while taking
into account ground interactions and atmospheric effects. At the moment the method models
sound propagation with a perfectly reflecting ground boundary condition. However most of the
time we do not deal with a solid ground, but with a vegetation covered ground such as grass, or
in the winter sometimes with a ground covered with snow. These ground types can be modelled
as an impedance boundary condition.
The property that indicates the amount of absorption and reflection of an incoming acoustic wave
is called the impedance. In reality there is a large range of ground types which all have different
impedances. The impedance depends on the material properties of the ground type and therefore
models are used to determine the impedance for a specific ground type. In the literature a wide
range of models can be found to describe the impedance of ground types. Some models were
derived analytically and others were fitted to experimental data.
The impedance is not only dependent on the ground type, but also on the frequency of the incom-
ing acoustic wave. The impedance is therefore defined in the frequency domain. However the
goal of this report is to implement an impedance boundary into the CAA method used by NLR
which calculates the propagation of sound in the time domain. Therefore the impedance needs to
be transformed to the time domain with the Fourier transform. Most models cannot be (easily)
transformed to the time domain and have to be approximated so that they can be transformed.
This can be done for a certain frequency (single band) or for a frequency band (broadband).
Several models and approximations are described and discussed in this report and a choice is
made for a model and approximation. There are different methods to numerically discretize the
5
NLR-TR-2012-349
chosen approximation. These methods have a different set-up and their accuracy and stability are
discussed with a one-dimensional simulation.
When the computational method for an impedance boundary is implemented, some test cases
can be simulated. In the final chapter of this report some two-dimensional simulations can be
found which were simulated with the aeroacoustic method of NLR. The first test case simulates
a source that produces a continuous acoustic wave of a certain frequency. The second test case
shows what an impedance boundary does to a Gaussian pulse, which contains a band of frequen-
cies.
2 Modelling of aeroacoustics
Aeroacoustics is the study of the propagation of sound in a fluid. Sound can be heard due to pres-
sure perturbations in fluids such as air. Therefore the study of the propagation of sound can be
seen as a fluid dynamics study. The equations that describe the propagation of sound can be de-
rived from the Euler equations for fluid dynamics problems with inviscid and compressible flow.
This will be shown in the next section.
2.1 Linearized Euler EquationsThe Euler equations for inviscid and compressible flow in conservation form are given as
∂ρ
∂t+
∂(ρuj)∂xj
= 0
∂(ρui)∂t
+∂(ρuiuj)
∂xj+
∂p
∂xi= 0
∂(ρE)∂t
+∂(ρHuj)
∂xj= 0. (1)
where p is the pressure, u the velocity vector and ρ the density. Also E is the total energy per
unit mass and H = E + pρ is the total enthalpy per unit mass (Ref. 7). For a calorically perfect
gas, the pressure can be eliminated from the Euler equations by using the following relation
p = (γ − 1)(ρE − 12ρuiui). (2)
Here γ is the ratio of specific heats and is written as γ = cp
cv, with cv and cp respectively the heat
capacity at constant volume and the heat capacity at constant pressure. For air, γ is equal to 1.4.
6
NLR-TR-2012-349
The density, the velocity and the energy can be split in a time-independent mean flow and time-
dependent perturbations of the mean flow as
ρ(x, t) = ρ0(x) + ρ′(x, t)
u(x, t) = u0(x) + u′(x, t)
E(x, t) = E0(x) + E′(x, t). (3)
Here the subscript 0 denotes the mean flow and the primes denote the perturbations. The param-
eter γ again appears, here in the relation for the speed of sound c0. This is given as c0 =√
γp0
ρ0,
where p0 is the pressure and ρ0 the density of the mean flow. The internal energy e for a calori-
cally perfect gas is dependent on the temperature T by e = cvT .
As said before, aeroacoustics is the study of the time-dependent perturbations of the mean flow.
The total flow has to satisfy the Euler equations. When substituting the flow variables (3) into the
Euler equations (1), and subtracting the steady Euler equations which have to hold for the mean
flow variables, a system for the perturbations is left. Assuming that the perturbations are small
compared to the mean flow variables, such that all quadratic or higher order terms of the pertur-
bations can be neglected, then the perturbations have to satisfy the linearized Euler equations
(LEE) in conservation form, given by
∂ρ′
∂t+
∂
∂xj(u0jρ
′ + ρ0u′j) = 0
∂(ρui)′
∂t+
∂
∂xj(u0j(ρui)′ + (ρ0u0i)u′j) +
∂p′
∂xi= 0
∂(ρE)′
∂t+
∂
∂xj(u0j(ρH)′ + (ρ0H0)u′j) = 0 (4)
with
(ρuj)′ = ρ0u′j + u0jρ
′
(ρE)′ = ρ0E′ + E0ρ
′
p′ = (γ − 1)((ρE)′ − u0i(ρui)′ +12(u0iu0i)ρ′)
(ρH)′ = (ρE)′ + p′. (5)
2.2 Boundary conditionsSound propagation is defined on a domain, thus boundaries for that domain have to be set. These
boundaries can have different types such as a solid wall or an outflow boundary. A few of these
boundaries will be described here for a 3-dimensional domain.
7
NLR-TR-2012-349
2.2.1 Solid wallA solid wall is called acoustically hard, i.e. there can be no flow through the solid wall, so the
normal momentum has to be zero. This is called a slip boundary condition. Say a vector n points
out of the boundary then the condition can be written as
(ρu)′ · n = 0. (6)
2.2.2 Inflow and outflow boundariesAnother possibility is an open boundary. At this boundary either the incident wave can enter or
leave the domain. This open boundary can be modelled as an inflow or outflow boundary, where
respectively an incident wave enters or leaves. The inflow and outflow boundary condition are
modelled as characteristic boundary conditions based on the one-dimensional characteristic the-
ory. This means that the solution is expressed in characteristic variables or Riemann invariants.
When n, s and t form an orthonormal system, with n pointing outside of the boundary, the Rie-
mann invariants are given as
s′ =cv
p0p′ − cp
ρ0ρ′
u′s = s · u′
u′t = t · u′
R′o =
12n · u′ + 1
2ρ0c0p′
R′i =
12n · u′ − 1
2ρ0c0p′
with c0, cv and cp as before.
The perturbations of the mean flow can be expressed in terms of the Riemann invariants as
ρ′ =1c20
p′ − ρ0
cps′
u′ = (R′o + R′
i)n + u′ss + u′tt
p′ = ρ0c0(R′o −R′
i).
For the inflow and outflow boundaries the incoming Riemann invariants have to be prescribed.
The inflow boundary has different incoming Riemann invariants than the outflow boundary. The
incoming Riemann invariants at a far field inflow boundary are s′, u′s, u′t and R′i and are set to
zero. At a far field outflow boundary the incoming Riemann invariant is R′i which is set to zero.
Because these characteristic boundary conditions are known to lead to significant reflections of
outward-going waves in 2D and 3D, a buffer zone is added. This however will not be discussed
here and more information can be found in Ref. 7.
8
NLR-TR-2012-349
2.2.3 Partially absorbing surfacesIn reality not all the boundaries are perfectly reflecting like a solid wall, or open like an outflow
boundary. Most natural surfaces, such as grasslands, are porous surfaces and have the property to
absorb a part of the incoming sound and reflect the rest. These surfaces are called impedance sur-
faces or boundaries and can be seen as acoustically soft. Impedance is the measure of reflection
and absorption by the boundary and is related to the pressure and velocity of the flow. In the next
chapter it is explained how the impedance boundary condition is defined, how the impedance is
modelled for different porous materials and surfaces, and how the impedance can be transferred
to the time domain.
3 Impedance ground surfaces
The impedance is defined as the ratio of the Fourier transforms of the pressure and the normal
velocity at the impedance boundary:
Z(ω) =P (ω)Un(ω)
, (7)
with ω the angular frequency. The impedance is a complex number and can be split up in a real
and imaginary part Z(ω) = R(ω) + iX(ω). The real part R(ω) is called the resistance and
reflects the loss of acoustic energy from the system. The imaginary part X(ω), which is called
the reactance, reflects how the kinetic energy is converted to potential energy.
3.1 Frequency vs. time domainAs can be seen in equation (7) the impedance Z is defined in the frequency domain and so are
the pressure P and normal velocity Un. However, our goal is to implement an impedance surface
in the CAA method which works in the time domain. Therefore the impedance needs to be trans-
formed to the time domain. This will be done with the inverse Fourier transform. The Fourier
transform F (ω) is defined as
F (ω) =∫ ∞
−∞f(t) e−iωtdt
so that the inverse Fourier transform f(t) is defined as
f(t) =12π
∫ ∞
−∞F (ω) eiωtdω.
The impedance in the frequency domain can be written as P (ω) = Z(ω)Un(ω) and the con-
volution integral is needed to transform the relation to the time-domain. The impedance in the
time-domain is now given by the relation
p′(t) =∫ ∞
−∞z(τ) (u′(t− τ) · n)dτ
9
NLR-TR-2012-349
where u′(t) · n is the normal velocity in the time domain and will be written as u′n(t) from now
on such that the previous equation becomes
p′(t) =∫ ∞
−∞z(τ) u′n(t− τ)dτ. (8)
3.2 Material dependenceThe impedance is dependent on which medium the air flows through. Different types of porous
material have various acoustical properties that reflect how easy it is for air to flow through the
material. The most important property is the flow resistivity σ which is the resistance that the
rigid frame offers an air flow through the material. This can be described as the drop in pressure
per unit length in the direction of the flow of air at unit speed (Ref. 1). The unit of flow resistiv-
ity is Pa sm2 . When the flow resistivity is high, like in asphalt, it is hard for air and thus for sound
to travel through the ground. When the flow resistivity is small it is very easy for sound to travel
through the ground, such as is the case with snow.
A difference is made between the flow resistivity σ of a material and the effective flow resistivity
σe. The effective flow resistivity takes into account that the pores in the material sometimes have
strange structures and that it is not possible for air to flow through all of the material. The struc-
ture of the pore shapes can be described as the tortuosity q2 which is the curviness of the pores
(Ref. 1). When the pores in the porous material are straight in the flow direction, the tortuosity is
equal to 1, when they have curves, lateral cavities or are in a different direction, the tortuosity is
larger than 1. Another material property is the porosity Ω which is the volume fraction of air in
the material (Ref. 14) and therefore lies between one and zero. Wilson et al. (Ref. 16) propose
the following relation for the effective flow resistivity σe
σe =σΩq2
(9)
which is smaller than the flow resistivity σ for porous materials because of the values of the tor-
tuosity and porosity.
The factor Ωq2 represents the air in the main pores per unit volume that gets moved by the pres-
sure gradient. In lateral cavities next to main pores the air does not move because the pressure
gradient is almost zero there, so not all the air in the main pores gets moved (Ref. 17). Some
impedance models use just the effective flow resistivity σe and others use the flow resistivity σ in
combination with other parameters such as the porosity Ω and tortuosity q2 in the models.
When impedance is mentioned in the rest of this report, the characteristic impedance is meant.
This means it is assumed that the material properties do not depend on the frequency of the inci-
dent acoustic waves.
10
NLR-TR-2012-349
3.3 Hard-backed layered groundWhen calculating the impedance of a ground it is assumed that the ground is semi-infinite. Most
of the ground types that are considered however have a layered structure, like grass-covered
and snow-covered fields. Here the lower layer is acoustically hard, which means totally reflect-
ing. For this case Rasmussen (Ref. 10) derived a formula for the impedance of a so called hard-
backed layer
Z(d, ω) =Z(ω)ρ0c0
coth(−ikd) (10)
where k is the complex wave number in the porous layer, d the thickness of the hard-backed
layer and Z(ω) the impedance of semi-infinite ground.
3.4 Physical validityImpedance is a physical property of a porous ground surface, therefore it must be physically
valid. For an impedance to be physically valid the next three physical conditions have to hold
(Ref. 12).
Because the pressure p′(t) and the normal velocity u′n(t) are real, the impedance in the time-
domain z(t) also has to be real. The consequence for the impedance in the frequency domain
Z(ω) describes the first physical condition, the reality condition
Z(−ω) = Z∗(ω) for all ω ∈ R (11)
where Z∗ denotes the complex conjugate of Z. Note that this condition holds when R is even
and X is odd in ω.
The second condition has to do with the energy in an acoustic flow. An impedance surface ab-
sorbs part of the acoustic energy and therefore cannot add energy to the acoustic flow. As men-
tioned before, the real part of the impedance represents a loss of acoustic energy if and only if
it is positive. This means that for the impedance surface to not add energy, the real part of the
impedance needs to be positive which leads to the passivity condition
R(ω) ≥ 0 for all ω ∈ R. (12)
The last condition for physical validity is the causality condition. This condition is based on the
requirement that the pressure p′(t) cannot depend on the normal velocity u′n(t) of the future.
When looking at the relation between the pressure, velocity and impedance in the time-domain
(8), note that for p′(t) to not depend on u′n(t) of the future, z(τ) needs to be equal to zero for
τ < 0. The result of this is that
Z(ω) is analytic in Im(ω) < 0.
11
NLR-TR-2012-349
Because it also holds that the velocity u′n(t) cannot depend on the pressure p′(t) of the future the
condition has to be expanded such that the impedance is non-zero in Im(ω) < 0. Therefore the
causality condition states that
Z(ω) is analytic and non-zero in Im(ω) < 0. (13)
A consequence of this condition is that the real and imaginary parts of the impedance are related
through the Hilbert transform H[.] through X(ω) = H[R(ω)].
3.5 Impedance modelsIn the last few decades several models to describe an impedance were devised. Some models
only use one parameter to describe the ground material like the Delany-Bazley and Miki models.
Others make use of more parameters, such as the phenomenological Zwikker-Kosten model, the
two-parameter Attenborough model and Wilson’s relaxation model.
3.5.1 Zwikker-KostenA way to describe the impedance Z is by the phenomenological Zwikker-Kosten model (ZK
model). This model was analytically derived by Zwikker and Kosten in 1949 by using linear
acoustic equations (the momentum and continuity equation) for porous materials and a harmonic
plane sound wave that satisfies the wave equation. The model is defined as
Z = Z∞
√1 + iωτ
iωτ(14)
with Z∞ = ρ0c0qΩ and τ = ρ0q2
σΩ . Here the three parameters of the model are the tortuosity q2, the
flow resistivity σ and the porosity Ω.
Below the analytical derivation for the one dimensional case is given. The linear acoustic equa-
tions for free air are
−∂u′
∂x=
1ρ0c2
0
∂p′
∂t(15)
−∂p′
∂x= ρ0
∂u′
∂t(16)
and the linear acoustic equations for porous materials are
−∂u′
∂x=
Ωρ0c2
0
∂p′
∂t(17)
−∂p′
∂x= ρ0
q2
Ω∂u′
∂t+ σu′. (18)
12
NLR-TR-2012-349
Here the first equation differs from the continuity equation for free air (15) through Ω. This is
because the available volume of air is a factor Ω smaller than in free air. The second equation
differs from the momentum equation for free air (16) in two ways. One of them uses Darcy’s
law to add a term that deals with the use of porous materials through the flow resistivity σ (Ref.
9). The flow resistivity σ is defined as the measure of the drop in pressure per unit length in the
direction of the flow of air at unit speed. In formula form this can be written as
σ =−∂p′
∂x
u′(19)
and can be found in the momentum equation for porous materials. The other difference comes
from the lack of flow in some parts of the porous material. In section 3.2 this was described with
the term Ωq2 which is also found in the momentum equation for porous materials.
From the linear acoustic equations for porous materials the impedance Z will be derived. Take a
harmonic plane sound wave travelling in the positive x-direction where the pressure and velocity
are given as
p′(x, t) =12q(x, k) eiωt +
12q∗(x, k) e−iωt (20)
with
q(x, k) = A e−ikx, (21)
and
u′(x, t) =12v(x, k) eiωt +
12v∗(x, k) e−iωt (22)
with
v(x, k) = B e−ikx. (23)
Here A and B are constants and the superscript ∗ denotes the complex conjugate. Then the impedance
is derived as Z = q(x,k)v(x,k) = A
B . When substituting the first (or second) term for the pressure and
velocity into the continuity equation (17), an equation for the complex wave number k expressed
in the impedance Z is gained
−∂u′
∂x=
Ωρ0c2
0
∂p′
∂t
⇒ −(−ik)(
12B e−ikx eiωt
)=
Ωρ0c2
0
(iω)(
12A e−ikx eiωt
)⇒ (ik)B =
Ωρ0c2
0
(iω)A
⇒ k =ωΩρ0c2
0
A
B=
ωΩρ0c2
0
Z (24)
13
NLR-TR-2012-349
When substituting the pressure and velocity into the momentum equation (18), an equation for
the impedance Z = AB expressed in the complex wave number k is given as
−∂p′
∂x=
q2ρ0
Ω∂u′
∂t+ σu′
⇒ −(−ik)(
12A e−ikx eiωt
)=
q2ρ0
Ω(iω)
(12B e−ikx eiωt
)+ σ
(12B e−ikx eiωt
)⇒ (ik)A =
(iρ0q
2ω
Ω+ σ
)B
⇒ Z =A
B=(
ρ0q2ω
Ω− iσ
)1k
(25)
Now combining the complex wave number (24) and the impedance (25) gives
Z =(
ρ0q2ω
Ω− iσ
)1k
=(
ρ0q2ω
Ω− iσ
)ρ0c
20
ωΩ1Z
⇒ Z2 =ρ20c
20q
2
Ω2− i
ρ0c20σ
ωΩ
⇒ Z = (ρ0c0)
√q2
Ω2− i
σ
ρ0ωΩ=
ρ0c0q
Ω
√1− i
1ωτ
= Z∞
√1 + iωτ
iωτ
with Z∞ = ρ0c0qΩ and τ = ρ0q2
σΩ . This is exactly the Zwikker-Kosten model (14).
When using the second term of the velocity and pressure, the impedance becomes its complex
conjugate and the reality condition (11) follows. The passivity condition (12) also holds because
Z∞ and τ are positive and so is the real part of the root. The root of the impedance can be writ-
ten as√1 + iωτ
iωτ=
√1− i
1ωτ
=
√√√√√1 + 1ω2τ2 + 1
2− i
√√√√√1 + 1ω2τ2 − 1
2,
where the last expression is just the root of a complex number split into a real and imaginary
part. From this it can be seen that the real part of the root is indeed bigger than zero and the pas-
sivity condition holds. The causality condition (13) is also satisfied because√
1+iωτiωτ is non-zero
for Im(ω) < 0 and also analytic, because the pole ω = 0 does not lie in the lower-half plane.
Therefore this description of the impedance is physically valid.
14
NLR-TR-2012-349
3.5.2 Delany-Bazley/MikiOne of the most popular and most-used models is the empirical one-parameter model that was
derived from experimental data by M.E. Delany and E.N. Bazley in 1969 (Ref. 3). Their model
for the impedance has the following form
Z = ρ0c0
(1 + a
(ρ0ω
2πσe
)b
+ ic(
ρ0ω
2πσe
)d)
(26)
where ρ0 and c0 are respectively the atmospheric pressure and speed of sound. In this report they
are taken as ρ0 = 1.225 kgm3 and c0 = 340m
s . Delany and Bazley experimentally derived the
following coefficients for their model (26)
a = 0.0595
b = −0.75
c = −0.0891
d = −0.73.
Here ρ0ω2πσ is dimensionless and thus the impedance has the same dimension as ρ0c0, which is
kgm2 s
. This model only has one parameter, σe, while the modified ZK model had three, σ, q and
Ω. Delany and Bazley based their model on experimental results of various fibrous absorbing
materials like glass-fibre and mineral-wool materials and assumed that these materials had porosi-
ties Ω and tortuosities q2 close to 1. In that case σe is replaced by σ in (26).
Usually absorbing materials do not have porosities and tortuosities close to 1 and therefore an
adapted parameter is used in the DB model, the effective flow resistivity σe. The effective flow
resistivity allows the porosity and tortuosity of a material to differ from 1 by the following rela-
tion for the effective flow resistivity as shown in Section 3.2: σe = σΩq2 .
Delany and Bazley did not only derive a model for the impedance, but also for the complex wave
number k
k =ω
c0
(1 + q
(ρ0ω
2πσe
)r
+ ip(
ρ0ω
2πσe
)s)(27)
where the coefficients are
q = 0.0989
p = 0.1972
r = −0.70
s = −0.59.
15
NLR-TR-2012-349
The complex wave number is needed when a hard-backed layered ground (section 3.3) is used.
In 1990 Yasushi Miki (Ref. 8) stated that not all the physical properties from the previous section
hold for the DB model because of the choice of coefficients. Delany and Bazley assumed that
the frequency only has positive real values. When that is assumed, the model for the impedance
(26) satisfies the reality condition (11). The passivity condition (12) is almost satisfied, as can be
seen in figure 1 for a flow resistivity σ of 200 kPa sm2 . The causality condition (13) is however not
satisfied.
Fig. 1 Relative impedance Zρ0c0
of the DB model for σ = 200 kPa sm2
Miki derived two relations between the coefficients which should hold for the model to be phys-
ically valid. He derived these from the consequence of the causality conditions (13). The con-
sequence says that X(ω) should be the Hilbert transform of R(ω). To require relations for the
coefficients from X(ω) = H[R(ω)], where H[.] is the Hilbert transform, the following relation
is used
H[|x|ν−1] = −cotνπ
2sgn(x)|x|ν−1. (28)
When applying this relation to the DB model (26) and assuming that ω is positive, the following
two relations for the coefficients are derived
d = b and c = −a cot(b + 1)π
2. (29)
Here it was assumed that the reality condition (11) holds and the passivity condition (12) holds
when a ≥ 0. With these relations in mind, Miki used the experimental data of Delany and Bazley
to come to the following coefficients:
16
NLR-TR-2012-349
a = 0.0794
b = −0.632
c = −0.1218
d = −0.632.
Miki also derived relations for the coefficients of the complex wave number of the DB model.
Miki expresses the complex wave number by a single complex power-law relation
k(ω) =ω
ic0(1 + iaf b).
When substituting the complex frequency ω = iω in the above relation
k(iω) =iωic0
(1 + iaibωb)
=ω
c0
(1 + ia cos(
bπ
2)ωb + i2a sin(
bπ
2)ωb
)=
ω
c0
(1− a sin(
bπ
2)ωb + ia cos(
bπ
2)ωb
)and comparing this to the equation for the complex wave number of Delany and Bazley (27)
k =ω
c0
(1 + q
( ρ0ω
2πσ
)r+ ip
( ρ0ω
2πσ
)s)=
ω
c0
(1 + q
( ρ0
2πσ
)rωr + ip
( ρ0
2πσ
)sωs)
, (30)
the assumption of Miki gives the following relations for the coefficients
s = r
and
q( ρ0
2πσ
)r= −a sin(
rπ
2)
= −(a cos(
rπ
2)) sin( rπ
2 )cos( rπ
2 ).
From equation (30), a cos( rπ2 ) should be equal to p
( ρ0
2πσ
)s such that the previous equation be-
comes
q( ρ0
2πσ
)r= −p
( ρ0
2πσ
)stan(
rπ
2)
= −p( ρ0
2πσ
)rtan(
rπ
2),
which gives
q = −p tanrπ
2.
17
NLR-TR-2012-349
From these relations and again the experimental data of Delany and Bazley, Miki derived the
following coefficients of the complex wave number
q = 0.1810
p = 0.1239
r = −0.618
s = −0.618.
3.5.3 Other modelsNext to these two models there are many more to be found in the literature. Some models make
use of more parameters to describe the porous materials and others make use of different param-
eters. Below two more examples are presented of models for the impedance of porous materials,
others can be found in additional literature like the book by Attenborough (Ref. 1).
The Attenborough two-parameter model (Ref. 1) makes use of the effective flow resistivity σe
and the effective rate of change of porosity with depth αe. He assumes that porous materials do
not have a stable porosity but they change in depth. The impedance is given by Attenborough as
Z = 0.436(1 + i)
√2πσe
ω+ 19.74i
2παe
ω. (31)
Another model was derived by Wilson et al. (Ref. 16). This so-called relaxation model has 4
parameters and is defined as
Z = ρ0c0q
Ω
[(1 +
γ − 1√1− iωτe
)(1− 1√
1− iωτv
)]−1/2
. (32)
τv and τe are the aerodynamic and thermodynamic characteristic times and are respectively given
by Wilson as
τv =2ρ0q
2
Ωσ
τe = NPR s2Bτv
with NPR the Prandtl number. The parameters q2, Ω, σ, and sB appear in the expression for the
aerodynamic and thermodynamic characteristic times. sB is the only parameter that has not been
seen before and denotes the pore shape factor.
This model mimics the DB model in the range 0.01 < ρ0fσe
< 1, but is also valid outside this
range when using the following approximations for the characteristic times
τv ≈2.1ρ0
σ
τe ≈3.1ρ0
σ.
18
NLR-TR-2012-349
3.6 ApproximationsTo calculate the pressure from the relation P (ω) = Z(ω)Un(ω) the inverse Fourier transforms of
the impedance
z(t) =12π
∫ ∞
−∞Z(ω) eiωtdω
and the normal velocity
u′n(t) =12π
∫ ∞
−∞Un(ω) eiωtdω
are needed. The pressure can be calculated with the convolution product (8), which can also be
written as
p′(t) =∫ ∞
−∞z(t− τ) · u′n(τ)dτ. (33)
However it is not straightforward to calculate the inverse Fourier transform of the impedance
models of the previous section. It is therefore useful to approximate the expression of the impedance.
There are several ways to do this.
3.6.1 PolesOne of the ways was described by Reymen et al. (Ref. 11) for a broadband model and is the sum
of S first-order systems with real poles and T second-order systems with complex conjugate
poles. The impedance can now be written as
Z(ω) =S∑
k=1
Zk(ω) +T∑
l=1
Zl(ω), (34)
where
Zk(ω) =Ak
λk + iω=
Akλk
λ2k + ω2
− iAkω
λ2k + ω2
(35)
and
Zl(ω) =Al
µl + iω+
Bl
µ∗l + iω=
Dl + iωCl
(αl + iω)2 + β2l
. (36)
Here λk ≥ 0 is called a real pole and µl and µ∗l are a pair of complex conjugate poles αl ± iβl,
with αl ≥ 0.
The inverse Fourier transforms of these systems are easy to calculate and are
zk(t) = Ak e−λktH(t) (37)
19
NLR-TR-2012-349
and
zl(t) = e−αlt
(Cl cos(βlt) +
Dl − αlCl
βlsin(βlt)
)H(t) (38)
where H(t) is the Heaviside function which is zero for t < 0 and 1 for t ≥ 0. Then the inverse
Fourier transform of the poles approximation becomes
z(t) =S∑
k=1
zk(t) +T∑
l=1
zl(t). (39)
Some restrictions on the parameters are needed for this approximation to be physically valid. In
the second-order system the coefficients Cl and Dl can be written as Cl = Al + Bl and Dl =
αl(Al + Bl) + iβl(Bl − Al). For the reality condition (11) to hold Ak, Cl and Dl need to be real,
which results in the condition Bl = A∗l . For the causality condition (13) to hold, the poles have
to satisfy λk ≥ 0 and αl ≥ 0 and the model is passive (12) when Ak ≥ 0 and for a good choice
of Cl and Dl. Ak ≥ 0 is a very strong restriction, because the model can also be passive when
not all the coefficients Ak are bigger than or equal to zero.
In all our models the imaginary part of the impedance is negative, therefore the imaginary part
of the second-order system needs to be negative. This results in constraints for Al and Bl and
subsequently for Cl and Dl.
The coefficients of this approximation can be found by a fitting method which fits this approxi-
mation to one of the models or a data set. Two fitting methods were given by Cotte et al. (Ref. 2)
One is the Vector Fitting technique proposed by Gustavsen and Semlyen (Ref. 5) and the other
makes use of the MATLAB function fmincon, an ε-constraint method. The first method can give
complex conjugate poles, but when keeping the number of poles relatively small (below 10),
only real poles appear. The second method assumes that only real poles are being used.
3.6.2 MSD+The same Reymen et al. described a special case of the pole approximation, which is an approx-
imation for a single frequency. This approximation is better known as the mass-damper-spring
approximation
Z(ω) =a−1
(iω)+ a0 + a1(iω).
In the article of Heutschi et al. (Ref. 6) this approximation was extended with an extra term
a−2(iω)−2 and in that way was made second order in ω. It turns out that this MSD+ approxi-
mation is a better approximation then the first order approximation of Reyman et al. because now
20
NLR-TR-2012-349
the real part of the impedance is described with two parameters instead of a single constant. The
MSD+ approximation is written as
Z(ω) = a−2(iω)−2 + a−1(iω)−1 + a0 + a1(iω). (40)
The inverse Fourier transform of the MSD+ approximation can be found by using the following
relations
∂y(t)∂t
F←→ (iω)Y (ω)∫ t
−∞y(τ)dτ
F←→ (iω)−1Y (ω).
The inverse Fourier transform then becomes
p′(t) = a−2
∫ t
−∞
∫ t′
−∞u′n(τ) dτ dt′
+ a−1
∫ t
−∞u′n(τ) dτ + a0 u′n(t) + a1
∂u′n(t)∂t
. (41)
Also for this approximation it needs to be checked which restrictions are needed for all the phys-
ical conditions to hold. The reality condition (11) follows directly from the approximation when
all the parameters are real. The passivity condition (12) is easily maintained when setting a−2 ≤0 and a0 ≥ 0. The causality condition (13) requires a bit more effort. The expression for the
impedance can be rewritten as a rational function
Z(ω) =a−2 + a−1(iω) + a0(iω)2 + a1(iω)3
(iω)2.
The frequency can now be written as a real and imaginary part as ω = ωR + iωI . For the
impedance to be causal it has to be analytic and non-zero in Im(ω) = ωI < 0. For the impedance
to be analytic in ωI < 0 the poles have to lie in the upper-half plane. It can easily be seen from
the previous equation that the poles of the impedance are located in 0, which is assumed to lie in
the upper-half plane. Also needed for causality is the impedance to be non-zero in ωI < 0. This
means that a−2 + a−1(iω) + a0(iω)2 + a1(iω)3 cannot be equal to zero. Already there are two
constraints on the parameters: a−2 ≤ 0 and a0 ≥ 0. Now the other two parameters a−1 and a1
need to be chosen in such a way that the zeroes lie in the upper-half plane.
Heutschi et al. suggest that finding the coefficients can be done by fitting the MSD+ approxima-
tion to a model by minimizing the error of the squared differences of the model and the MSD+
approximation.
21
NLR-TR-2012-349
3.7 ConclusionsThis chapter gave an overview of some different ground impedance models and ways to approxi-
mate these models. For the next part of the report, the numerical discretization of the approxima-
tions will be used and therefore a choice of model and approximation will be made.
First let us discuss the different models that were presented. These models were the Zwikker-
Kosten model (14) and the Delany-Bazley model (26) with two choices for the coefficients. The
ZK model is based on a mathematical derivation, but has more parameters. The DB model has
a simple form, and with the Miki coefficients the model is also physically valid. In figure 2 the
three different models for a range of frequencies and flow resistivity of σ = 200 kPa sm2 , porosity
of Ω = 0.5 and the tortuosity of q2 = 1.3 were plotted. In the Miki and Delany-Bazley method
the effective flow resistivity σe is used as parameter. The effective flow resistivity depends on the
tortuosity and porosity and in this case is equal to σe ≈ 76.92 kPa sm2
(a) Real part of the relative impedance for σ =
200 kPa sm2 , Ω = 0.5 and q2 = 1.3
(b) Imaginary part of the relative impedance for σ =
200 kPa sm2 , Ω = 0.5 and q2 = 1.3
Fig. 2 Relative impedance for the DB, Miki and ZK models
As can be seen, the models do not differ that much. Therefore the choice of model is mainly one
of convenience. For this reason the DB model with Miki coefficients is chosen, because it only
has one parameter to work with and is a physically valid model.
Which approximation to use is another choice that has to be made. The choice is between the
pole approximation (34), which is a broadband model, and the MSD+ approximation (40), which
is a small-band model. Both models are easily transferred to the time-domain and can both be
made physically valid by the right choice of coefficients. The discretization of both approxima-
tions has not yet been discussed, but also plays a role in the decision which approximation to use.
In sections 4.4.3 and 4.4.4 two discretizations for the pole approximation are discussed and two
22
NLR-TR-2012-349
discretization for the MSD+ approximation can be found in appendix B. As can be seen in ap-
pendix B, all the possible discretization methods for the MSD+ approximation have some fault.
From instability issues, needing to approximate the derivative of the normal velocity to division
by zero. A discretization method of the pole approximation also needs approximations, but one
method does seem to work well without any concerns. Therefore the pole approximation is pre-
ferred.
Also important is the broad usability of the approximations. Because the pole approximation
can be fitted for a large band of frequencies, our preference goes to this approximation. However
only real poles for the approximation are considered such that the pole approximation is written
as
Z(ω) =S∑
k=1
Ak
λk + iω=
S∑k=1
Akλk
λ2k + ω2
− iAkω
λ2k + ω2
. (42)
4 Numerical method
To acquire solutions for an acoustical problem with an impedance boundary, the problem has to
be transformed to a computational problem. In the first section the two-dimensional discretized
form of the linearized Euler equation is described which can be adapted for one-dimensional
and higher-dimensional problems. In the next two sections the numerical methods for the time
integration and spatial discretization as used by NLR are discussed. In section 4.4 two ways to
discretize an impedance boundary condition are discussed. The two ways are then put to the test
in a one-dimensional test case in terms of the accuracy and stability of the discretized impedance
boundary conditions.
4.1 Computational problemAssuming a two-dimensional domain D ∈ R2, the linearized Euler equations (4) prescribe the
acoustic flow on this domain. Before the acoustical problem can be solved numerically it has to
be defined as a computational problem.
As seen in section 2.1, the flow can be split into the mean flow and the perturbations of the mean
flow (3). The computational problem considers a prescribed mean flow with velocity u0 =
(u0, v0). Also prescribed are the density ρ0 of the mean flow and the speed of sound c0. Because
it is assumed that the domain only contains air, which is assumed to be a calorically perfect gas,
the pressure p0 of the mean flow can be calculated from
c0 =√
γp0
ρ0⇒ p0 =
c20ρ0
γ.
23
NLR-TR-2012-349
Here γ is the heat capacity ratio which is 1.4 for air. The energy per unit mass of the mean flow
for a calorically perfect gas E0 is calculated from the relation (2) as
E0 =p0
(γ − 1)ρ0+
12(u2
0 + v20).
The mean flow variables ρ0, p0, E0, c0 and u0 are prescribed, but not necessarily constant through-
out the domain. Another mean flow variable that can be seen in the linearized Euler equations (4)
is the total enthalphy per unit mass H0, which can be calculated from the energy of the mean
flow E0 by
H0 = E0 +p0
ρ0.
Now having all the mean flow variables prescribed, the linearized Euler equations (4) in conser-
vative form of the 2D-domain D can be rephrased as
∂U
∂t= ∇ · F(U) (43)
with U the flow state vector
U =
ρ′
(ρux)′
(ρuy)′
(ρE)′
.
Here ∇ denotes the two-dimensional gradient and F(U) is defined as
F(U) =
u0ρ
′ + ρ0u′x v0ρ
′ + ρ0u′y
u0(ρux)′ + (ρ0u0)u′x + p′ v0(ρux)′ + (ρ0u0)u′yu0(ρuy)′ + (ρ0v0)u′x v0(ρuy)′ + (ρ0v0)u′y + p′
u0(ρH)′ + (ρ0H0)u′x v0(ρH)′ + (ρ0H0)u′y
. (44)
When integrating the conservative linearized Euler equations (43) over a fixed control volume
Ω ⊂ D and applying Gauss’ divergence theorem the integral form of the equations is obtained:∫Ω
∂U
∂tdV +
∫δΩF(U) · n dA = 0, (45)
where δΩ is the boundary of the fixed volume Ω and n is the unit normal pointing outside of the
volume. The entire domain D can be discretized into grid cells Ωi,j . When applying the integral
form of the equations to a grid cell, the discretized form of the integral form of the linearized
Euler equations (45) is written as:
Vi,j
(∂U
∂t
)i,j
+ Bi,j = 0 (46)
24
NLR-TR-2012-349
where Vi,j is the volume of the grid cell Ωi,j :
Vi,j =∫
Ωi,j
dV
and Bi,j is the discretized form of∫δΩi,j
F(U) · n dA
which is called the flux balance. In section 4.2 time-integration methods are discussed and in
section (4.3) the derivation of Bi,j is discussed.
4.2 Time integrationIn this section the time integration of the discretized linearized Euler equations (46) is discussed.
Reorganizing the discretized equations (46) gives(∂U
∂t
)i,j
= −Bi,j
Vi,j= f(t, U).
There are various methods to solve this differential equation and a well-known and used range of
solutions methods are the Runge-Kutta methods. The general structure of a Runge-Kutta method
is
Un+1 = Un + ∆t
s∑i=1
bif(tn + ci∆t, Yi) (47)
where
Yi = Un + ∆t
s∑j=1
aijf(tn + cj∆t, Yj) for i = 1, ..., s. (48)
A special case of the Runge-Kutta method is used by NLR to discretize the time-integration.
This is the standard fourth-order low-storage Runge-Kutta method and is given as
Un+1 = Y4
and
Yi = Un + γi∆t f(tn + γi−1∆t, Yi−1) for i = 1, ..., 4 (49)
with coefficients given as
γ0 = 0, γ1 =14, γ2 =
13, γ3 =
12, γ4 = 1. (50)
Each step of the fourth-order low-storage Runge-Kutta is linear with a different time step de-
pending on the coefficients, but together they result in a fourth-order method for linear equations.
The method is called low-storage because only the previous stage has to be stored to calculate
the next one. The stability region of the standard fourth-order low-storage Runge-Kutta is the
same as that of the classical fourth-order Runge-Kutta method.
25
NLR-TR-2012-349
4.3 Spatial schemes and artificial diffusionThe second part of the discretized equations (46) that needs to be discussed is the spatial scheme.
Consider a one-dimensional grid which is uniformly divided into grid cells of length ∆x. On this
grid a function f(x) is defined. A finite-difference approximation of the first derivative of the
function f(x) at point x is given as
∂f
∂x(x) ≈ 1
∆x
M∑j=−N
ajf(x + j∆x) (51)
where the coefficients aj determine the kind of approximation and the order of accuracy. Exam-
ples of finite difference approximations are the forward difference approximation of order 1∂f
∂x(x) ≈ 1
∆x(f(x + ∆x)− f(x))
and central difference approximation of order 2
∂f
∂x(x) ≈ 1
∆x
(12f(x + ∆x)− 1
2f(x−∆x)
).
The coefficients of these and other examples and their order of accuracy are found in table 1.
Accuracy a−3 a−2 a−1 a0 a1 a2 a3
1 -1 1
2 -12 0 1
2
4 112 -2
3 0 23 - 1
12
6 - 160
320 -3
4 0 34 - 3
20160
Table 1 The coefficients of some finite difference approximations for a first-order derivative
and their order of accuracy
The finite-difference approximation that is used by NLR is the central 7-points dispersion-relation-
preserving (DRP) scheme of Tam and Webb (Refs. 15, 7 and further in this section). The coeffi-
cients for this approximation are
a0 = 0
a1 = −a−1 = 0.770882380518
a2 = −a−2 = −0.166705904415
a3 = −a−3 = 0.0208431427703.
The coefficients are chosen in such a way that the approximation is fourth-order accurate and low
dispersive. The low dispersiveness of the method means that the effective numerical wave num-
ber approximates the real wave number of the solution for waves longer than 8 mesh spacings
(Ref. 13).
26
NLR-TR-2012-349
∆x
∆ y
(i+1/2,j−1/2)(i−1/2,j−1/2)
(i−1/2,j+1/2) (i+1/2,j+1/2)
(i,j)
x
y
n
nn
n
Fig. 3 2D-grid cell Ω∆i,j
Recall the integral form of the linearized Euler equations (45) for a grid cell Ωi,j of the domain
D ∈ R2∫Ωi,j
∂U
∂tdV +
∫δΩi,j
F(U) · n dA = 0
Consider a rectangular grid cell Ω∆i,j with cell center at (i, j), length ∆x in the x-direction and
∆y in the y-direction. Other grid cells can be used such as curvilinear grid cells (Ref. 7), but that
will not be discussed in this report. The integral over the boundary of the grid cell Ω∆i,j can be
written as∫δΩ∆
i,j
F(U) · n dA =∫
rightF(U) · n dA +
∫leftF(U) · n dA
+∫
upperF(U) · n dA +
∫lower
F(U) · n dA =
= F1(U∆i+1/2,j) ·∆y −F1(U∆
i−1/2,j) ·∆y
+ F2(U∆i,j+1/2) ·∆x−F2(U∆
i,j−1/2) ·∆x
= F∆i+1/2,j − F∆
i−1/2,j + F∆i,j+1/2 − F∆
i,j−1/2
= B∆i,j (52)
where F1 denotes the first column of F and F2 denotes the second column of F from equation
(44). B∆i,j is the flux balance over the grid cell Ω∆
i,j .
A flux balance of a bigger grid cell Ω3∆i,j which is three times the size of Ω∆
i,j , rectangular with
cell centre (i, j) and size (3∆x) by (3∆y), can also be made. For this grid cell, the flux balance
B3∆i,j is written as
B3∆i,j = F 3∆
i+3/2,j − F 3∆i−3/2,j + F 3∆
i,j+3/2 − F 3∆i,j−3/2. (53)
27
NLR-TR-2012-349
F is a flux on a cell face, but depends on the state vector U which only has values in the cell cen-
tres. Therefore the fluxes on the cell faces have to be approximated by fluxes in the cell centres.
This can be done in various ways such as averaging cell centres
F∆i+1/2,j =
Fi,j + Fi+1,j
2
F 3∆i+3/2,j =
Fi,j + Fi+3,j
2
or by an optimized approximation
F∆i+1/2,j =
1− α
2(Fi,j + Fi+1,j) +
α
2(Fi−1,j + Fi+2,j)
F 3∆i+3/2,j =
1− β
2(Fi,j + Fi+3,j) +
β
2(Fi+1,j + Fi+2,j) (54)
with
α = −0.22227453922
β = −9α.
Here α and β were chosen in such a way that the final scheme for the flux balance Bi,j gives the
earlier presented DRP scheme. This will now be shown.
NLR uses the combination of the ∆-flux balance B∆i,j and the 3∆-flux balance B3∆
i,j which re-
sults in the flux balance Bi,j that is used in the integral form of the linearized Euler equations
(45). The flux balance is defined as
Bi,j =172
(81B∆i,j −B3∆
i,j ) (55)
= Fi+1/2,j − Fi−1/2,j + Fi,j+1/2 − Fi,j−1/2, (56)
with the fluxes defined as
Fi+1/2,j =98F∆
i+1/2,j −172
(F 3∆
i−1/2,j + F 3∆i+1/2,j + F 3∆
i+3/2,j
). (57)
When substituting the flux balances for the grid cells Ω∆i,j and Ω3∆
i,j , respectively equation (52)
and (53), into the previous equation for the flux balance (55) and using the approximation for the
fluxes on the cell faces (54) the flux balance results in the following scheme
Bi,j = a1(Fi+1,j − Fi−1,j) + a2(Fi+2,j − Fi−2,j) + a3(Fi+3,j − Fi−3,j)
+ a1(Fi,j+1 − Fi,j−1) + a2(Fi,j+2 − Fi,j−2) + a3(Fi,j+3 − Fi,j−3). (58)
28
NLR-TR-2012-349
with
a1 =9− 15α
16= 0.770882380518
a2 =3α
4= −0.166705904415
a3 = −1 + 9α
48= 0.0208431427703
the coefficients of the DRP scheme.
4.3.1 Artificial diffusionThe DRP spatial scheme cannot handle acoustic waves shorter than five grid-cells and they need
to be eliminated from the numerical solution. This can be done by damping the short waves by
an artificial diffusion term Bai,j and adding this to the discretized linearized Euler equations (46):
Vi,j
(∂U
∂t
)i,j
+ Bi,j −Bai,j = 0.
The artificial diffusion term Bai,j can be written in the same way as the flux balance in the previ-
ous section (55)
Bai,j = F a
i+1/2,j − F ai−1/2,j + F a
i,j+1/2 − F ai,j−1/2. (59)
To preserve the fourth-order accuracy of the discretized linearized Euler equations with artificial
diffusion, the artificial diffusion needs to be represented by a sixth-order derivative. For this to
hold the fluxes F a are described by a fifth-order difference scheme of the flow variables by
F ai+1/2,j =
k(6)
256(|u0|+ c0)i+1/2,j∆y(Ui+3,j − 5Ui+2,j + 10Ui+1,j − 10Ui,j
+ 5Ui−1,j − Ui−2,j)
F ai,j+1/2 =
k(6)
256(|v0|+ c0)i,j+1/2∆x(Ui,j+3 − 5Ui,j+2 + 10Ui,j+1 − 10Ui,j
+ 5Ui,j−1 − Ui,j−2).
Typical values for the artificial diffusion coefficient k(6) are 0,12 and 2.
4.4 Discretized boundary conditionsSo far only inner grid cells Ωi,j of a computational domain D ∈ R were discussed. But because 7
grid cells in both x and y-direction are needed to apply the DRP scheme to these inner grid cells,
grid cells outside the computational domain are needed. These extra cells are called dummy
cells. The value in the dummy cells depends on the type of boundary condition near the dummy
cells. If no boundary condition is defined, the dummy cells will be filled by regular fourth-order
29
NLR-TR-2012-349
extrapolation. When the labelling of the inner grid cells starts from 1, the dummy cells take the
labels of −2, −1 and 0 and the values of the dummy cells are assigned as
f0,j = 4f1,j −6f2,j + 4f3,j −f4,j
f−1,j = 4f0,j −6f1,j + 4f2,j −f3,j
f−2,j = 4f−1,j−6f0,j + 4f1,j−f2,j . (60)
In the case that there is a boundary condition which assigns a value fw to the boundary, the so-
called Dirichlet boundary condition, a different extrapolation to the first dummy cell is needed
which is given as
f0,j =165
fw − 3f1,j + f2,j −15f3,j . (61)
The other two dummy cells are again extrapolated by the regular fourth-order extrapolation. This
extrapolation can also be used to calculate the value on the boundary fw when the inner grid cell
values are known. The first dummy cell f0,j can be calculated from the regular fourth-order ex-
trapolation and is used in the previous equation to calculate the value on the boundary such that
fw =3516
f1,j −3516
f2,j +2116
f3,j −516
f4,j . (62)
4.4.1 Solid wallAt a solid wall a slip boundary condition is applied as discussed earlier in section 2.2.1. This
means that the normal momentum is equal to zero at the boundary (ρu)′n = 0 and the Dirichlet
boundary condition is applied to the normal momentum. The other flow variables, ρ′, p′, (ρE)′
and the components of (ρu)′ not in the normal direction, are set in the dummy cells by regular
fourth-order extrapolation.
Because the normal velocity at a solid wall is equal to zero, also the convective flux through the
wall needs to be equal to zero. This is not satisfied by just setting the normal momentum to zero.
Say there is a wall at (12 , j). Then the convective flux through the wall is F1/2,j and should be
equal to zero, i.e. equation (57) should be equal to zero:
F1/2,j =98F∆
1/2,j −172
(F 3∆−1/2,j + F 3∆
1/2,j + F 3∆3/2,j
)= 0.
The convective flux is equal to zero if the fluxes of ∆ and 3∆ on the boundary are set to zero and
the flux at the face between the first 2 dummy cells, F 3∆−1/2,j , is set to the negative value of F 3∆
3/2,j :
F 3∆−1/2,j = −F 3∆
3/2,j .
30
NLR-TR-2012-349
4.4.2 Symmetry boundaryThere is another way of implementing a solid wall when the grid only consists of rectangular
grid cells. This way uses symmetric and anti-symmetric ways of filling the variables of the dummy
cells and it is therefore called a symmetry boundary. As in the previous section, the normal mo-
mentum is equal to zero at the symmetry boundary at (12 , j). The normal momentum on the sym-
metry boundary can be calculated in the same way as for the flux at a boundary with equation
(57). From this the normal momentum in the dummy cells need to be defined anti-symmetric as
(ρu)′n(0, j) = −(ρu)′n(1, j)
(ρu)′n(−1, j) = −(ρu)′n(2, j)
(ρu′n(−2, j) = −(ρu)′n(3, j).
For the other variables than the normal velocity, the normal derivative needs to be equal to zero
at a symmetry boundary. The normal derivatives are calculated with the DRP scheme given in
section 4.3:
∂U
∂x(12, j) = a3(U1,j − U0,j) + a2(U2,j − U−1,j) + a1(U3,j − U−2,j),
with the coefficients that can also be found in section 4.3. From this it is easily seen that the
derivative is zero at the boundary when the variables are set symmetrically around the boundary.
4.4.3 Impedance boundary with recursive convolutionIn section 3.7 the pole approximation (42) was chosen to describe the pressure on an impedance
wall. There are two ways to implement the pole approximation into the computational problem.
One of them is based on a recursive convolution proposed by Reymen et al. (Ref. 11), which
is discussed in this section. The other is based on writing the impedance boundary as a system
ODEs and is discussed in the next section. Both ways assume that the velocity on the impedance
boundary is known due to extrapolation, and from that the pressure on the impedance boundary
p′(t) can be calculated.
In the time-domain the relation between the pressure and the velocity on the impedance bound-
ary can be written as a convolution integral (8)
p′(t) =∫ ∞
−∞z(τ) u′n(t− τ)dτ
which is in discrete form
p′(n∆t) =∫ ∞
−∞z(τ) u′n(n∆t− τ)dτ =
∞∑m=−∞
∫ (m+1)∆t
m∆tz(τ) u′n(n∆t− τ)dτ.
31
NLR-TR-2012-349
Reymen et al. make an assumption on the velocity u′n to approximate the integral. This can be
done by assuming that the velocity is piecewise constant or piecewise linear within a time step
∆t. For the next step the velocity is assumed to be piecewise constant within a time step such
that
u′n(t) = u′n(m∆t) for t ∈ [m∆t, (m + 1)∆t].
The pressure can be split into S individual and independent parts p′k(t) corresponding to the S
poles of the impedance. This is done as follows. Remembering that p′(t) is written as the convo-
lution product (8) and using that the impedance z(τ) is the sum of S parts zk(τ) (39), the pres-
sure can be written as
p′(t) =∫ ∞
−∞z(τ) u′n(t− τ)dτ =
∫ ∞
−∞
S∑k=1
zk(τ) u′n(t− τ)dτ
=S∑
k=1
∫ ∞
−∞zk(τ) u′n(t− τ)dτ =
S∑k=1
p′k(t) (63)
with the individual parts of the pressure
p′k(t) =∫ ∞
−∞zk(τ) u′n(t− τ)dτ =
∫ ∞
−∞Ake
−λkτH(τ) u′n(t− τ)dτ
= Ak
∫ ∞
0e−λkτ u′n(t− τ)dτ. (64)
So far it was just a matter of rewriting the equation for the pressure. Reymen et al. writes the
pressure as
p′k(t) = Akφk(t)
with
φk(t) =∫ ∞
0e−λkτ u′n(t− τ)dτ
a so-called accumulator. When it is assumed that the velocity is piecewise constant as defined
above, Reymen et al. suggests that the accumulator φk(t) can be approximated by a discrete re-
cursive relation
φk(t) = φmk = (u′n)m 1− e−λk∆t
λk+ φm−1
k eλk∆t.
Implementing this recursive relation in the standard fourth-order low-storage Runge-Kutta method
(49), the accumulator is rewritten for each stage i as
φik = (u′n)i 1− e−λk(γi−γi−1)∆t
λk+ φi−1
k eλk(γi−γi−1)∆t,
32
NLR-TR-2012-349
where γi are the coefficients of the standard fourth-order low-storage Runge-Kutta (50). (u′n)i
is the normal velocity at the impedance boundary at stage i, which is assumed to be known by
extrapolating the normal velocity in the grid cells to the boundary.
The problem which arises from this is how to pick the initial accumulator at time zero: φk(0).
Picking the initial accumulator can be done from the analytical solution of the kth part of the
pressure at the impedance boundary p′k(t) at time zero, which depends on the analytic solution
of the normal velocity u′n(t) at the impedance boundary through the convolution product. As-
suming the normal velocity at the impedance boundary is known, p′k(0) is known and so is the
accumulator:
φk(0) =1
Akp′k(0).
When no analytic solution is available, the velocity is assumed zero at t ≤ 0, so that φk(0) = 0.
4.4.4 Impedance boundary with a system of ODEsA new way of discretizing the pole approximation is by a system of ODEs. As in the previous
section the pressure can be split into S individual and independent parts (63)
p′(t) =S∑
k=1
p′k(t),
but now the parts will not be approximated by recursive convolution but by solving a discretized
ODE. Rewriting the individual parts of the pressure (64) as the other way of writing the convolu-
tion product (33), the expression for the pressure becomes
p′k(t) =∫ ∞
−∞zk(t− τ) u′n(τ)dτ
=∫ ∞
−∞Ake
−λk(t−τ)H(t− τ) u′n(τ)dτ
= Ak e−λkt
∫ t
−∞u′n(τ) eλkτ dτ.
Taking the derivative of this gives
dp′kdt
(t) = −λk Ak e−λkt
∫ t
−∞u′n(τ)eλkτ dτ + Ak e−λkt eλkt u′n(t)
= −λk p′k(t) + Ak u′n(t) (65)
which is an ODE for p′k. The velocity on the boundary u′n(t) is again assumed to be known from
extrapolation. The individual S parts can now easily be calculated via the standard fourth-order
low-storage Runge-Kutta method (49) and then be added to get the pressure p′(t) on the bound-
ary.
33
NLR-TR-2012-349
4.5 One-dimensional test caseThe next step is to find out which of the discretizations of the impedance boundary works better:
the recursive convolution or the ODE discretization. To decide this, calculations are performed
on a one-dimensional domain [0, X] which is split into N grid-cells of size ∆x. The domain has
a symmetry boundary on the left side x = 0 and an impedance boundary on the right side x =
X . The final time of the simulation is T with time steps ∆t. The discretized one-dimensional
linearized Euler equations in conservative form for grid cell i are
Vi
(∂U
∂t
)i
+ Bi = 0 (66)
where the flow state vector U is defined as
U =
ρ′
(ρux)′
(ρE)′
and
Vi = ∆x. (67)
The time integration(
∂U∂t
)i
and spatial integration Bi are done by respectively the standard fourth-
order low-storage Runge-Kutta method from section 4.2 and the fourth-order DRP scheme from
section 4.3. Artificial diffusion was used to make the simulation stable. The artificial diffusion
coefficient of k(6) = 2 was used with the scheme given in section 4.3.1. Two programmes were
made in Matlab, one for the impedance boundary with recursive convolution (section 4.4.2) and
one for the impedance boundary with ODE implementation (section 4.4.3). The ODE implemen-
tation also uses the standard fourth-order low-storage Runge-Kutta method to calculate pk.
The symmetry boundary is simulated as in section 4.4.2. The impedance boundary simulated is
grass with flow resistivity σ = 100 kPa sm2 , where the impedance was calculated with the Miki
model (26). Cotte et al. (Ref. 2) fitted the coefficients Ak and λk of the pole approximation (42)
to the Miki model and got
A1 = 1.414390450609 · 106 kgm2s2
λ1 = 5.233002301836 · 101 1s
A2 = 1.001354674975 · 106 kgm2s2
λ2 = 4.946064975401 · 102 1s
A3 = −3.336020206713 · 106 kgm2s2
λ3 = 1.702517657290 · 103 1s
A4 = 5.254549668250 · 106 kgm2s2
λ4 = 1.832727486745 · 103 1s
A5 = 3.031704943714 · 107 kgm2s2
λ5 = 3.400000000000 · 104 1s.
34
NLR-TR-2012-349
(a) Real part of the relative impedance (b) Imaginary part of the relative impedance
Fig. 4 Relative impedance of the Miki model and its pole approximation for grass (σ = 100 kPa sm2 )
In figure 4 the real and imaginary part of the relative impedance ( Zρ0c0
) of the Miki model with
σ = 100 kPa sm2 and the relative impedance of its pole approximation are shown. ρ0 and c0 and the
other mean flow variables are taken as
u0 = 0m
sc0 = 340
m
s
ρ0 = 1.225kgm3
p0 =ρ0 c2
0
γ= 101150 Pa
E0 =p0
ρ0(γ − 1)= 206429
m2
s2H0 = E0 +
p0
ρ0= 289000
m2
s2,
with γ = 1.4. The length of the domain is chosen as X = 1m.
The calculations are all done in a dimensionless way. This means that some reference variables
have to be set so that the mean flow variables and the perturbations of the mean flow can be made
dimensionless. The reference values are chosen as
ρref = ρ0 pref = p0 Lref = X
uref =√
pref
ρref=
c0√γ
tref =Lref
uref=
X√
γ
c0.
The mean flow variables and the perturbations of the mean flow are both made dimensionless
by the same reference values. The velocities of the variables are made dimensionless by uref ,
the density by ρref , the pressure by pref and the energy and total enthalpy by pref
ρref. The final
simulation time T and time step ∆t are made dimensionless by tref and the position x and grid
size ∆x by Lref . Also the impedance coefficients have to be made dimensionless. The Ak are
made dimensionless by pref
Lrefand the λk by 1
tref. For the rest of this section all the variables will
be dimensionless.
35
NLR-TR-2012-349
The initial value for the calculations is taken from the dimensionless analytical solution of the
linearized Euler equations with an impedance and solid boundary by setting t = 0. The analyti-
cal solution is given as
p′(x, t) = A(eiωtcos(kx) + e−iω∗tcos(k∗x)
)ρ′(x, t) =
1c20
p′(x, t)
(ρu)′(x, t) = −iA(
k
ωeiωtsin(kx)− k∗
ω∗e−iω∗tsin(k∗x)
)(ρE)′(x, t) =
H0
c20
p′(x, t). (68)
The superscript ∗ denotes the complex conjugate, A is equal to 1 and the mean flow variables
are taken as the dimensionless quantities mentioned before. The wave number k and frequency
ω are solutions of the dispersion relation and relation between the frequency, wave number and
impedance, which are given as
k2 =ω2
c20
(69)
k tan(kX) =iρ0ω
Z(ω). (70)
For this particular case the eigenmode
k = 12.36888 + i0.19252
ω = 14.63505 + i0.22779.
was used. The reason for picking this eigenmode is because the real part of the frequency is
about f = 669 Hz, which is a fairly common audible frequency. In figure 5 more eigenmodes
(a) Wave number k (b) Frequency ω
Fig. 5 Eigenmodes
36
NLR-TR-2012-349
can be found. The eigenmodes were calculated using Newtons method to solve the dispersion
relation (69) and the relation between the frequency, wave number and impedance (70).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5Pressure
x
p/pr
ef
Fig. 6 Initial value of the pressure
The initial value of the pressure on the domain is plotted in figure 6 for ∆x = 1/50.
4.5.1 AccuracyFor the decoupled system (the linearized Euler equations without boundary conditions) the com-
bination of the standard fourth-order low-storage Runge-Kutta method and the DRP spatial scheme
gives a fourth-order accurate result. For several boundaries, such as a solid wall or symmetry
boundary, it has been investigated that the entire system is indeed fourth order. Now the accuracy
of the one-dimensional problem stated in the previous section is investigated. This will be done
by comparing the numerical solution with the analytical solution given in equation (68).
At a final time of T = 14.36762, the root-mean-square (rms) of the difference between the nu-
merical and analytical solution at time T is calculated which is defined as
rms =
(1N
N∑i=1
(fi − fi)2) 1
2
. (71)
The root-mean-square of the difference is of orders ∆ta and ∆xb
rms = O(∆ta) +O(∆xb). (72)
When taking the mesh size ∆x linearly dependent of ∆t and assuming that b ≥ a, the rms is
now of order ∆ta. To see what the order of accuracy a is, the natural logarithm of the root-mean-
square has to be taken:
rms = O(∆ta) = c∆ta + higher order terms ≈ c∆ta
⇒ ln(rms) ≈ ln(c∆ta) = ln(c) + a ln(∆t).
37
NLR-TR-2012-349
−7 −6.5 −6 −5.5 −5 −4.5 −4−18
−16
−14
−12
−10
−8
−6
−4
ln(dt)
ln(r
ms)
Order of accuracy
fourth order lineoderecursive constantrecursive linear
Fig. 7 Fourth-order accuracy of LEE with symmetry and impedance boundary
When plotting the ln of the time step ∆t to the ln of the rms and measuring the slope of the plot,
a is acquired. In table 4.5.1 the rms of the difference between the numerical and analytical solu-
tion at time T for different time steps ∆t is given. Here the ODE discretization was used for the
impedance boundary at x = X .
In figure 7 the plot of the natural logarithm of the different time steps is plotted against the natu-
ral logarithm of the rms for the ODE discretization The same is plotted for the recursive convolu-
tion with a constant approximation of the velocity and for the recursive convolution with a linear
approximation of the velocity. Also the line for a fourth-order accuracy was plotted.
It can be seen that the ODE discretization is fourth-order accurate. The implementation with con-
stant approximation of the velocity for the recursive convolution is not even near fourth-order
accurate, but seems of first order. The approximation of the velocity is also first order, so that
seems to work through into the entire system. The discetization with linear approximation of
∆t ∆x rms of Ui − Ui
0.011494 0.04 8.433291 · 10−3
0.0057470 0.02 2.42378 · 10−4
0.0028735 0.01 1.19729 · 10−5
0.0014368 0.005 4.39244 · 10−7
Table 2 The rms of the difference between the numerical and analytical solution at time T for
different time steps ∆t
38
NLR-TR-2012-349
Fig. 8 Stability region of the standard fourth-order low-storage Runge-Kutta method with eigen-
values of periodic boundaries times time step ∆t = 1/50
the velocity for the recursive convolution is of higher order, but not fourth order. For small time
steps, figure 7 shows that the discretization is about second order, which is also the order of the
approximation of the velocity. From figure 7 it can also be seen that the ODE discretization gives
the best accuracy for all time steps used and therefore is preferred as discretization method.
4.5.2 StabilityIn this section the numerical stability of the one-dimensional computational problem described
before is discussed and in particular what an impedance boundary condition does for the numer-
ical stability. One way to investigate the numerical stability of a system of ODE’s is by looking
at the eigenvalues of the system matrix A. The system of discretized linearized Euler equations
with boundary conditions can be written as
ddt
U = AU. (73)
Here the standard fourth-order low-storage Runge-Kutta method is used for the time-integration
of the system which is stable when the eigenvalues µ of the system matrix A, multiplied with
the time step ∆t, lie in the stability region of figure 8. In the figure the eigenvalues of the system
matrix with periodic boundary, ∆x = 1/50 and no artificial diffusion are plotted with ∆t =
1/50. Note that now CFL = 1.
The numerical eigenvalues of other systems can be plotted as well. For different boundary condi-
tions the eigenvalues of the system matrix are plotted in figure 9. Artificial diffusion was added
here and a mesh size ∆x = 1/50 was used.
39
NLR-TR-2012-349
(a) Double solid boundary and solid and impedance
boundary
(b) Double symmetry boundary and symmetry and
impedance boundary
(c) Double outflow boundary and outflow and
impedance boundary
Fig. 9 Eigenvalues of system matrices with different boundary conditions
It can easily be seen that the impedance boundary creates some extreme eigenvalues and that it
does not matter for the value of these extreme eigenvalues which boundary condition is used on
the other boundary. In figure 10 the eigenvalues of a system matrix with symmetry and impedance
boundary are scaled by ∆t = 1/100 (CFL = 1/2) so that they fit into the stability region of the
standard fourth-order low-storage Runge-Kutta method of figure 8.
It is also nice to check how the coefficients of the impedance boundary affect the extreme eigen-
values that arise. Because the last coefficients A5 and λ5 are also by far the largest, these will
have the most effect on the extreme eigenvalues as shown in figure 11. The effect of the next
largest coefficients A4 and λ4 can be seen in figure 11 when the coefficients are set to zero. In
the figure the effect of the largest coefficients is shown in the eigenvalues of the system matrix
with symmetry and impedance boundary. For λ5 the effect is quite clear. When dividing the co-
40
NLR-TR-2012-349
Fig. 10 Eigenvalues of system matrix with symmetry and impedance boundary times ∆t =
1/100.
efficient by two, the real part of the extreme eigenvalues will also become twice as small. The
imaginary part however does not differ much. So the coefficient λk has a big effect on the real
part of the extreme eigenvalues. When changing A5 then the extreme eigenvalues also change.
This can also be viewed in figure 11.
The eigenvalues of the system matrix shown before were derived numerically. However, for
small values of the mesh space ∆x, the numerical eigenvalues should approximate the analyti-
cal eigenvalues. The analytical eigenvalues are found as follows. The linearized Euler equations
including a symmetry boundary and impedance boundary are again written as a system equation
(73). From the eigenvalue theory AU can be written as AU = λU , where λ is an eigenvalue with
Fig. 11 Eigenvalues of system matrix with A5 and λ5 changed
41
NLR-TR-2012-349
corresponding eigenvector U . An eigenmode of the linearized Euler equation is
U = U0 eikx+iωt, (74)
with k and ω satisfying the dispersion relation (69) and relation between the frequency, wave
number and impedance (70). When assuming this solution is an eigenvector and substituting this
into the system equation, we get
∂
∂tU = AU = λU
→ iω U0 eikx+iωt = λ U0 eikx+iωt
→ λ = iω.
This means that for very small values of ∆x the eigenvalues should resemble iω, where ω is the
frequency of an eigenmode.
Fig. 12 Eigenvalues vs iω
In figure 12 numerical eigenvalues of the system matrix were plotted for ∆x = 1/400 as blue
circles. Also iω, with ω that correspond to a wave number k smaller than 50π, are plotted as
green circles. The green circles lie on top of the blue circles and therefore the values match.
From the figure it can easily be seen that the analytical eigenvalues correspond to some of the
numerical eigenvalues. The extreme numerical eigenvalues that can be found in multiple figures
in this section do not seem to be analytical. The Newton method used to calculate the analytical
eigenmodes does not give values close to the extreme eigenvalues.
In the last part of the stability section the need for artificial diffusion is discussed. So far most
of the figures use artificial diffusion with a diffusion coefficient k(6) = 2, except for figure 8
42
NLR-TR-2012-349
which uses no artificial diffusion. In figure 13 the eigenvalues of a system with symmetric and
impedance boundary are shown. For the scaled eigenvalues to lie in the stability region of figure
8 it is convenient that the real part of the unscaled eigenvalues are at least negative or equal to
zero. In figure 13 it can be seen that when no artificial diffusion, or not enough artificial diffusion
is used, not all the eigenvalues have real parts less or equal to zero. This is better seen in the right
figure of figure 13, which is zoomed in. Artificial diffusion is therefore neccesary for the system
to be numerically stable.
Fig. 13 Influence of artificial diffusion on the system with symmetry and impedance boundary
conditions
5 Two-dimensional problems
As shown in the previous chapter the one-dimensional method to implement the impedance
boundary condition seems accurate enough and stable for small enough time steps. Now the
method is extended to higher dimensions and implemented in ENSOLV, NLR’s computational
method. ENSOLV calculates in dimensionless variables. The flow variables are again taken and
made dimensional in the same way as in section 4.5. Now the reference values are taken as
ρref = ρ0 pref = p0 Lref = 1m
uref =√
pref
ρref=
c0√γ
tref =Lref
uref=√
γ
c0.
All variables are assumed dimensionless in this chapter, unless stated otherwise. More about
ENSOLV can be read in the appendix.
In the next two sections two-dimensional calculations are done in ENSOLV where the lower
boundary is an impedance boundary of either grass with σ = 100 kPa sm2 , or of a snow covered
43
NLR-TR-2012-349
ground with σ = 10 kPa sm2 . The coefficients for the pole approximation used for these impedance
boundaries were given by Cotte et al. (Refs. 2,4). The grass ground uses five poles, while the
snow uses six poles and these are given respectively for grass and snow as:
A1 = 1.414390450609 · 106 kgm2s2
λ1 = 5.233002301836 · 101 1s
A2 = 1.001354674975 · 106 kgm2s2
λ2 = 4.946064975401 · 102 1s
A3 = −3.336020206713 · 106 kgm2s2
λ3 = 1.702517657290 · 103 1s
A4 = 5.254549668250 · 106 kgm2s2
λ4 = 1.832727486745 · 103 1s
A5 = 3.031704943714 · 107 kgm2s2
λ5 = 3.400000000000 · 104 1s.
and
A1 = −2.984324740698 · 106 kgm2s2
λ1 = 1.854449527167 · 100 1s
A2 = 4.036520521121 · 106 kgm2s2
λ2 = 2.211765605322 · 100 1s
A3 = 7.883025344426 · 105 kgm2s2
λ3 = 2.765078273277 · 103 1s
A4 = −1.075888617178 · 107 kgm2s2
λ4 = 6.483391113779 · 103 1s
A5 = 2.190201745924 · 107 kgm2s2
λ5 = 1.294856146826 · 104 1s.
A6 = 1.171170292401 · 106 kgm2s2
λ6 = 1.700000000000 · 104 1s.
In figure 14 the real and imaginary parts of the relative impedance ( Zρ0c0
) are shown for these
(a) Real part of the relative impedance (b) Imaginary part of the relative impedance
Fig. 14 Relative impedance of pole approximation for grass and snow
44
NLR-TR-2012-349
Fig. 15 Domain
coefficients. For ENSOLV, the coefficients Ak and λk are made dimensionless by respectivelypref
Lrefand 1
tref.
Two different types of simulations have been performed. The first simulations are done with an
acoustic wave of a single frequency, while the second set of simulations are done with a Gaus-
sian pulse as initial condition. In both types of simulations, artificial diffusion of k(6) = 2 is
used.
5.1 Single frequencyIn this section a long two-dimensional grid with a range in the x-direction of [−50, 150] and a
range in the y-direction of [0, 5], is used. The mesh sizes are both equal to ∆x = ∆y = 0.1. A
source term with a single dimensionfull frequency of f = 500 Hz is situated at x0 = (0, 2). The
source term is added to the right-hand side of equation (4) and is given by
RHS = Aω
c20
sin(ωt)e−‖x−x0‖2/d2
1
u0
v0
H0
, (75)
with A the pressure amplitude at x0 and the length d chosen as d2 = 2(∆x)2/ ln 2. The di-
mensionless angular frequency is given as ω = 10.933 and the mean flow velocity is equal to
u0 = v0 = 0. When 50 time steps per period are taken, the time step is ∆t = 0.0115. At y = 0
an impedance boundary, solid boundary or far-field boundary is applied, while the other bound-
aries are all far-field boundaries. At the far-field boundaries a buffer zone of length 1 is applied.
In figure 16 the pressure at height y = 2.05 and after 320 periods is shown, which is equivalent
to t = 95.2. In figure 17 a contour plot of the pressure over a small part of the domain is plotted
at the same time. This time is chosen because then the wave has travelled more than the distance
45
NLR-TR-2012-349
(a) Pressure at x ∈ (−50, 150) (b) Pressure at x ∈ (−20, 20)
(c) Pressure at x ∈ (100, 110)
Fig. 16 Pressure near source height (y = 2.05) at t = 320 periods
of the grid and the pressure has become periodic in time. From subfigure (b) it can be seen that
both impedance boundaries reflect a little different than the other and the solid boundary. Apart
from the obvious difference in absorption, this is presumably caused by a phase shift, which
can be viewed in subfigure (c) of figure 16. The phase shift occurs because the reflection at an
impedance boundary is different than a reflection at a solid boundary, which is in full anti-phase.
The wave length does not seem to change and stays the same for all boundaries. It is strange that
the reflecting wave of an impedance boundary for snow seems to be stronger than a reflecting
wave of an impedance boundary for grass. But this does not necessarily mean that the sound that
is heard (sound pressure level) is louder as well.
Now the sound pressure level (SPL) will be calculated for different boundaries at y = 0. The
46
NLR-TR-2012-349
(a) Pressure with impedance boundary for grass (b) Pressure with impedance boundary for snow
Fig. 17 Pressure at x ∈ (100, 110) after t = 320 periods
sound pressure level is defined as
SPL = 20 log10
(prms
pref
), (76)
where prms is the root mean square value of the pressure perturbations over one periode and pref
is the reference pressure. The reference pressure is defined as pref = 2 · 10−5 Pa. The sound
pressure level of the tone at x0 = (0, 2) is chosen to be SPL = 130 dB, so that the amplitude of
the tone is equal to A = pref · 10130/20.
In figure 18 the sound pressure level is plotted for different values of y. At the boundary it can be
seen that the impedance boundary for snow has (almost everywhere) the lowest sound pressure
level. This is in line with what was expected, because snow should damp the most sound. The
outflow boundary was left out of this picture because the buffer zone starts at y = 1 which damps
the sound wave.
Overall the sound pressure level in figure 18 is not as smooth as an analytical one would be. The
reason for this lies in how the root mean square value is taken. In this case, the variance of the
pressure perturbations over one period of time is used as the root mean square value. However,
this is only valid when the mean of the pressure perturbations over one period of time is equal to
zero. This is not the case here.
In figure 19 the relative sound pressure level, ∆SPL, is given at height y = 2.00. The relative
sound pressure level is the sound pressure level of the reflecting wave and is defined as
∆SPL = 20 log10
(pimp
pfree
), (77)
47
NLR-TR-2012-349
(a) SPL at boundary (y = 0) (b) SPL at y = 2.00
(c) SPL at y = 3.00 (d) SPL at y = 5.00
Fig. 18 Sound pressure level at several y
where pimp is the root mean square of the pressure over one period when an impedance boundary
is set at y = 0 and pfree is the root mean square of the pressure for an far-field boundary. Both
impedance boundaries have a relative sound pressure level that is mostly above zero, but some-
times falls below zero.
5.2 Gaussian pulseFor the next simulation instead of a source with a constant frequency, a Gaussian pulse is used as
an initial condition at x0 = (0, 2). The Gaussian pulse is made up of a band of frequencies. This
is where the advantage of the pole approximation is seen. Because the pole approximation is a
broadband approximation, whereas the MSD+ approximation is a single frequency approxima-
48
NLR-TR-2012-349
(a) Impedance boundary with grass (b) Impedance boundary with snow
Fig. 19 Sound pressure level of two different impedance boundaries at y = 2.00
tion used as a broadband model. The pulse is defined asρ′
(ρux)′
(ρuy)′
(ρE)′
(x, y) =
1
0
0
1
A · 12
min(x2+(y−2)2
2,40)
.
In this section the influence of an impedance boundary on the pressure is shown and compared to
the influence of a slip (solid) boundary. Again a long two-dimensional grid is used of dimensions
[-50,150]x[0,5] and mesh space ∆x = ∆y = 0.1 to simulate the propagation of the pulse in
time. The boundary at y = 0 is an far-field, solid or impedance boundary, while the other bound-
Fig. 20 Pressure near boundary (y = 0.05) at t = 0.4226
49
NLR-TR-2012-349
aries are again far-field boundaries. In figures 20 and 21 the pressure is shown for the different
types of boundaries at y = 0. In these simulations a time step of ∆t = 0.0084 is used.
Figure 20 shows the pressure after a small period of time (t = 0.4226), when the reflection just
occured. Here the reflected wave can easily be recognized. The blue line is the pressure when
there is an outflow boundary at y = 0, so it is just an outgoing wave. The other lines signify
boundaries where there is a reflected wave, which is added to the outgoing wave. It can easily be
seen that both impedance boundaries have a smaller pressure than the solid wall. This is because
of the damping of the impedance boundary. What can also be seen is that a snow impedance
boundary damps more than an impedance boundary for grass.
In figure 21 the pressure is plotted at several times. The grid ends at x = 150, which is almost
the distance that the wave has travelled at time t = 118.3216 (subfigure (d)). The pressure is
plotted at a height of y = 2.05, which is about the same height as the source. What can already
be viewed in subfigure (b), and becomes more apparent in subfigures (c) and (d), is that the wave
length changed and phase shift occured for a wave that reflects on an impedance boundary of
(a) Pressure at t = 2.5355 (b) Pressure at t = 16.9031
(c) Pressure at t = 67.6123 (d) Pressure at t = 118.3216
Fig. 21 Pressure near source height (y = 2.05) at several times
50
NLR-TR-2012-349
both grass and snow (green and red line).
6 Conclusions and recommendations
In this thesis the modelling of a ground impedance boundary condition in NLR’s CAA method
was investigated. The modelling depends on three layers. First the modelling of the impedance
in the frequency domain, then transforming the frequency to the time domain and finally dis-
cretizing the time-domain model.
The first layer describes the impedance by models in the frequency domain that depend on the
ground type. Several models are to be found in the literature such as the analytically-derived
Zwikker-Kosten model and the experimentally-derived Delany-Bazley model. All the discussed
models do not differ much and the experimentally-derived and physically-valid Miki model was
used for further modelling of the impedance boundary.
All the models discussed are dependent on the frequency and are therefore defined in the fre-
quency domain. The CAA method of NLR calculates in the time domain and so transformations
have to be done. However this is not straightforward. The Miki model can indeed be transformed
to the time-domain, but not without approximating part of the impedance in the time-domain.
(Ref. 2). So the most convenient way to transform the model to the time domain is by first ap-
proximating the model over a frequency range. This can be done by either the MSD+ or the pole
approximation. The MSD+ approximation is basically a single-frequency approximation, but can
also be used to approximate the model over a limited frequency range. The pole approximation
is a broadband approximation. The pole approximation is preferred because it can be fitted to a
larger frequency range.
In the literature a recursive convolution was proposed to discretize the pole approximation. Rey-
men et al. (Ref. 11) show that their approach works well, but uses approximations on the veloc-
ity. However when investigating the pole approximation more closely, the idea for a more natural
discretization arose. Here the impedance is described by a system of ordinary differential equa-
tions and then solved by the fourth-order low-storage Runge-Kutta time-integration method.
The discretization with ODE’s was investigated on accuracy and stability and compared to the
recursive discretization. This was done with a one-dimensional method implemented in Matlab.
The ODE was expected to be fourth-order accurate, because the same time-integration method
was used for the system and the ODE discretization. It is observed that the ODE discretization is
51
NLR-TR-2012-349
indeed fourth-order accurate. The recursive discretization does not reach fourth order with either
constant or linear approximation of the velocity. Thus the ODE discretization is more accurate
than the recursive implementation. There is a restriction for the method to be stable, which is that
the method uses small enough time steps and artificial diffusion.
The ODE discretization was used to implement the impedance boundary condition in NLR’s
computational aeroacoustic method. Once the impedance boundary condition was implemented
two-dimensional simulations were done. The first simulations are done with a source that gener-
ates acoustic waves of constant frequency and the propagation of the acoustic wave was viewed
over a large domain. The second set of simulations uses an initial pulse to view the difference
between a solid wall, an impedance boundary and an far-field boundary. It could be seen that the
impedance boundary does not only damp the reflecting wave, but also causes a change in wave
length and a phase shift.
So far the calculations that were done with the one-dimensional and two-dimensional methods
were done with coefficients derived by Cotte et al. (Refs. 2,4). They only gave two sets of co-
efficients. One for a hard-backed layer with snow of σ = 10 kPa sm2 and the other for grass of
σ = 100 kPa sm2 . The coefficients were derived by Cotte by fitting the pole approximation to the
Miki model for impedance over a frequency band. The same can be done for surfaces with a dif-
ferent flow resistivity. However this was left out of this thesis due to time constraints.
There is another way of implementing the mean flow at an impedance boundary in NLR’s com-
putational method. In the simulation in this report a mean flow of zero was used and the mean
flow in the dummy cells was computed with a slip condition. However when there is a mean
flow not equal to zero, the choice can be made to implement a boundary layer. At a boundary a
no-slip condition is needed to simulate the mean flow at the impedance boundary. This was not
investigated due to time constraint and is for future investigation.
52
NLR-TR-2012-349
References
1. K. Attenborough, K.M. Li and K. Horoshenkov. Predicting Outdoor Sound. 2007.
2. B. Cotte, P. Blanc-Benon, C. Bogey and F. Poisson. Time-domain impedance boundary con-
dition simulations of outdoor sound propagation. AIAA Journal, 47(10): 2391-2403, October
2009.
3. M.E. Delany and E.N. Bazley. Acoustical properties of fibrous absorbent materials. Applied
Acoustics, Vol 3: 105-116, Elsevier, 1970.
4. D. Dragna, B. Cotte, P. Blanc-Benon and F. Poisson. Time-domain simulations of out-
door sound propagation with suitable impedance boundary conditions. In 15th AIAA/CEAS
Aeroacoustics Conference, May 2009, AIAA 2009-3306.
5. B. Gustavsen and A. Semlyen. Rational approximation of frequency domain responses by
vector fitting. IEEE Transactions on power delivery, 14(3): 1052-1061, July 1999.
6. K. Heutschi, M. Horvath and J. Hofmann. Simulation of ground impedance in finite differ-
ence time domain calculations of outdoor sound propagation. Acta Acustica United with
Acustica, Vol 91: 35-40, 2005.
7. A computational aeroacoustic method for aircraft noise propagation. J.C. Kok and R.J. Ni-
jboer. National Aerospace Laboratory, December 2003. NLR-CR-2003-629.
8. Y. Miki. Acoustical properties of porous materials, Modifications of Delany-Bazley models.
Journal of the Acoustical Society of Japan (E), 11(1): 19-24, 1990.
9. D.A. Nield and A. Bejan. Convection in porous media. 2006.
10. K.B. Rasmussen. Sound propagation over grass covered ground. Journal of Sound and Vi-
bration, 78(2): 247-255, 1981.
11. Y. Reymen, M. Baelmans and W. Desmet. Time-domain impedance formulation based on
recursive convolution. In 12th AIAA/CEAS Aeroacoustics Conference, May 2006, AIAA
2006-2685.
12. S.W. Rienstra. Impedance models in time domain including the extended Helmholtz res-
onator model. In 12th AIAA/CEAS Aeroacoustics Conference, May 2006, AIAA 2006-2686.
13. W. Rozema. Modelling of acoustic liners in a time-domain CAA method. National
Aerospace Laboratory, August 2010. NLR-TR-2010-579.
14. E.M. Salomons. Computational atmospheric acoustics. 2001.
15. C.K.W. Tam and J.C. Webb. Dispersion-relation-preserving finite difference schemes for
computational acoustics. In Journal of Computational Physics, Vol 107: 262-281, 1993.
16. D.K. Wilson, S.L. Collier, V.E. Ostashev, D.F. Aldridge, N.P. Symons and D.H. Marlin.
Time-domain modelling of the acoustic impedance of porous surfaces. Acta Acustica United
with Acustica, Vol 92: 965-975, 2006.
53
NLR-TR-2012-349
17. C. Zwikker and C.W. Kosten. Sound absorbing materials. 1949.
54
NLR-TR-2012-349
Appendix A ENSOLV routines
The two-dimensional problems from section 5 were done in a time-domain programme of NLR
called ENSOLV (Ref. 7) which is programmed in Fortran. ENSOLV solves the linearized Euler
equations using the fourth-order accurate and low-dispersive DRP-scheme for spatial integra-
tion and the fourth-order low-storage Runge-Kutta method for time integration. To implement
the impedance boundary condition two subroutines were added to ENSOLV: pol.h to specify the
global parameters needed for the impedance boundary and bcpole.F for the implementation of
the impedance boundary. The two subroutines can be found in the next two sections. Other sub-
routines also had to be adapted to implement the impedance boundary. These subroutines are
dim.h, bcond.F, rdbcd.F and updbcd.F. In the last four sections it is explained and/or showed
how the subroutines differ now that the impedance boundary is included.
A.1 pol.hcommon / p o l / n p o l e s ( nefmax ) ,
. a p o l e ( polmax , nefmax ) , l p o l e ( polmax , nefmax ) ,
. i p o l e ( nfbcm , nefmax , nlmax )
r e a l apo le , l p o l e
i n t e g e r npo le s , i p o l e
C−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−CB COMMON BLOCK NAME: POL ( l o c a l l y r e a c t i n g P o le model p a r a m e t e r s )
C−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−CF COMMON VARIABLES :
CF NAME DESCRIPTION
CF −−−−−−−−−−−− −−−−−−−−−−−CF APOLE C o e f f i c i e n t s f o r p o l e model f o r LEE
CF LPOLE C o e f f i c i e n t s f o r p o l e model f o r LEE
CF NPOLES Number of p o l e s needed f o r p o l e model f o r LEE
CF IPOLE P o i n t e r to BCD f o r p o l e model f o r LEE
C−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
A.2 bcpole.Fs u b r o u t i n e b c p o l e ( f ,m, l , n i , n j , nk , nq , eord , mstag , dt ime ,
. var , p , varp , pp ,
. nx , ny , nz ,
. npo l e s , apo le , l p o l e ,
. lmpm , lmpi )
# i n c l u d e ” imp . h ”
i n t e g e r f ,m, l , n i , n j , nk , nq , eord , mstag
r e a l d t ime
55
NLR-TR-2012-349
r e a l v a r ( 0 : n i + 1 , 0 : n j + 1 , 0 : nk + 1 , 5 ) , p ( 0 : n i + 1 , 0 : n j + 1 , 0 : nk + 1) ,
. va rp ( 0 : n i + 1 , 0 : n j + 1 , 0 : nk + 1 , 5 ) , pp ( 0 : n i + 1 , 0 : n j + 1 , 0 : nk +1)
r e a l nx ( nq ) , ny ( nq ) , nz ( nq )
i n t e g e r n p o l e s
r e a l a p o l e ( n p o l e s ) , l p o l e ( n p o l e s )
r e a l lmpm ( nq∗ n p o l e s ) , lmpi ( nq∗ n p o l e s )
C c a l l in bcond
C
C c a l l b c p o l e ( i e f , i f b , l , n i1 , nj1 , nk1 , nq , eord ,m, dt ime ,
C . v r e f , p r e f , varb , p ,
C . bcd ( i r x ) , bcd ( i r y ) , bcd ( i r z ) , n p o l e s ( i e f ) ,
C . a p o l e ( i e f , n p o l e s ) , l p o l e ( i e f , n p o l e s ) ,
C . bcd ( iqm ) , bcd ( iqm+nq∗ n p o l e s ) )
C
C
C∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗C NATIONAL AEROSPACE LABORATORY (NLR)
C THE NETHERLANDS
C
CA PROGRAM : ENSOLV (3D E u l e r / Navier−S t o k e s m u l t i−block f low s o l v e r )
C−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−C
CB NAME : BCPOLE ( ground impedance p o l e model f o r LEE)
CC REFERENCE : NLR CR 93190 L , S o f t w a r e d e s i g n of ENSOLV
CC NLR CR 93152 L , Numer ica l d e f i n i t i o n o f ENSOLV
CD FUNCTION : S e t t h e dummy−c e l l v a l u e s f o r a f a c e a t a l o c a l l y
CD r e a c t i n g s u r f a c e f o r t h e l i n e a r i z e d E u l e r e q u a t i o n s by
CD i n t e g r a t i o n o f t h e ground impedance p o l e model .
C
CE UPDATE HISTORY :
CE VSN DATE AUTHOR SUMMARY OF CHANGES
CE 1 0 3 / 0 4 / 1 2 Pol i n i t i a l co d in g
C
C−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−C
CF ARGUMENTS:
CF I /O NAME DESCRIPTION
CF −−− −−−−−−−−−−−− −−−−−−−−−−−CF I F e l e m e n t a r y f a c e numberCF I M i n d e x of f ace−block c o n n e c t i o n
CF I L g r i d l e v e l
CF I NI , NJ ,NK block dimensionCF I NQ dimension a l o n g f a c e
56
NLR-TR-2012-349
CF I EORD o r d e r o f e x t r a p o l a t i o n ( 1 . . 4 )
C ( q : I c o u l d throw t h i s away b e c a u s e I only do 4 t h )
CF I MSTAG Runge−K u t t a s t a g e (0 i f r e s t r i c t i o n )
CF I FIRST f i r s t t ime s t e p
CF I DTIME t ime s t e p o f n e x t RK s t a g e
CF I /O VAR flow v a r i a b l e s in one blockCF I /O P p r e s s u r e in one blockCF I /O VARP p e r t u r b a t i o n v a r i a b l e s in one blockCF I /O PP p r e s s u r e p e r t u r b a t i o n in one blockCF I NX,NY, NZ u n i t normal v e c t o r
CF I NPOLES number of p o l e s o f impedance
CF I APOLE, LPOLE c o e f f i c i e n t s f o r t h e p o l e s o f impedance
CF I /O LMPM p a r t s o f t h e p r e s s u r e a t impedance a t o l d / n e x t RK s t a g e
CF I /O LMPI i n i t i a l p a r s o f t h e p r e s s u r e a t impedance a t o l d / n e x t RK s t a g e
C
C−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−C
CG SHORT DESCRIPTION OF THE ALGORITHM:
CG s e t t h e e x t r a p o l a t i o n c o e f f i c i e n t s
CG i f n e c e s s a r y i n i t i a l i z e t h e i n t e g r a t i o n v a r i a b l e s
CG f o r each c e l l f a c e :
CG c a l c u l a t e c e l l i n d i c e s
CG e x t r a p o l a t e t h e p r e s s u r e to dummy c e l l t h r o u g h t h e p r e s c r i b e d
CG v a l u e
CG e x t r a p o l a t e o t h e r p e r t u r b a t i o n s to dummy c e l l
CG c a l c u l a t e t h e r e f e r e n c e d e n s i t y a t f a c e
CG c a l c u l a t e t h e p r e s s u r e p e r t u r b a t i o n a t f a c e
CG c a l c u l a t e t h e i n t e g r a t i o n v a r i a b l e s in t h e n e x t s t e p
C
C∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗# i n c l u d e ” dim . h ”
# i n c l u d e ” f t p . h ”
# i n c l u d e ” f a r . h ”
# i n c l u d e ” uns . h ”
C
i n t e g e r q , t , s , i0 , j0 , k0 , i1 , j1 , k1 , i2 , j2 , k2 , i3 , j3 , k3 , i4 , j4 , k4 , i p o
r e a l dc1 , dc2 , dc3 , dc4 , ddw , dd1 , dd2 , dd3 , fc1 , fc2 , fc3 , fc4 ,
. rhe , pe , rue , rve , rwe , ree ,
. r r b a , uuba , vvba , wwba ,
. pb , rb , rub , rvb , rwb , ub , vb , wb , qen ,
. rba , uba , vba , wba , respm
C
C−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−C
57
NLR-TR-2012-349
C S e t t h e 4 th−o r d e r e x t r a p o l a t i o n c o e f f i c i e n t s
C ( q : I only i n c l u d e 4 th−o r d e r )
C
i f ( eo rd . eq . 4 ) thenC s e t t h e c o e f f i c i e n t s f o r s t a n d a r d e x t r a p o l a t i o n to dummy c e l l
dc1 = 4
dc2 = −6
dc3 = 4
dc4 = −1
C s e t t h e c o e f f i c i e n t s f o r e x t r a p o l a t i o n to dummy c e l l t h r o u g h a
C p r e s r i b e d v a l u e a t f a c e
ddw = 3 . 2
dd1 = −3
dd2 = 1
dd3 = −0.2
C s e t t h e c o e f f i c i e n t s f o r s t a n d a r d e x t r a p o l a t i o n to f a c e
f c 1 = 2 .1875
f c 2 = −2.1875
f c 3 = 1 .3125
f c 4 = −0.3125
e n d i fC
C
i f ( mstag . l e . 1 ) thenC t h e i n i t i a l i n t e g r a t i o n v a r i a b l e s f o r t h e n e x t RK s t a g e a r e s e t
do q =1 , nq
do i p o =1 , n p o l e s
lmpi ( ( q−1)∗ n p o l e s + i p o ) = lmpm ( ( q−1)∗ n p o l e s + i p o )
enddoenddo
e n d i fC
C loop ove r t h e c e l l f a c e s
C
do 60 , q =1 , nq
C
C c a l c u l a t e t h e c e l l i n d i c e s
C
t = ( q−1) / ns (m, f , l )
s = q−1 − t ∗ns (m, f , l )
C
i 1 = i b (m, f , l ) + s∗ i s (m, f ) + t ∗ i t (m, f )
j 1 = j b (m, f , l ) + s∗ j s (m, f ) + t ∗ j t (m, f )
k1 = kb (m, f , l ) + s∗ks (m, f ) + t ∗ k t (m, f )
58
NLR-TR-2012-349
C
i 0 = i 1 + i r (m, f )
j 0 = j 1 + j r (m, f )
k0 = k1 + kr (m, f )
C
i 2 = i 1 − i r (m, f )
j 2 = j 1 − j r (m, f )
k2 = k1 − kr (m, f )
i 3 = i 2 − i r (m, f )
j 3 = j 2 − j r (m, f )
k3 = k2 − kr (m, f )
i 4 = i 3 − i r (m, f )
j 4 = j 3 − j r (m, f )
k4 = k3 − kr (m, f )
C
C e x t r a p o l a t e a l l p e r t u r b a t i o n v a r i a b l e s to dummy c e l l
C
r h e = dc1∗ va rp ( i1 , j1 , k1 , 1 ) + dc2∗ va rp ( i2 , j2 , k2 , 1 ) +
. dc3∗ va rp ( i3 , j3 , k3 , 1 ) + dc4∗ va rp ( i4 , j4 , k4 , 1 )
r u e = dc1∗ va rp ( i1 , j1 , k1 , 2 ) + dc2∗ va rp ( i2 , j2 , k2 , 2 ) +
. dc3∗ va rp ( i3 , j3 , k3 , 2 ) + dc4∗ va rp ( i4 , j4 , k4 , 2 )
r v e = dc1∗ va rp ( i1 , j1 , k1 , 3 ) + dc2∗ va rp ( i2 , j2 , k2 , 3 ) +
. dc3∗ va rp ( i3 , j3 , k3 , 3 ) + dc4∗ va rp ( i4 , j4 , k4 , 3 )
rwe = dc1∗ va rp ( i1 , j1 , k1 , 4 ) + dc2∗ va rp ( i2 , j2 , k2 , 4 ) +
. dc3∗ va rp ( i3 , j3 , k3 , 4 ) + dc4∗ va rp ( i4 , j4 , k4 , 4 )
r e e = dc1∗ va rp ( i1 , j1 , k1 , 5 ) + dc2∗ va rp ( i2 , j2 , k2 , 5 ) +
. dc3∗ va rp ( i3 , j3 , k3 , 5 ) + dc4∗ va rp ( i4 , j4 , k4 , 5 )
C
C c a l c u l a t e t h e p r e s s u r e p e r t u r b a t i o n a t f a c e
C
pb = 0 . 0
do i p o =1 , n p o l e s
pb = pb + lmpm ( ( q−1)∗ n p o l e s + i p o )
enddoC
C e x t r a p o l a t e t h e p r e s s u r e t h r o u g h t h e p r e s c r i b e d p r e s s u r e
C
pe = ddw∗pb + dd1∗pp ( i1 , j1 , k1 )
. + dd2∗pp ( i2 , j2 , k2 ) + dd3∗pp ( i3 , j3 , k3 )
C
C c a l c u l a t e t h e r e f e r e n c e v a r i a b l e s a t f a c e
C
r r b a = 0 . 5∗ ( v a r ( i0 , j0 , k0 , 1 ) + v a r ( i1 , j1 , k1 , 1 ) )
uuba = 0 . 5∗ ( v a r ( i0 , j0 , k0 , 2 ) + v a r ( i1 , j1 , k1 , 2 ) ) / r r b a
59
NLR-TR-2012-349
vvba = 0 . 5∗ ( v a r ( i0 , j0 , k0 , 3 ) + v a r ( i1 , j1 , k1 , 3 ) ) / r r b a
wwba = 0 . 5∗ ( v a r ( i0 , j0 , k0 , 4 ) + v a r ( i1 , j1 , k1 , 4 ) ) / r r b a
C
C e x t r a p o l a t e p e r t u b a t i o n v a r i a b l e s to f a c e
C
rb = f c 1 ∗ va rp ( i1 , j1 , k1 , 1 ) + f c 2 ∗ va rp ( i2 , j2 , k2 , 1 )
. + f c 3 ∗ va rp ( i3 , j3 , k3 , 1 ) + f c 4 ∗ va rp ( i4 , j4 , k4 , 1 )
rub = f c 1 ∗ va rp ( i1 , j1 , k1 , 2 ) + f c 2 ∗ va rp ( i2 , j2 , k2 , 2 )
. + f c 3 ∗ va rp ( i3 , j3 , k3 , 2 ) + f c 4 ∗ va rp ( i4 , j4 , k4 , 2 )
rvb = f c 1 ∗ va rp ( i1 , j1 , k1 , 3 ) + f c 2 ∗ va rp ( i2 , j2 , k2 , 3 )
. + f c 3 ∗ va rp ( i3 , j3 , k3 , 3 ) + f c 4 ∗ va rp ( i4 , j4 , k4 , 3 )
rwb = f c 1 ∗ va rp ( i1 , j1 , k1 , 4 ) + f c 2 ∗ va rp ( i2 , j2 , k2 , 4 )
. + f c 3 ∗ va rp ( i3 , j3 , k3 , 4 ) + f c 4 ∗ va rp ( i4 , j4 , k4 , 4 )
C
C c a l c u l a t e normal v e l o c i t y on f a c e
C
ub = ( rub − uuba∗ rb ) / r r b a
vb = ( rvb − vvba∗ rb ) / r r b a
wb = ( rwb − wwba∗ rb ) / r r b a
C
C t a k e i n n e r p r o d u c t to e x t r a c t normal momentum on f a c e
C
qen = ub∗nx ( q ) + vb∗ny ( q ) + wb∗nz ( q )
C
C c a l c u l a t e t h e i n t e g r a t i o n v a r i a b l e s in t h e n e x t s t e p
C
C −−− c a l c u l a t e r e s i d u e s , make t h e t ime s t e p
do i p o =1 , n p o l e s
respm = a p o l e ( i p o )∗ qen − l p o l e ( i p o )∗ lmpm ( ( q−1)∗ n p o l e s + i p o )
lmpm ( ( q−1)∗ n p o l e s + i p o ) = lmpi ( ( q−1)∗ n p o l e s + i p o )
. + d t ime ∗ respm
enddoC
C S e t t h e dummy c e l l v a l u e s
va rp ( i0 , j0 , k0 , 1 ) = r h e
va rp ( i0 , j0 , k0 , 2 ) = r u e
va rp ( i0 , j0 , k0 , 3 ) = r v e
va rp ( i0 , j0 , k0 , 4 ) = rwe
va rp ( i0 , j0 , k0 , 5 ) = r e e
pp ( i0 , j0 , k0 ) = pe
C
60 c o n t i n ue
60
NLR-TR-2012-349
end
A.3 dim.hIn dim.h the maximum size of some parameters is defined. It is convenient to not have too many
poles to describe the impedance, so a maximum for the amount of poles is defined as polmax and
set to 10.
A.4 bcond.FThis subroutine calculates the dummy-cell variables for the selected boundary condition. This
means that when an impedance boundary condition is used (bc = 12), the subroutine bcpole is
called for the perturbations (when eqm1 = 1). The mean flow is modelled as a solid wall and
therefore the slip condition (bcwal1) is called for the mean flow variables. The calls are:
e l s e i f ( bc ( i e f ) . eq . 1 2 ) theni f ( eqm1 . eq . 1 ) theniqm = i p o l e ( i f b , i e f , l )
c a l l b c p o l e ( i e f , i f b , l , n i1 , nj1 , nk1 , nq , eord ,m, dt ime ,
. v r e f , p r e f , varb , p ,
. bcd ( i r x ) , bcd ( i r y ) , bcd ( i r z ) , n p o l e s ( i e f ) ,
. a p o l e ( 1 , i e f ) , l p o l e ( 1 , i e f ) ,
. bcd ( iqm ) , bcd ( iqm+nq∗ n p o l e s ( i e f ) ) )
e l s ec a l l bcwal1 ( i e f , i f b , l , n i1 , n j1 , nk1 , nq , eqm1 , eord ,
. va rb ( 1 , 1 ) , va rb ( 1 , 2 ) , va rb ( 1 , 3 ) ,
. va rb ( 1 , 4 ) , va rb ( 1 , 5 ) , p ,
. bcd ( i r x ) , bcd ( i r y ) , bcd ( i r z ) , bcd ( i v x ) )
e n d i f
No new parameters have to be defined for this subroutine. However pol.h is included such that
global parameters ipole, npoles, apole and lpole can be used.
A.5 rdbcd.FThe subroutine rdbcd.F reads the boundary conditions and the accompanying parameters from
the BCDAT-file and writes these to the global parameters described in pol.h. So pol.h is again in-
cluded in the subroutine. Extra parameters are needed for this subroutine that do not have to be
used globally, so they are defined in the subroutine itself. One of the parameters is called tnpole
and signifies the amount of poles needed. Also a counter called ipo is defined here. Both param-
eters are integers. Two other parameters defined in the subroutine are the vectors tapole(polmax)
and tlpole(polmax) of length polmax and are real.
61
NLR-TR-2012-349
In the subroutine some actions have to be taken. First the coefficients of the pole approximation,
apole(ipo,f) and lpole(ipo,f), are set to default for the face f, which is zero.
do i p o = 1 , polmax
a p o l e ( ipo , f ) = 0 . 0
l p o l e ( ipo , f ) = 0 . 0
enddo
Apole and lpole are vectors of length polmax = 10 for face f, which include all the possible coef-
ficients needed for the pole approximation.
The local parameters tnpole, tapole and tlpole are first set to zero.
t n p o l e = 0
C −−−do i p o = 1 , polmax
t a p o l e ( i p o ) = 0 . 0
t l p o l e ( i p o ) = 0 . 0
enddo
Then they are read from the BCDAT-file.
i f ( i b c t . eq . 1 2 ) thenread ( lub , ∗ ) t n p o l e
i f ( t n p o l e . g t . polmax ) thenw r i t e ( 2 , 9 8 5 ) polmax
s top ’ERROR’
e n d i fC −−−
do i p o = 1 , t n p o l e
read ( lub , ∗ ) t a p o l e ( i p o ) , t l p o l e ( i p o )
enddoe n d i f
Finally the global parameters apole, lpole and npoles are written by respectively the local param-
eters tapole, tlpole and tnpole for face f.
n p o l e s ( f ) = t n p o l e
do i p o =1 , t n p o l e
a p o l e ( ipo , f ) = t a p o l e ( i p o )
l p o l e ( ipo , f ) = t l p o l e ( i p o )
enddo
62
NLR-TR-2012-349
A.6 updbcd.FThe last subroutine that has to be altered to allow the impedance boundary is updbcd. This sub-
routine reserves space for parameters by setting pointers and has to be extended so that the impedance
boundary parameters also have room to be stored. Again the global parameters have to be in-
cluded via pol.h, and one local integer parameter is defined, tnpole, to make the implementation
a little easier.
Some general pointers and parameters have to be set such as the cell face centre coordinates, the
unit normal vectors, cell-face centers and grid velocity times unit normal.
In the next part the specific pointers for the impedance boundary are set.
i f ( bc ( f ) . eq . 1 2 ) theni f ( f i r s t ) then
C −−− I n i t i a l i z e t h e p o i n t e r to t h e p o l e datat n p o l e = n p o l e s ( f )
i p o l e (m, f , l ) = iws
iws = iws + 2∗ t n p o l e ∗nq
lw = lw − 2∗ t n p o l e ∗nq
e n d i fC −−− Check i f t h e r e i s enough memory l e f t
i f ( lw . l t . 0 ) c a l l w s e r r ( ’UPDBCD’ , lw )
i f ( f i r s t ) thenC −−− S e t t h e i n t e g r a t i o n p a r a m e t e r s to z e r o
iqm = i p o l e (m, f , l )
do i =iqm , iqm+2∗ t n p o l e ∗nq−1
ws ( i ) = 0 . 0
enddoe n d i f
e n d i f
63
NLR-TR-2012-349
Appendix B Discretization methods for the MSD+ approximation
In section 3.6.2 the MSD+ approximation was discussed. The MSD+ approximation was defined
as
Z(ω) = a−2(iω)−2 + a−1(iω)−1 + a0 + a1(iω). (78)
From the relation between the impedance, normal velocity and pressure on the impedance bound-
ary,
Z(ω) =P (ω)Un(ω)
,
the MSD+ approximation can be written as a relation between the pressure and the normal veloc-
ity in the frequency domain
P (ω) = a−2(iω)−2 Un(ω) + a−1(iω)−1 Un(ω) + a0 Un(ω) + a1(iω) Un(ω). (79)
Using the relations for the inverse Fourier transform,
∂y(t)∂t
F←→ (iω)Y (ω)∫ t
−∞y(τ)dτ
F←→ (iω)−1Y (ω), (80)
the MSD+ approximation is transformed to the following relation between the pressure and nor-
mal velocity in the time domain
p′(t) = a−2
∫ t
−∞
∫ t′
−∞u′n(τ) dτ dt′
+ a−1
∫ t
−∞u′n(τ) dτ + a0 u′n(t) + a1
∂u′n(t)∂t
. (81)
This appendix discusses the discretization of the MSD+ approximation. There are three possi-
ble methods to discretize the MSD+ approximation. One method is found in the literature and
uses sums to discretize the integrals of expression (81). The second method makes use of auxil-
iary parameters and ordinary differential equations and the third method is based on a system of
ordinary differential equations. The three discretizations can be found in the next sections.
B.1 Integration discretizationThe first discretization method was proposed by Heutschi et al. (Ref. 6). The time-domain ex-
pression for the MSD+ approximation (81) uses integrals. Heutschi et al. propose to discretize
64
NLR-TR-2012-349
these integrals as summations∫ t
−∞u′n(τ) dτ = ∆t
M∑m=−∞
(u′n)m
∫ t
−∞
∫ t′
−∞u′n(τ) dτ dt′ = (∆t)2
M∑m′=−∞
m′∑m=−∞
(u′n)m.
This is a very straightforward idea in numerical mathematics, however Heutschi et al. notice
that instabilities may occur. These instabilities occur when the double sum starts to dominate the
other terms. A weak damping in the double sum is then introduced by Heutschi et al. to keep the
double sum from dominating. This does not seem to be a very elegant way to avoid instabilities,
so two other discretization methods were thought of.
B.2 Discretization with auxiliary parametersFor this discretization method two auxiliary parameters g and h are defined as
g(t) =∫ t
∞u′n(τ)dτ (82)
h(t) =∫ t
∞
∫ t′
∞u′n(τ)dτdt′ =
∫ t
∞g(τ)dτ. (83)
The MSD+ approximation in the time domain is now written as
p′(t) = a−2 h(t) + a−1 g(t) + a0 u′n(t) + a1∂u′n(t)
∂t.
Now h(t) and g(t) can be solved by taking the derivative
dg
dt= u′n(t)
dh
dt= g(t),
and then solving this with an time-integration method such as the fourth-order low-storage Runge-
Kutta method. In the MSD+ approximation a derivative of the normal velocity is found. There is
no expression for the derivative of the normal velocity and therefore the derivative has to be ap-
proximated.
Heutschi et al. give some examples of the coefficients for several flow resistivities. In all of the
examples the coefficient a1 is taken to be zero. This would solve the problem of the derivative of
the normal velocity. However, it is preferred that the discretization method works in general, so
an approximation of the derivative of the normal velocity is still needed.
65
NLR-TR-2012-349
B.3 System of ODEsFor the last discretization method the MSD+ approximation is transformed to a system of ODEs
which can then be solved by the fourth-order low-storage Runge-Kutta method. First the MSD+
approximation in the frequency domain (79) is multiplied by (iω)2, which gives
(iω)2P (ω) = a−2Un(ω) + a−1(iω)Un(ω) + a0(iω)2Un(ω) + a1(iω)3Un(ω). (84)
Remembering the relations for the inverse Fourier transform (80) this is transformed to the fol-
lowing time-domain expression
∂2p′(t)∂t2
= a−2 u′n(t) + a−1∂u′n(t)
∂t+ a0
∂2u′n(t)∂t2
+ a1∂3u′n(t)
∂t3. (85)
The time-domain expression will now be rewritten as a system of ODEs. First an auxiliary pa-
rameter k is set
k = a0 u′n + a1∂u′n∂t− p′.
After scrambling this, an expression for the first derivative of the normal pressure arises
∂u′n∂t
=1a1
(k + p′ − a0 u′n). (86)
Now two more auxiliary parameters l and m are set
l =∂k
∂t= a0
∂u′n∂t
+ a1∂2u′n∂t2
− ∂p′
∂t
m =∂l
∂t= a0
∂2u′n∂t2
+ a1∂3u′n∂t3
− ∂2p′
∂t2.
From the time-domain expression (85), m can be written as
m = −a−2 u′n − a−1∂u′n∂t
.
The expression for the first derivative (86) is used to again rewrite the expression for m to
m = −a−2 u′n −a−1
a1(k + p′ − a0 u′n).
Now using m = ∂l∂t and l = ∂k
∂t , the MSD+ approximation is written as a system of non-
homogeneous ordinary differential equations
∂
∂t
u′n
k
l
=
−a0
a1
1a1
0
0 0 1
−a−2 + a0a−1
a1−a−1
a10
u′n
k
l
+
1a1
0
−a−1
a1
p′. (87)
66
NLR-TR-2012-349
This system can be solved by applying the time-integration fourth-order low-storage Runge-
Kutta method.
As discussed before, Heutschi et al. give values for the coefficients where a1 = 0. This causes a
problem with the system of ODEs because devision by a1 is used. So the coefficients of Heutschi
et al. cannot be used.
67
