bodony-2009

47th Aerospace Sciences Meeting and Exhibit

5–8 January 2009, Reno, NV

Characteristic boundary conditions fornon-orthogonal, moving meshes

Daniel J. Bodony∗

Department of Aerospace EngineeringUniversity of Illinois at Urbana-Champaign

Urbana, IL 61801

Boundary conditions to the compressible Navier-Stokes equations are developed for the case of deformable,generalized coordinates. The general theory is based on a idea of Halpern [SIAM J. Math. Analy., Vol. 22(5),pp. 1256–1283, 1991] which, in the inviscid case, reduces tostandard characteristic treatment and thus logicallyextends the work of Thompson [J. Comput. Phys., vol. 68, pp. 1–24, 1987], Poinsot & Lele [ibid, vol. 101,pp. 104–129, 1992], and Kim & Lee [AIAA J., vol. 42(1), pp. 47–55, 2004]. The issue of well-posedness isconsidered. The developed boundary conditions are applicable to fluid problems with moving boundaries ininviscid and viscous fluids. Several verification problems are presented to demonstrate accuracy.

I. Introduction

The simulation of unsteady fluid flows has had a profound influence on the development of boundary conditionsthat can accurately represent the required boundary data. One of the requirements that all boundary condition imple-mentations should satisfy is the correct specification of the physical boundary conditions as determined by analysis,1, 2

with the remaining numerical boundary conditions specifiedaccordingly.For inviscid flows the boundary conditions may be constructed based on characteristics, such as considered by

Thompson3, 4 and by Poinsot & Lele.5 For flows with non-zero viscosity the notion of characteristics fails as the gov-erning equations are no longer hyperbolic, often being called incompletely parabolic. As such characteristic boundaryconditions are not strictly applicable but, in practice, have been applied with success in a number of cases. In theseinstances the issue of well-posedness is not certain and thefailure of a simulation to remain stable, in the sense ofKreiss,6 suggests that it may be the inconsistent boundary conditionformulation causing the instability.

More complete boundary conditions for the compressible viscous governing equations have been developed byHesthaven & Gottlieb,7 Svardet al.,8 and Svard & Nordstrom,9 among others, based on a penalization technique(called Simultaneous Approximation Term, or SAT) combinedwith very specific internal finite difference operators.These formulations, which are based on local data, have beenshown to be well-posed2 but introduce at least oneartificial time scale into the problem through the penalty parameter. These schemes have also been shown to preservethe formal accuracy of the overall numerical method. These conditions are imposed weakly in the equations as

∂q∂t= · · · + σ [

(A+ + ǫB)q − g]

whereσ = σI + ǫσV is usually a scalar with specific bounds to ensure stability.A+ is the “incoming” part of thehyperbolic characteristics,B is a matrix associated with the viscous influence on the boundary condition through thesmall parameterǫ, andg are the boundary data to be imposed. The reader is referred toRefs. 8,9 for the details.

The introduction of SAT-based boundary conditions into an existing code requires the implementation of newfinite difference schemes following the Summation-by-parts (SBP) property.10 Together, the SBP-SAT formulationcan be shown to yield provably stable numerical discretizations for the compressible Navier-Stokes equations onuniform Cartesian meshes. For non-uniform meshes only the diagonal SBP schemes up to and including 3rd order

∗AIAA [email protected] c© 2009 by D. J. Bodony. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

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American Institute of Aeronautics and Astronautics Paper 2009-0010

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-10

Copyright © 2009 by Daniel J. Bodony. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

admit an energy norm-based stability proof. When one is interested in a numerical method higher than 3rd orderor if one wishes to use alternative finite difference schemes there is no advantage to the SBP-SAT approachotherthan through the existence of well-posed boundary conditions. As such one must search for other formulations. Asmentioned earlier the inviscid equations are well-posed when coupled with characteristic-based boundary conditions,such as developed by Thompson.3, 4 The later development by Poinsot & Lele5 improved on Thompson’s approach,and included the viscous terms in anad hoc manner. Further work by Kim & Lee12, 13 partially generalized the one-dimensional approach by including the metric transformation for non-uniform meshes but also altered the mannerin which the wave amplitudes are calculated through the inclusion of the transverse terms, possibly violating wellposedness.

The approach taken by Poinsot & Lele and by Kim & Lee with respect to the viscous terms was arbitrary butguided by the work of Engquist & Majda,14 Rudy & Strikwerda,15 and by Oliger & Sundstrom.1 Absent any proof ofwell posedness the failure of these boundary conditions, inthe sense of numerical instability, is never precisely known.

Thus the purpose of this work is to develop boundary conditions which are well posed and which may be cast inthe form consistent with Thompson’s approach, in the sense that time-dependent equations are solved on the boundaryalong with the interior scheme without the need for implementation of the SBP-SAT schemes. This is not a criticismof SBP-SAT but an alternative to it. The continuous equations will be developed following the approach of Halpern.16

Since it may be shown that this approach is equivalent to the characteristic-based boundary conditions the discussionwill start with the more traditional development in the style of Thompson. Later the viscous boundary conditions willbe stated.

II. Derivation of the characteristic relations for an inviscid fluid

The conservation form of the governing equations of a compressible fluid, in Cartesian form, may be writtena

∂Q∂t+∂Fi

∂xi= S (1)

whereQ = [ρ ρu ρE]T is the vector of mass, momentum, and energy density per unit volume. We takeu to be the ve-locity vector inN-dimensions. TheFi are the fluxes in theith direction and contain only the inviscid contributionsFI

i .The Cartesian coordinates (x, t) can be mapped to another coordinate system (ξ, τ) via the time-dependent mappings

x = X(ξ, τ) with inverse ξ = Ξ(x, t) (2)

whereX−1 = Ξ and we only consider non-singular mappings such thatX−1 exists and is well defined. Moreover wetaket = τ. The Jacobian of the transformation is defined asJ = det(∂Ξi/∂x j) and is strictly positive.

Under these conditions and with simple application of the chain rule it can be shown17 that Eq. (1) tranforms to

∂

∂τ

(QJ

)

+∂Fi

∂ξi=

SJ

(3)

after using the identities

∂

∂ξ j

(

1J

∂ξ j

∂xi

)

= 0 for i = 1, . . . ,N

∂

∂τ

(

1J

)

+∂

∂ξ j

(

1J

∂ξ j

∂t

)

= 0.

(4)

If we define the weighted metricξi = J−1(∂ξ/∂xi) and contravariant velocityU = u jξ j + ξt, with similar expressionsfor the remaining components, then the inviscid fluxesF I

i are

FI1 =

ρU

ρuU + pξx

ρvU + pξy(ρE + p)U − ξt p

and FI2 =

ρV

ρuV + pηx

ρvV + pηy

(ρE + p)V − ηt p

(5)

aThe summation convention is used where repeated indices aresummed from 1 toN.

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in two dimensions and

FI1 =

ρU

ρuU + pξx

ρvU + pξyρwU + pξz

(ρE + p)U − ξt p

, FI2 =

ρV

ρuV + pηx

ρvV + pηy

ρwV + pηz

(ρE + p)V − ηt p

, and FI3 =

ρW

ρuW + pζxρwV + pζyρwW + pζz

(ρE + p)W − ζt p

(6)

in three dimensions.Following Thompson we write Eq. (3) as

∂Q∂τ+ J∂F

Ij

∂ξ j= S − Q

∂

∂τ

(

1J

)

≡ S (7)

For the case with the coordinateξ = ξ1 is not in the plane of the boundary, but is not necessarily normal to theboundary, we will move the boundary-parallel fluxes to the right-hand-side of Eq. (7) as

∂Q∂τ+ J∂F

I1

∂ξ= S − J

N∑j=2

∂FIj

∂ξ j(8)

and expandFI

into its Cartesian components,

FI = ξ jFIj + ξtQ (9)

so that Eq. (8) becomes∂Q∂τ+ J

ξ j∂F j

∂ξ+ ξt∂Q∂ξ

= S − J

F j∂

∂ξ(ξ j) + Q

∂

∂ξ(ξt)

. (10)

Upon introducing the matrixA

A = J

ξ j

∂FIj

∂Q+ ξtI

(11)

with I the (N + 2)× (N + 2) identity matrix we can write Eq. (10) in the desired form

∂Q∂τ+ A∂Q∂ξ= ˜S (12)

where˜S = S − J

F j∂

∂ξ(ξ j) + Q

∂

∂ξ(ξt)

. (13)

Equation (12) is the final form of the governing equations. Itis nothing more than a rearrangment of theξ-fluxes tothe left-hand-side and the remaining terms to the right-hand-side.

From Eq. (12) we now splitQ into its characteristic components. To do this we note that the submatrix

Jξ j

∂FIj

∂Q

is diagonalized by the matrixP according to

Jξ j

∂FIj

∂Q= PΛP−1

where the matrixP and its inverse may be found in Hirsch.18 The current matrixA contains the additional termξtI.

SinceP diagonalizesJξ j∂FI

j

∂Q we may write

Λ = P−1

Jξ j

∂FIj

∂Q+ JξtI − JξtI

P

= P−1AP − JξtP−1IP

= P−1AP − JξtI. (14)

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On definingU = JU = J(u jξ j + ξt) to be theJ-scaled contravariant velocity associated withξ we find that (fromEq. (14)) we have

Λ = P−1AP (15)

whereΛ = diag(U,U,U,U + c|∇xξ|,U − c|∇xξ|) (16)

and where∇xξ is the gradient ofξ with respect to the Cartesianx coordinate.UsingP andΛ we now diagonalize Eq. (12) as

∂R∂τ+ Λ∂R∂ξ= S c (17)

where the variations (in three dimensions) are

δR =(

δρ − δp/c2, δW, δV, δp/(ρc) + δU, δp/(ρc) − δU)

(18)

with δU = ξx j u j the contravariant component of the velocity at the wallwhich does not include the term ξt. Hereξx j = ξx j/|∇xξ| is the jth component of the unit vector in the direction of∇xξ. Likewise we write variationsδV =−ξxδv + ξyδu andδW = ξxδw − ξzδu. Equation (17) is the characteristic form of the governing equations to which weapply the appropriate boundary conditions. The elements ofthe vector

L = Λ∂R∂ξ

(19)

can be associated (under suitable approximations) with thewave amplitudes of the incoming and outgoing componentsof R.5 The expression forL in Eq. (19) reduces to the similar result found by Kim & Lee12 when the grid is fixed,i.e.,whenξt ≡ 0. When, in addition, we takeSC = 0 we recover the equivalent expression found by Poinsot & Lelewhenexpressed in curvilinear coordinates.

III. Boundary conditions for the viscous compressible Navier-Stokes

As mentioned in the introduction the use of characteristicsfor developing boundary conditions does not ensurewell posedness for the viscous equations, although in practice such approaches have been known to work reasonablywell for high Reynolds numbers. For example, consider the linear advection diffusion equationut + aux − ǫuxx =

0, u(x, t = 0) = u0 for which the Cauchy problem is well posed. Split thex-axis into two parts, and consider theproblem to be solved on the left half domainΩ− = (x, t)| ∈ R− × R+. Since the domain is semi-infinite one mustapply “transparent” boundary conditions onΓ = x = 0 to mimic the right-half real line. Loheac19 showed that onecould close the problem by solvingut + aux = 0 for x ∈ Γ and get an error that is of orderǫ2 asǫ → 0. Such aresult qualitatively explains the experience regarding characteristic-based boundary conditions for viscous problems.For reacting flows Sutherland & Kennedy20 found that ignoring the viscous flux on the boundary, though stable, wasunsuitably inaccurate.

Thus one is left to search for more accurate, and well posed, boundary conditions. For viscous non-slip wallsSvard & Norstrom9 have shown that specifying the wall velocity and the wall temperature is well posed such that oneonly needs to solve for the density (or other suitable thermodynamic variable) on the wall. The case when the normaltemperature gradient is specified is treated similarly.

When the boundary represents an open one, in the sense that “waves” may pass in or out as necessary, the situationin more complex. Thus, following Halpern,16 consider perturbationsw = [u′ T ′ ρ′]T about a uniform base state[u T ρ]T . These perturbations satisfy

∂w∂t= A( j) ∂w

∂x j+ ǫP( j,k) ∂w

∂x j∂xk(20)

whereA( j) andP( j,k) aren × n square matrices. The particular choice ofw has been made so that theA andP may bewritten

A( j) =

B( j) C( j)

D( j) A( j)

, P( j,k) =

P( j,k)

0

0 0

where∂t − A( j)∂ j is strictly hyperbolic, andA

( j)is diagonal withp negative eigenvalues. The rank ofP

( j,k)is r.

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In what follows a right boundary atx = 0 is considered on a Cartesian mesh. Other boundaries and mesh nonuni-formity can be included without difficulty. The essential idea of the boundary conditions is written out as the followingsteps

1. Fourier-Laplace transform iny = (x2, x3, . . . )T and in time, respectively. The Fourier variable isη; the Laplacevariable iss.

2. Use the transmission condition that

ǫP( j,1)∂w∂x

is continuous overΓ

w is continuous overΓ

to develop boundary conditions in the Fourier-Laplace domain.

3. Expand the conditions in series ofǫ andη, the transverse wavenumber

4. “Invert” the boundary conditions to local space-time following Engquist & Majda14, 21

Thenon-local, exact andwell posed boundary conditions may be written

ǫP(1, j) ∂wI

∂x j= ǫ

r+p∑i=m+1

P(1,1)

r+p∑j=1

ζiN−1i j Ψ

ijw j (21)

wk =

r+p∑i=1

r+p∑j=1

N−1i j Ψ

ikw j, r + p + 1 ≤ i ≤ n (22)

wherewI = [u′ T ′]T , (ζi,Ψi) solve a generalized eigenvalue problem to be given below, and the columns ofN areconstructed from portions ofΨ. Observe that the first row corresponds to boundary conditions specific to viscousflows. Likewise it may be shown that the second row, whenǫ = 0, is identical to characteristic boundary conditions.

Construction of (ζi,Ψi) is as follows. Consider the generalized eigenvalue problem

(ξA(1) + ǫξ2P(11) − sI)Φ = 0. (23)

For the 1-D compressible Navier-Stokes equations this is quintic equation inξ. To each of the then eigenvaluesλi andeigenvectorsΛi of A(1) are associated, for the case whenη = 0,

ξi(s, ǫ) = ζi(s) + ǫχis2 + O(ǫ2), ζi = s/λi

andΦi(s, ǫ) = Ψi + ǫΞi s + O(ǫ2), Ψi = Λi.

for i ≤ m wherem is the number of negative eigenvaluesλi.The viscous component satisfies

ξi(s, ǫ) = ζi(s) + χis + O(ǫ), ζi = θi/ǫ

andΦi(s, ǫ) = Ψi + ǫΞi s + O(ǫ2), Ψi = Θi.

where (θ,Θ) satisfy (P(1,1)θ + A(1))Θ = 0. Observe that the eigenvaluesξi and eigenvectorsΦi have been expandedin terms of the small parameterǫ and thatη = 0 has been assumed. The latter condition, which assumes wavefrontsparallel to boundary, can be relaxed to include obliquely traveling waves.

For the one-dimensional Navier-Stokes equations we find that

A(1) =

−u −a2/(γT ) −a2

/γ

−(γ − 1)T −u 0

−1 0 −u

and P(1,1) = diag(4/3, γ/Pr, 0)

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from which is may be shown that

ξ1 =s

−a − u+

3α(γ − 1)+ 4γ6γ(a + u)3

ǫs2 + O(ǫ2), Φ1 =

a

(γ − 1)T

1

+ sǫ

3α(γ−1)+4γ6γ(a+u)(γ−1)Tαa(a+u)

0

+ O(ǫ2) (24)

ξ2 =s−u+α

γu3ǫs2 + O(ǫ2), Φ2 =

0

T

−1

+ sǫ

αγu

0

0

+ O(ǫ2) (25)

ξ3 =s

a − u+

3α(γ − 1)+ 4γ6γ(a − u)3

ǫs2 + O(ǫ2), Φ3 =

a

−(γ − 1)T

−1

+ sǫ

−3α(γ−1)−4γ6γ(a−u)(γ−1)Tαa(a−u)

0

+ O(ǫ2) (26)

ξ4 =θ1

ǫ+

34θ1

s + O(ǫ), Φ4 =

u(γ−1)T uαθ1−u

−1

+ sǫ

0

1

0

+ O(ǫ2) (27)

ξ5 =θ2

ǫ+

34θ2

s + O(ǫ), Φ4 =

u(γ−1)T uαθ2−u

−1

+ sǫ

0

1

0

+ O(ǫ2) (28)

whereα = γ/Pr andθ1,2 satisfy

θ1θ2 =3

4α(u2 − a2), θ1 + θ2 =

34α(u2 − a2/γ) + (4/3)u2

αu.

To date the only the zeroth-order form of the boundary conditions have been constructed, following Halpern. Forthe case of non-reflecting subsonic outflow one has

ǫ∂u∂x=θ1a u

Z

(

γ

au − T

T− ρρ

)

(29)

ǫ∂T∂x=θ1aU

Z

(

γ

au − T

T− ρρ

)

(30)

whereZ = γu − a(U/T − 1) andU = (γ − 1)Tu/(αθ1 − u). Observe that whenǫ = 0 one recovers the characteristiccondition thatγu/au − T/T − ρ/ρ = 0.

IV. Specific boundary conditions

To focus the general formulation presented in the previous sections on specific boundary types (walls, etc.) anumber of boundary conditions are derived based on Eq. (17) in a manner analogous to that proposed by Thompson.3, 4

Not all of the possible boundary conditions will be presented but a detailed derivation of a useful subset of the possibleconditions will be given. Additional boundary conditions can be easily derived in a similar manner. It is further statedthat when the grid is stationary the present boundary conditions resort to those found in Kim & Lee.12, 13

A. Inviscid deformable wall

Consider the case where a non-rigidly moving wall bounds oneportion of the fluid domain. A schematic of aξ-constantwall is given in Figure 1, with unit normaln and wall velocity vector ˙xwall. On this surface we require that

u · n = xwall · n (31)

be satisfied; we denote ˙xwall · n by vw. We observe that the diagonal ofΛ = (0, 0, 0, |c|,−|c|). This fact results from theconformally-deforming mesh which has metrics that satisfyξt = −vwall · ∇xξ = −vw for all time such thatU + ξt = 0.Hence only theL4 element enters the domain and it must be chosen such that

∂U∂τ= vw. (32)

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This requirement suggests that the replacement

L4← L5 − 2∂U∂τ

(33)

be used.

Figure 1: Wall lying on a constantξ-plane.

B. Viscous, isothermal deformable wall

As for the inviscid deformable wall we assume the geometry ofFigure 1. For a viscous, isothermal deformable wallwe require that

u∣

∣

∣

wall= xw (34)

T∣

∣

∣

wall= Tw (35)

where the terms on the right-hand-sides of the above equations represent the wall velocity and temperature, respec-tively. The wall density is updated according to the continutity equation

∂

∂τ

(

ρ

J

)

+∂

∂ξ j

(

ρU j

)

= 0. (36)

The momentum and total energy density are updated using Eq. (36) according to

∂

∂τ(ρu) = xw

∂ρ

∂τ+ ρ∂xw

∂τ(37)

∂

∂τ(ρE) =

(

CvTw +12

xw · xw

)

∂ρ

∂τ+ ρCv

∂Tw

∂τ+ ρxw ·

∂xw

∂τ(38)

a calorically perfect gas with a specific heat at constant volume ofCv has been assumed.

C. Inviscid subsonic inflow

In this application we take an inflow boundary to exist along aξ-plane and that the boundary does not move. Thislatter assumption is not restrictive but is the form most commonly applied. Under these conditions we observe thatthe first four elements ofΛ are positive, assumingU is strictly positive, implying we are to enforce four conditions toestimate the amplitudesL1, . . . ,L4. Choosing to specify the three velocity componentsU, V, andW, along with thetemperatureT , following PL, suggests the following substitutions

L4← L5 − 2∂U∂τ

(39)

L2← −∂W∂τ

(40)

L3← −∂V∂τ

(41)

L1← +(γ − 1)ρ

2c(L4 +L5) +

ρ

T∂T∂τ

(42)

whereL5 is computed using information from the domain interior.

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V. Applications

In this section a series of example applications are presented to demonstrate the boundary conditions in variouscontexts. In cases where possible the numerical data is compared to analytical solutions. These problems have beenchosen because the body motion is thesole source of unsteadiness and thus test specifically the ability of the boundaryconditions to handle deforming boundaries.

The first two examples consider moving cylinders placed in a quiescent environment whose motion is known togenerate acoustic waves.22 The third example presents a fluid-structure coupled problem of an initially displacedcircular elastic ring, emersed in a quiescent fluid, into which the ring radiates energy as sound.23 The fourth andfinal example considers the one-dimensional nonlinear problem of an impulsively displaced piston in a constant areachannel. In the first three problems an analytical solution is available in the linear limit while a nonlinear analyticalsolution, based on the method of characteristics, is available for the fourth problem until shock formation.

In all simulations the same numerical code is used which implements the boundary conditions formulated inSections II and III. The code was developed by the author and is based on the work of Visbal & Gaitonde.24 Theconservation equations of continuity, momentum, and energy are written in conservative form in the transformed spaceξ and transformed timeτ.17 The spatial derivatives use the sixth order standard compact finite difference stencil11 andthe temporal integration uses the standard fourth order Runge-Kutta scheme. For the boundary conditions the matrixP and its inverseP−1 of Eq. (15) are computed at each point along the boundary and the boundary value updates areimplemented as Eq. (17). No filtering of the variables is performed throughout the solution history.

A. Translating cylinder

Consider a circular cylinder of fixed radiusa and center (x0(t), 0) whose time-dependent motion is constrained alongthex axis. The cylinder is immersed in a quiescent fluid of uniformdensityρ∞, uniform pressurep∞, and zero velocity.If the translation in thex-direction is of the form

x0(t) = ǫw(t) sinωt (43)

wherew(t) is a suitably smooth ramp function, such thatw(0) = 0 andw(t → ∞) = 1, then whenǫ ≪ a andω≪ c∞/athe fluid response may be regarded as linear. If, in addition,we assume the fluid to be inviscid, then the linear waveequation for the pressure

∂2p∂t2− a2

∞∂2p∂xi∂xi

= 0 (44)

holds with the boundary condition∂p∂r

∣

∣

∣

∣

∣

r=a= −ρ∞

∂xw

∂t· r = −ρ∞

∂vr

∂t, (45)

with r the unit outward radial normal, applied atr = a after linearization.Together with the Sommerfeld radiation condition the time-harmonic solution of the above problem is easily found

using separation of variables and is given by

p(x, ω) =∞∑

n=0

An(ω) cos(nθ)H(1)n (kr) (46)

wherek = ω/c∞ is the acoustic wavenumber andH(1)n is the Hankel function, of ordern, of the first kind.25 For motion

only along thex axis onlyA1 is nonzero and is given by

A1(ω) =2iρ∞c∞Vc(ω)

H(1)0 (ka) − H(1)

2 (ka)(47)

where

Vc(ω) =

∞∫

−∞

x0(t)eiωt dt (48)

is the Fourier transform of the center velocity. For the present case of nonharmonic motion the pressure is given bythe inverse Fourier transform of Eq. (46) as

p(x, t) =12π

∞∫

−∞

p(x, ω)e−iωt dω. (49)

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A single block structured grid was used to describe the translating cylinder, as depicted in Fig. 2. The instantaneouscylinder surface is located on the inner boundaryRi while the fixed, outer surface of radius 5a is atR0. The mesh isnonorthogonal throughout the domain: the origin of the radial lines of the mesh are offset vertically by an amounty0 = a/2 from the instantaneous center. For a specific cylinder motion we take

w(t) =12

[1 + tanh(5(t − 2))] and ω = 1 (50)

in Eq. (43) consider and two different amplitudes ofǫ = 0.05 andǫ = 0.0005.The resulting response taken at (x, y) = (x0(t) + a, 0), i.e., at the point on the cylinder surface that lies on thex

axis, is shown in Fig. 3. For a purely linear response all three curves shown in the figure should collapse on to a singlecurve. This is approximately the case but one can discern that the ǫ = 0.0005 response is closer to the analyticalresponse than is the high amplitude motion, indicating somepresence of nonlinearity.

-6 -4 -2 0 2 4 6

-4

-2

0

2

4

x/a

y/a

Ro

Ri

y0

Figure 2: Non-orthogonal mesh for translating and “breathing” cylinders.

0 2 4 6 8−1.5

−1

−0.5

0

0.5

1

ta/a∞

[(p−

p ∞)/

p ∞]/ǫ

Figure 3: Pressure at (x, y) = (x0(t) + a, 0) for the translating cylinder. Legend: ——, analytical solution; ——,ǫ = 0.05; —∗—, ǫ = 0.0005.

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B. “Breathing” cylinder

In this section we allow the radius of the cylinder to be a function of time and keep the center fixed at (0, 0). The radiusat timet is given by

r(t) = a(1+ ǫw(t) cosωt) (51)

with w(t) andω as in Eq. (50) and again consider two amplitudesǫ = 0.05 andǫ = 0.0005. In a manner similar to thatgiven in IV. A. the analytical pressure response is determined by the inverse Fourier transform of Eq. (49) but with

p(x, ω) = − iρ∞c∞Vr

H(1)1 (ka)

H(1)0 (kr) (52)

for the pressure Fourier amplitude and

Vr(ω) =

∞∫

−∞

r(t)eiωt dt (53)

for the radial velocity Fourier transform.Using the same single block structured grid shown in Fig. 2 the results for the pressure on the cylinder surface at

(x, y) = (a(t), 0) are shown in Fig. 4. Again we note that in the purely linear case all three curves should collapse upona single curve and that the simulation data approximately demonstrate this behavior, with theǫ = 0.0005 being thecloser of the two simulations to the anaytical solution.

0 2 4 6 8−1

−0.5

0

0.5

1

1.5

2

tc∞/a∞

[(p−

p ∞)/

p ∞]/ǫ

Figure 4: Pressure at (x, y) = (a(t), 0) for the breathing cylinder. Legend: ——, analytical solution; ——, ǫ = 0.05;—∗—, ǫ = 0.0005.

C. Fluid-structure interaction

A slightly more interesting validation case concerns the coupled motion of a linearly elastic thin cylindrical shell inaquiescent medium. Consider the situation when the cylinder, which has a mean radiusa at equilibrium, is symmetri-cally displaced in the outward radial direction by an amountw0 as shown in Fig. 5. Under the restriction ofh/a ≪ 1and axisymmetric displacement the motion of the solid is described by linear, second order ordinary differential equa-tion whose solution depends on pressure load distribution.The pressure likewise depends on the acceleration of thecylinder’s surface as seen in the previous two examples.

As before the requirement thatw0 ≪ a lends the problem to linearization and, hence, to analytical treatment.Because the problem of interest is an initial boundary valueproblem the Laplace transform is used in lieu of theFourier transform. Details of the solution are provided in the Appendix; the relvant detail is that the early response ofthe cylinder displacement,w(t), from the equilibrium condition is given by the simple expression

w(t) = exp−tτ/2 cosβt (54)

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whereτ = [(1 − ν)2c∞ρ∞]/[(1 − 2ν)hρs], β = Cph/a, ν is Poisson’s ratio,h is the shell thickness (see Fig. 5),ρs is thedensity of the solid, andCph =

√

E/[(1 − ν2)ρs] is the wavespeed of the solid. The rate of energy transmission fromthe solid to the fluid is proportional toζ = τ/(2β) and is a fundamental quantity in the structural-acoustic response ofthe system.

The specific case under consideration is a thin circular shell of thicknessh/a = 0.01, a Poisson’s ratio ofν = 0for plane strain, a density ratio ofρs/ρ∞ = 2700, and a phase speed ratio ofCph/c∞ = 17.9. These values roughlycorrespond to an aluminum cylinder of 1 m radius. The structure will vibrate with a frequency ofβ = 970 Hz and hasa damping ratio ofζ = 0.00134. Figure 6 shows the agreement between the numerical and the analytical solution forthe displacement history of the cylinder. (The analytical solution involves a numerically inverted Laplace transformof Eq. (60) derived in the Appendix.) It is apparent that the damping rate of the two solutions is quite similar. Thenumerical estimate of the damping rate is subject to uncertainities as there are a variety of methods to determine it.Using the peak-to-peak decrement in thew(t) history shown in Fig. 6 givesζnum = 0.013± 0.005, which is consistentwith the expected value.

h

a

Displaced shell att = 0

ρ∞, c∞

w0

Figure 5: Coupled fluid-structure motion of aradially-displaced thin-shelled cylinder.

t

ww0

0

0

1

−1

−0.2

−0.4

−0.6

−0.8

0.2

0.4

0.6

0.8

0.02 0.04 0.06

Figure 6: Vibration response of a thin-shelled cylin-der immersed in a fluid. Legend: ——, analyticalsolution; ——, numerical solution.

VI. Piston motion within a channel

The last example problem to be presented is that of an accelerating piston enclosed in a one-dimensional channel.As shown in Fig. 7 a tighly fitted piston in a constant area channel (of diameterD) is moves to the right with positionhistoryX(t). Before the time any shock forms one can use the method of characteristics to determine the flow propertiesas a function ofx andt. For a fixed positionx0 the density may be expressed as a simple function of the piston motion.

When the channel is filled initially with a quiescent fluid of densityρ0 and pressurep0 is straightfoward to showthat the density at a fixed pointx0 is given by the piecewise expression

ρ(t; x0) =

ρ0 t ≤ x0/c0[

c2

γs0

]1/(γ−1)t ≥ x0/c0

(55)

wheres0 = p0ρ−γ0 andc(t) = [(γ−1)/2]X(τ)+c0 whereτ(t) is given implicitly byx0 = X(τ)+[(γ+1)/2]X(τ)+c0(t−τ).

The results from the specific case ofX(t) = [A/2]t2, A = 1/10 andx0 = 6 are given in Fig. 8 where the agreementbetween the two solutions is good.

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x

X(t)

D

Piston

Figure 7: Motion of piston through a one-dimensional channel.

0 2 4 6 80.8

1

1.2

1.4

1.6

1.8

2

tc0/D

ρ/ρ

0

Figure 8: Density response at a point due to pistonmotion. Legend: ——, analytical;, numerical.

VII. Conclusions

An extension to grids of arbitrary deformation of the characteristics-based boundary conditions was proposed. Byformulating the problem in curviliner coordinates with a time-dependent metric inclusion of the grid motion into theboundary conditions is straight forward and closely follows the structural form developed by Thompson3, 4and Poinsot& Lele.5 When the grid is rigid the present boundary conditions naturally resort to those given by Kim & Lee.12, 13 Forviscous problems the well posed method of Halpern16 was presented and carried out for the case of a subsonic viscousinflow for a zeroth order condition; further work is needed toderive the 1st order boundary condition.

Verification of the boundary conditions by five example problems with analytical solutions shows their utilityin various cases, including linearized acoustics, fully coupled fluid-structure interactions, and nonlinear waves. Thesimulations highlight the accuracy of the boundary formulation and did not exhibit numerical instability. However,since we do not have proofs of stability, via an energy norm2 or otherwise, we cannot claim these boundary condtionsare strictly stable in all cases.

A. Analytical solution of the vibration of thin-shell cylin der in a fluid

A derivation of the solution for the coupled response of a thin-shelled cylinder immersed in a quiescent fluid isgiven here. Under the assumptions listed in§ IV.C the governing equation for the solid is23

C−2ph

d2w

dt2+

w

a2=

(1− ν2)(p − p∞)Eh

(56)

whereE is the material’s Youngs modulus and we consider only the influence of the fluid outside of the cylinder. Thefluid is governed by the usual inviscid wave equation given inEq. (44). These equations are supplemented with theinitial and boundary conditions

w(0) = w0, p ≡ p∞ at t = 0 (57)

ρ∞d2wdt2+∂p∂r

∣

∣

∣

∣

∣

r=a= 0 (58)

limr→∞

√r

(

∂p∂r+

1a∞

∂p∂t

)

= 0. (59)

The last condition given is the Sommerfeld radiation condition for a two-dimensional acoustic medium.On using the Laplace transform

f (s) =

∞∫

0

f (t) exp−stdt

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the Sommerfeld condition immediately implies that ˆp(x, s) = A(s)H(2)0 (−isr/a∞) whereA(s) is an unknown function

to be determined from the remaining equations and boundary conditions. Upon combining the Laplace-transformedequations Eq. (56) and Eq. (44) and the boundary conditions,we find that the Laplace-transformed radial displacementof the solid is given by

w(s)w0=

sC2

ph+ i (1−ν2)ρ∞c∞

EhH(2)

0 (κ)

H(2)1 (κ)

s2

C2ph+ 1

a2 + is(1−ν2)ρ∞c∞C2

ph

EhH(2)

0 (κ)

H(2)1 (κ)

(60)

whereκ = −isa/c∞.The inverse Laplace transform

f (t) =1

2πi

∫

Γ

f (s) expstds

involves the Bromwich contourΓ which must lie to the right of any poles off (s). It is straightforward to verify thatEq. (60) does not have any poles in the right half of the complex s-plane thus we can deformΓ to the imaginarys axis.At present we do not have an analytical form of the time domainexpression of the radial displacementw(t), aside fromthe inverse Laplace transform, which much be performed numerically. However, the inverse Laplace transform of thelimit of w(s) ass → ∞ gives an estimate to the early time response, which is instructive. In taking thes → ∞ limitwe find that Eq. (60) simplifies to (using the expressions forτ andbeta below Eq. (54))

w(s)w0=

s + τs2 + τs + β2

(61)

which has the simple inverse Laplace transform of

w(s)w0= e−

tτ2 cos

βt

√

1−(

τ

2β

)2

+τ/(2β)

√

1−(

τ2β

)2e−(τ/2)t sin

βt

√

1−(

τ

2β

)2

. (62)

For the 1 m Aluminum shell paramaters given in IV. C we have that τ/β ≪ 1 such that the approximate early timeresponse is given by Eq. (54). Note that the analytical curvein Fig. 6 uses the numerically calculated inverse Laplacetransform of Eq. (60).

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16Halpern, L., “Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems,”SIAM J. Math. Analy., Vol. 22,No. 5, 1991, pp. 1256–1283.

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