[american institute of aeronautics and astronautics 15th aiaa/ceas aeroacoustics conference (30th...

Direct Computation of Noise Generated by Complex

Geometries Using a High-Order Immersed Boundary

Method

Roberto F. Bobenrieth Miserda ∗

Universidade de Brasılia, Brasılia, DF, 70910-900, Brasil

Rudner Lauterjung Queiroz †

Instituto Tecnologico de Aeronautica, Sao Jose dos Campos, SP, 12230-090, Brasil

Ana L. Maldonado ‡ Isadora D. Ribeiro ‡ Kerson Godoy ‡ and Oswaldo G. Neto ‡

Universidade de Brasılia, Brasılia, DF, 70910-900, Brasil

The objective of this work is the development of a fourth-order immersed boundarymethod for compressible and viscous flows and to apply it to directly compute the noiseproduced by flows over complex geometries. The unsteady and compressible Navier-Stokesequations are numerically solved using a finite volume discretization where the fluxes arecomputed using the skew-symmetric form of Ducros explicit fourth-order numerical scheme.The time marching process is achieved using a third-order Runge-Kutta scheme proposedby Shu. The immersed boundary method is based on a discrete forcing approach wherethe boundary conditions are directly imposed in the control volumes that contain the im-mersed boundary points, resulting in a sharp representation of the solid boundary since anull velocity is imposed in the boundary volumes. At these volumes, pressure and temper-ature are obtained by imposing a null value for the spatial derivatives of these variables inthe outward normal direction from the solid wall. The spatial derivatives at the boundaryvolumes are calculated with a fourth-order accuracy, and thus preserving the overall spatialaccuracy of the numerical scheme. In order to avoid the numerical oscillations resultingfrom the discrete forcing approach applied at initial conditions, a pseudo-force and its as-sociated pseudo-work are introduced in the right-hand side of the momentum and energyequations in order to gradually accelerate, using a non-inertial frame of refence, the entireflow field from the stagnation condition to the free-flow condition. Numerical results arepresented and compared with analytical results for the case of the subsonic laminar flowover a flat plate, the supersonic laminar flow over a double-wedge airfoil and the supersonicturbulent flow over a circular cylinder, with all cases showing very good agreement with thetheoretical results. Aeroacoustics results are presented, in terms of the overall sound pres-sure level (OASPL) field, for the case of the laminar subsonic flow over a circular cylinderand the turbulent subsonic flow over two circular cylinders in tandem configuration.

Nomenclature

β Inclination angle of an oblique shockβM Visualization variable based on the Mach number gradientβT Visualization variable based on the temperature gradientβ Inclination angle of an oblique shockC2 Second gas constant in Sutherland’s formulacp Constant-pressure specific heat

∗Associate Professor, Departamento de Engenharia Mecanica, [email protected], AIAA Member.†Postgraduate Student, Divisao de Engenharia Aeronautica, [email protected], AIAA Member.‡Graduate Student, Departamento de Engenharia Mecanica.

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American Institute of Aeronautics and Astronautics

15th AIAA/CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference)11 - 13 May 2009, Miami, Florida

AIAA 2009-3181

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

cv Constant-volume specific heatD Diameter of the circular cylinder∆t Time stepδij Kronecker’s delta functione Internal energy per unit massek Kinetic energy per unit masseT Total energy per unit massΦ Jameson’s sensorγ Ratio of specific heatsi Unit vector in the x-directionj Unit vector in the y-directionk Unit vector in the z-directionk Thermal conductivityL Characteristic lengthµ Dynamic viscosityM Mach numbern Unit vector in the normal directionPr Prandtl numberp Pressureqs Volumetric fluxqi Heat-flux density vectorρ DensityR Gas constantRe Reynolds numberSij Strain tensorS Surface vectorsx Component of the surface vector (x -direction)sy Component of the surface vector (y-direction)sz Component of the surface vector (z -direction)τij Stress tensorT Temperaturet Timeta Acceleration time from stagnation to free-flow conditionsU Velocity magnitudeu Component of the velocity vector (x -direction)u Velocity vectorui Component of the velocity vector (i -direction)V Volumev Component of the velocity vector (y-direction)w Component of the velocity vector (z -direction)Ψ Ducros’ sensorx First spatial coordinatey Second spatial coordinatez Third spatial coordinate

Subscript∞ Free-flow properties

Superscript∗ Dimensional property

I. Introduction

The term immersed boundary method was first used in reference to a method developed by Peskin1

to simulate cardiac mechanics and associated blood flow. The distinguishing feature of this method was

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that the entire simulation was carried out on a Cartesian grid, which did not conform to the geometry ofthe heart, and a novel procedure was formulated for imposing the effect of the immersed boundary on theflow. Since Peskin introduced this method, numerous modications and refinemets have been proposed anda number of variants of this approach now exist. Initially, the immersed boundary methods were developedfor incompressible viscous flows. Only recently, immersed boundary methods for compressible viscous flowshas been developed, and examples this type of application are the works of de Tullio2 et al., Cho3 et al.,Liu4 and Vasilyev, and Ghias5 et al. In all these work the resulting numerical schemes are second-orderaccurate in space and time. The objective of this work is to propose an immersed boundary method thatis fourth-order accurate in space and third-order accurate in time, in order to apply this methodology toaeroacoustic problems that involve complex geometries.

II. Governing Equations

The unsteady compressible Navier-Stokes equations, for a non-inertial frame of reference, can be writtenin a conservation-law form as:

∂ρ

∂t+

∂

∂xi(ρui) = 0, (1)

∂

∂t(ρui) +

∂

∂xj(ρuiuj) = − ∂p

∂xi+∂τij∂xj

+ fi, (2)

∂

∂t(ρeT ) +

∂

∂xi(ρeTui) = − ∂

∂xi(pui) +

∂

∂xi(τijuj)−

∂qi∂xi

+ fiui. (3)

where the nondimensional variables used in the above equations are defined as:

x =x∗

L∗, y =

y∗

L∗, z =

z∗

L∗, t =

t∗

L∗/U∗∞, u =

u∗

U∗∞, v =

v∗

U∗∞, w =

w∗

U∗∞,

p =p∗

ρ∗∞ (U∗∞)2, ρ =

ρ∗

ρ∗∞, T =

T ∗

T ∗∞, e =

e∗

(U∗∞)2, µ =

µ∗

µ∗∞, f =

f∗

ρ∗∞ (U∗∞)2 /L∗. (4)

The viscous stress tensor is given by

τij =1

Re∞(µSij) =

1Re∞

{µ

[(∂ui∂xj

+∂uj∂xi

)− 2

3δij∂uk∂xk

]}, (5)

where the Reynolds number is defined as

Re∞ =ρ∗∞U

∗∞L∗

µ∗∞. (6)

The total energy is given by the sum of the internal and kinetic energy as

eT = e+ ek = cvT +ui ui

2(7)

and the heat-flux density vector is

qi = − µ

(γ − 1) M2∞Re∞ Pr

(∂T

∂xi

), (8)

where the Mach and the Prandtl numbers are respectively defined as

M∞ =U∗∞a∗∞

=U∗∞√γ R∗ T ∗∞

, Pr =c∗pk∗∞

µ∗∞. (9)

In this work, the Prandtl number is considered a constant with the value 0.72. For a thermally andcalorically perfect gas, the equation of state can be written as

p = (γ − 1) ρe (10)

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and

T =γM2

∞ p

ρ. (11)

The molecular viscosity is obtained using Sutherlands formula

µ = C1T 3/2

T + C2, C1 =

[(T ∗∞)1/2

µ∗∞

]C∗1 , C2 =

C∗2T ∗∞

, (12)

where C1 and C2 are constants.In order to avoid the numerical oscillations resulting from the discrete forcing approach used by the

immersed boundary methodology, a volume pseudo-force (fi) and its associated volume pseudo-work (fiui)are introduced in the right-hand side of the momentum and energy equations in order to continuouslyaccelerate, using a non-inertial frame of reference, the entire flow field from the stagnation condition to thefree-flow condition during during the acceleration time, ta. For a free-flow velocity aligned with the Cartesianx-direction (u∞ = U∗∞i) the only component of the pseudo-force fi is

fx =f∗x

ρ∗∞ (U∗∞)2 /L∗=

ρ∗ (U∗∞/t∗a)

ρ∗∞ (U∗∞)2 /L∗=(ρ∗

ρ∗∞

)(L∗/U∗∞t∗a

)=

ρ

ta, (13)

for t 6 ta. After this acceleration time, the value of the pseudo-force fi must be zero, since the free-flowconditions are achieved, resulting in

fx = 0, (14)

for t > ta.

III. Numerical Method

In order to numerically solve the unsteady compressible Navier-Stokes equations using a finite volumeapproach, Eqs. (1), (2) and (3) are written in the vector form

∂U∂t

+∂E∂x

+∂F∂y

+∂G∂z

= R, (15)

where the vectors U, E, F, G and R are given by

U =

ρ

ρu

ρv

ρw

ρeT

, (16)

E =

ρu

ρu2 + p− τxxρuv − τxyρuw − τxz

(ρeT + p)u− uτxx − vτxy − wτxz + qx

, (17)

F =

ρv

ρvu− τxyρv2 + p− τyyρvw − τyz

(ρeT + p) v − uτxy − vτyy − wτyz + qy

(18)

G =

ρw

ρwu− τxzρwv − τyz

ρw2 + p− τzz(ρeT + p)w − uτxz − vτyz − wτzz + qz

, (19)

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R =

0fx

00fxu

. (20)

Defining tensor Π asΠ = E⊗ i + F⊗ j + G⊗ k, (21)

Eq. (15) is rewritten as∂U∂t

+∇ ·Π = R. (22)

Integrating the above equation over the control volume V , and applying the divergence theorem to the firstterm of the right-hand side results

∂

∂t

∫V

UdV = −∫V

(∇ ·Π) dV +∫V

RdV = −∫S

(Π · n) dS +∫V

RdV, (23)

Defining the volumetric mean of vectors U and R in the control volume V as

U ≡ 1V

∫V

UdV (24)

andR ≡ 1

V

∫V

RdV, (25)

respectively, Eq. (23) is written as∂U∂t

= − 1V

∫S

(Π · n)dS + R. (26)

For the volume (i, j, k), the first-order approximation of the temporal derivative is given by(∂U∂t

)i,j,k

=∆Ui,j,k

∆t+ O (∆t) , (27)

and the temporal approximation of Eq. (26) for a hexahedral control volume is

∆Ui,j,k = − ∆tVi,j,k

[∫Si+1/2

(Π · n) dS +∫Si−1/2

(Π · n) dS+∫Sj+1/2

(Π · n) dS +∫Sj−1/2

(Π · n) dS+

∫Sk+1/2

(Π · n) dS +∫Sk−1/2

(Π · n) dS

]+ ∆tR, (28)

where Si+1/2, Si−1/2, Sj+1/2, Sj−1/2, Sk+1/2 e Sk−1/2 are the surfaces that define the hexahedral controlvolume and Si+1/2 is the common surface between volume (i, j, k) and volume (i+ 1, j, k).

Considering that the value of tensor Π is constant over the control surfaces, it is possible to defineF(U)i,j,k as a function of the flux of tensor Π over the control surfaces as

F(U)i,j,k = − ∆tVi,j,k

[(Π · S)i+1/2 + (Π · S)i−1/2+

(Π · S)j+1/2 + (Π · S)j−1/2+

(Π · S)k+1/2 + (Π · S)k−1/2

]+ ∆tR, (29)

and the resulting spatial approximation of Eq. (28) is

∆Ui,j,k = F(U)i,j,k

+D(U)i,j,k

(30)

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where D(U)i,j,k is an explicit artificial dissipation. In order to calculate F(U)i,j,k, the flux of tensor Π overthe control surfaces must be calculated. For the control surface Si+1/2, this flux is given by

(Π · S)i+1/2 =

(Π · S)1(Π · S)2(Π · S)3(Π · S)4(Π · S)5

i+1/2

. (31)

The first component of the vector defined by the above equation is associated to the continuity equationand given by

(Π · S)1 = ρi+1/2 (qs)i+1/2 , (32)

where the volumetric flux is

(qs)i+1/2 = ui+1/2 · Si+1/2 = ui+1/2 (sx)i+1/2 + vi+1/2 (sy)i+1/2 + wi+1/2 (sz)i+1/2 . (33)

The second, third, and fourth components are associated to the three components of the momentum equationand are respectively given by

(Π · S)2 = (ρu)i+1/2 (qs)i+1/2 + pi+1/2 (sx)i+1/2 −[µi+1/2 (Sxx)i+1/2

](sx)i+1/2 −

[µi+1/2 (Sxy)i+1/2

](sy)i+1/2 −[

µi+1/2 (Sxz)i+1/2

](sz)i+1/2 , (34)

(Π · S)3 = (ρv)i+1/2 (qs)i+1/2 + pi+1/2 (sy)i+1/2 −[µi+1/2 (Sxy)i+1/2

](sx)i+1/2 −

[µi+1/2 (Syy)i+1/2

](sy)i+1/2 −[

µi+1/2 (Syz)i+1/2

](sz)i+1/2 , (35)

and,

(Π · S)4 = (ρw)i+1/2 (qs)i+1/2 + pi+1/2 (sz)i+1/2 −[µi+1/2 (Sxz)i+1/2

](sx)i+1/2 −

[µi+1/2 (Syz)i+1/2

](sy)i+1/2 −[

µi+1/2 (Szz)i+1/2

](sz)i+1/2 . (36)

The fifth component is associated with the energy equation and given by

(Π · S)5 = (ρeT )i+1/2 (qs)i+1/2 + pi+1/2 (qs)i+1/2 − ui+1/2 (sx)i+1/2

[µi+1/2 (Sxx)i+1/2

]−

vi+1/2 (sy)i+1/2

[µi+1/2 (Syy)i+1/2

]− wi+1/2 (sz)i+1/2

[µi+1/2 (Szz)i+1/2

]−[

ui+1/2 (sy)i+1/2 + vi+1/2 (sx)i+1/2

] [µi+1/2 (Sxy)i+1/2

]−[

vi+1/2 (sz)i+1/2 + wi+1/2 (sy)i+1/2

] [µi+1/2 (Syz)i+1/2

]−[

ui+1/2 (sz)i+1/2 + wi+1/2 (sx)i+1/2

] [µi+1/2 (Sxz)i+1/2

]−[

ki+1/2 (∂T/∂x)i+1/2

](sx)i+1/2 −

[ki+1/2 (∂T/∂y)i+1/2

](sy)i+1/2 −[

ki+1/2 (∂T/∂z)i+1/2

](sz)i+1/2 . (37)

In order to calculate the flux (Π · S) according to Eqs. (32) to (37), it is necessary to approximate thevalues of the variables at the control surface Si+1/2 from the mean values of the conservative variables in

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the control volumes, given by the vector

Ui,j,k =

ρ

ρu

ρv

ρw

ρeT

i,j,k

, (38)

In order to obtain the momentum and energy primitive variables, the Favre mean is used to calculatethe mass-averaged momentum and energy primitive variables as

u =ρu

ρ, v =

ρv

ρ, w =

ρw

ρ, eT =

ρeTρ. (39)

The mean of the total energy is given by

eT = e+ ek = e+uu+ vv + ww

2. (40)

Since it is not possible to directly calculate the mass-averaged kinetic energy, given by the second term ofthe right-hand side of the above equation, the internal energy is calculated as

e = eT − ek = eT −u u+ v v + w w

2, (41)

and the mean of the thermodynamic variables p, T , µ e k are calculated as

p = (γ − 1) ρ e, T =γM2 p

ρ, µ = C1

T 3/2

T + C2and k = −

µ

(γ − 1) M2 Re∞ Pr. (42)

It is important to note that the first terms in the right-hand side of Eqs. (32), (34), (35), (36), and (37)are the fluxes of mass, momentum and total energy through surface Si+1/2 and the other terms are fluxesthat are functions of the right-hand sides of the momentum and total energy equations. In order to evaluateall this terms at that surface, in this work is used the fourth-order skew-symmetric scheme proposed byDucros7 et al. where

ui+1/2 =23

(ui + ui+1)− 112

(ui−1 + ui + ui+1 + ui+2) (43)

for the primitive variables, exemplified in the above equation by the x−direction component of the velocity,and where

(ρu)i+1/2 =13(ρi + ρi+1

)(ui + ui+1)−

124(ρi−1ui−1 + ρi+1ui−1 + ρiui + ρi+2ui + ρi+1ui+1+

ρi−1ui+1 + ρi+2ui+2 + ρiui+2

)+

13

[12(ρi+1ui+1 + ρiui

)− 1

4(ρi+1 + ρi

)(ui+1 + ui)

]. (44)

for the conservative variables, also exemplified by the x−direction component of the specific momentum.The scheme proposed by Eqs. (43) and (44) is a centered one, and therefore, an explicit artificial viscosity

was previously included in Eq. (30). In order to enhance the numerical method with shock-capturingcapabilities and the ability to cope with steep gradient regions, this artificial dissipation uses the basic ideaproposed by Jameson8 et al. given by

D(U) = [di+1/2(U)− di−1/2(U)] + [dj+1/2(U)− dj−1/2(U)] + [dk+1/2(U)− dk−1/2(U)], (45)

wheredi+1/2(U) = ε

(2)i+1/2[Ui+1 −Ui]− ε(4)i+1/2[Ui+2 − 3Ui+1 + 3Ui −Ui−1]. (46)

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The first and second terms of Eq. (46) are second-order and fourth-order dissipation operators, respec-tively. The first one acts in the shock and the second one acts over steep gradient regions, like the viscousregions. The coefficients of Eq. (46) are given by

ε(2)i+1/2 = K (2)max (ΨiΦi,Ψi+1Φi+1) , ε

(4)i+1/2 = max

[0,(

K (4) − ε(2)i+1/2

)], (47)

whereK (2) = 1/4, K (4) = 1/256. (48)

Sensors Ψi and Φi are given by

Ψi =|pi+1− 2p

i+ p

i−1|

|pi+1|+ |2p

i|+ |p

i−1|, Φi =

(∇ · u)2

(∇ · u)2 + |∇ × u|2 + ε, ε = 10−30. (49)

The sensor Ψi, proposed by Jameson8 et al., is pressure-based and it is intended to detect shock waves. Thefunction of sensor Φi, proposed by Ducros6 et al., is to inhibit sensor Ψi in regions were the divergent is low,but the rotational of the velocity field is high, like a pure vortex wake. In regions were the divergent andthe rotational are high, like the vortex-shock interaction, the inhibiting capacity of sensor Φi decreases.

In order to advance Eq. (30) in time, a third-order Runge-Kutta is used as proposed by Shu and reportedby Yee.9 This yield to the following three steps:

U1

= Un −

[F(Un)−D

(Un)], (50)

U2

=34Un

+14U

1 − 14

[F(U

1)−D

(U

1)], (51)

Un+1

=13Un

+23U

2 − 23

[F(U

2)−D

(U

2)]. (52)

As used in this work, the resulting numerical method is fourth-order accurate in space and third-orderaccurate in time.

This numerical method has been applied successfully in the solution of the problems of the numerical sim-ulation of vortex-shock interactions in laminar flows,10 unsteady aerodynamic forces over circular cylindersin transonic flow,11 the effect of plunging and pitching motions over an airfoil in transonic laminar flow,12

and the effect of the plunging velocity over an airfoil in subsonic laminar flow.13 It was also recently appliedin the direct computation of the noise generated by subsonic, transonic, and supersonic cavity flows.14

IV. Immersed-Boundary Technique

The approach used in this work for imposing the boundary conditions at the boundary volumes, definedas the control volumes that contain one or more surface-grid points, is a discrete forcing one where theboundary conditions are directly imposed directly to the boundary volumes. In all the control volumes, themean values of the conservative variables are given by

Ui,j,k =

ρ

ρ u

ρ v

ρ w

ρ eT

i,j,k

. (53)

In the boundary volumes, the no-slip condition directly results in the boundary values

u = v = w = 0. (54)

Since the total energy is the sum of the internal and kinetic energy, the application of the no-slip conditionin Eq. (41) results in

eT = e, (55)

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bearing, for the boundary volumes

Ub

i,j,k =

ρ

000e

b

i,j,k

, (56)

where the superscript b indicates that the finite volume (i, j, k) is a boundary volume. It is important tonote that the number of boundary volumes is less or equal to the number of surface points, since more thanone surface points can lie within one boundary volume.

In order to obtain the boundary values for the density, ρ, and the internal energy, e, the averaged equationof state,

p =(

1γM2∞

)ρ T , (57)

is derived in the normal outward direction from the solid wall. With this objective, it is defined n as a unitvector with a direction that is normal to the wall with outward sense, where the Cartesian components aren = nxi + nyj + nzk and the magnitude is |n| =

√n2x + n2

y + n2y = 1. Depending on the resolution of the

Cartesian and surface grids, more than one surface point can lie within a boundary volume, and in this caseit is used the mean among all normal unit vectors associated to the grid points that lie within the boundaryvolume. With the normal direction defined by one unit vector in the boundary volume or by an averagedunit vector over the boundary volume, the derivative in this direction is given by

∂p

∂n=(

1γM2∞

)∂

∂n(ρ T ) =

(1

γM2∞

)(ρ∂T

∂n+ T

∂ρ

∂n

). (58)

For an adiabatic wall, ∂T/∂n = 0, and considering the boundary-layer approximation, ∂p/∂n = 0, Eq. (58)yields

∂ρ

∂n= 0, (59)

and sincee =

1γ(γ − 1)M2

∞T , (60)

the adiabatic wall condition results in∂e

∂n= 0. (61)

Defining n as a unit vector with a direction that is normal to the solid wall and a sense that is outwardwith Cartesian components n = nxi + nyj + nzk, the derivates of the averaged density and internal energyare written as

∂ρ

∂n=∂ρ

∂x

∂x

∂n+∂ρ

∂y

∂y

∂n+∂ρ

∂z

∂z

∂n= nx

∂ρ

∂x+ ny

∂ρ

∂y+ nz

∂ρ

∂z(62)

and∂e

∂n=∂e

∂x

∂x

∂n+∂e

∂y

∂y

∂n+∂e

∂z

∂z

∂n= nx

∂e

∂x+ ny

∂e

∂y+ nz

∂e

∂z. (63)

For the boundary volumes, Eqs. (59) and (61) apply and result in

0 = nx

(∂ρ

∂x

)bi,j,k

+ ny

(∂ρ

∂y

)bi,j,k

+ nz

(∂ρ

∂z

)bi,j,k

(64)

and

0 = nx

(∂e

∂x

)bi,j,k

+ ny

(∂e

∂y

)bi,j,k

+ nz

(∂e

∂z

)bi,j,k

. (65)

If nx > 0, in regular region of the Cartesian grid the derivative ∂ρ/ ∂x in the boundary volumes can becalculated with fourth-order spatial precision using a forward finite-difference approach as(

∂ρ

∂x

)bi,j,k

=1

12∆x[−25ρ bi,j,k + 48ρi+1,j,k − 36ρi+2,j,k + 16ρi+3,j,k − 3ρi+4,j,k +O(∆x)4

]. (66)

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Defining the difference operator

D+i ρ =

125(48ρi+1,j,k − 36ρi+2,j,k + 16ρi+3,j,k − 3ρi+4,j,k

), (67)

Eq. (66) is written as (∂ρ

∂x

)i,j,k

=25

12∆x[−ρ bi,j,k +D+

i ρ+O(∆x)4]. (68)

If n = i (nx = 1, ny = 0 and nz = 0), Eq. (64) gives

0 =(∂ρ

∂x

)bi,j,k

, (69)

and introducing this result in Eq. (68) yields

ρ bi,j,k = D+i ρ+O(∆x)4. (70)

Following the same line of reasoning, if n = j (nx = 0, ny = 1 and nz = 0),

ρ bi,j,k = D+j ρ+O(∆y)4, (71)

and if n = k (nx = 0, ny = 0 and nz = 1),

ρ bi,j,k = D+k ρ+O(∆z)4. (72)

For the generalized case, where n = nxi + nyj + nzk, the averaged density is calculated in the boundaryvolumes as the weighted value

ρ bi,j,k =|nx|Diρ+ |ny|Djρ+ |nz|Dkρ

|nx|+ |ny|+ |nz|. (73)

Following an analogous procedure, since ∂ρ/∂n = ∂e/∂n = 0, the averaged internal energy is calculated asthe weighted value

e bi,j,k =|nx|Die+ |ny|Dje+ |nz|Dke

|nx|+ |ny|+ |nz|, (74)

where the difference operators (Di, Dj and Dk) can be in the forward direction (D+i , D+

j and D+k ), if the

values of nx, ny and nz are positive, or in the backward direction (D−i , D−j and D−k ), if the values of nx,ny and nz are negative. For the case of the averaged density, the operator D+

i is given by Eq. (67), and theother forward and backward difference operators are given by

D+j ρ =

125(48ρi,j+1,k − 36ρi,j+2,k + 16ρi,j+3,k − 3ρi,j+4,k

), (75)

D+k ρ =

125(48ρi,j,k+1 − 36ρi,j,k+2 + 16ρi,j,k+3 − 3ρi,j,k+4

), (76)

D−i ρ =125(48ρi−1,j,k − 36ρi−2,j,k + 16ρi−3,j,k − 3ρi−4,j,k

), (77)

D−j ρ =125(48ρi,j−1,k − 36ρi,j−2,k + 16ρi,j−3,k − 3ρi,j−4,k

), (78)

D−k ρ =125(48ρi,j,k−1 − 36ρi,j,k−2 + 16ρi,j,k−3 − 3ρi,j,k−4

). (79)

In this manner, the conservative variables vector for the boundary volumes is given by

Ub

i,j,k =

ρ bi,j,k

000

e bi,j,k

, (80)

where the first and last components are given by Eqs. (73) and (74), respectively.

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V. Numerical Results

V.A. Subsonic Laminar Flow over a Flat Plate

Since the incompressible laminar flow over a flat plate has the Blasius15 analytical solution, it is interestingto analyze this flow in order to compare the boundary layer thickness and the velocity profiles given by theimmersed boundary methodology with Blasius’ analytical solution. The case studied consists in a subsoniclaminar flow with a Reynolds number of 2.5 × 102 in order to avoid the transition to turbulence along thewhole flat-plate length, and a Mach number of 0.1 in order to avoid compressibility effects.

The Cartesian grid has 840,000 volumes overall, being 480,000 volumes at the uniform region only, whilethe surface grid has 2,029 points that define the flat plate geometry. For this geometry, the thickness tolength ratio is 0.014, where 14 control volumes where used to resolve the flat-plate thickness, as shownis figure 1, since the immersed boundary method is applied to fourth-order numerical scheme that has afive-element stencil.

Figure 1. Grid refinement at the leading edge of the flat plate, showing the Cartesian grid (grey) and the surface grid(black).

According to the Blasius solution, the boundary-layer thickness is given by

δ(x)x

=4.99√Rex

, (81)

represented by the white dashed line in figure 2. The contour variable represented in this figure is

βM = |∇M |120 . (82)

Figure 2. Subsonic laminar boundary layer over a flat plate, where the plotted variable is βM (Eq. (82)). The tracedwhite line in the upper side represents the the boundary layer thickness given by the Blasius solution(Eq. (81)).

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This variable is a good visualization option for this case, because the magnitude of the Mach numbergradient clearly shows the limits of the regions of strong velocity gradients associated to the boundary layer.Figure 2 shows a very good agreement between the calculated boundary-layer thickness and the analyticalsolution, also showing a very sharp definition of the solid boundary. This is also the case for figure 3, thatshows the comparison between the numerical results, obtained at postion x/L = 0.5, and Blasius’ analyticalsolution for the velocity profile. It is important to notice that the surface grid is located at y/L = 0.008,were the numerical solution gives a zero value for the velocity, and sharply simulating the solid wall. This isalso the case for the points under the surface grid (y/L < 0.008).

Figure 3. Velocity profile, obtained at position x/L = 0.5, given by the immersed boundary method (squares) andBlasius’ analytical solution (continuous line). The flat-plate surface is located at y/L = 0.008.

V.B. Supersonic Laminar Flow over a Double-Wedge Airfoil

In this case, the laminar flow over a double-wedge profile is studied, in order to compare the oblique-shock propierties obtained using the immersed boundary method with the analytical results reported byAnderson.16 The double-wedge airfoil has a maximum thickness of 12% of the chord, with a Reynoldsnumber of 2.8×102, based on the chord length, and a free-flow Mach number of 2. The level of refinement ofthe Cartesian grid is presented at figure 4, where the horizontal axis represents the nondimensional length,based on the chord of the profile. The Cartesian grid has 1,890,000 volumes overall, being 1,600,000 volumesat the uniform region while the surface grid has 2017 points that define the geometry.

Figure 4. Grid refinement at the leading edge of the diamond wedge airfoil.

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According to the θ − β −M relation,16

tan θ = 2 cotβ[

M2 sin2 β − 1M2(γ + cos 2β) + 2

], (83)

for a free-flow Mach number M = 2 and an oblique-shock angle β = 45.8◦, there should be, just after theshock, a flow-deflection angle θ = 15.3◦. For this case, the immersed boundary method gives a deflectionangle θ = 14.9◦. Figure 5 presents this comparison, being the black dashed line the theoretical value for theflow-deflection angle θ (Eq. (83)) and the blue continuous line the numerical stream line.

Figure 5. Mach number levels over the diamond wedge airfoil. The black dashed line represents de theoretical flowdeflection after the oblique shock and the continuous blue line is the calculated streamline.

V.C. Supersonic Turbulent Flow over a Circular Cylinder

For the case of the supersonic turbulent flow, a detached shock wave is formed upstream of the circularcylinder. It is known16 that the curved shock goes through all possible conditions allowed for oblique shocksfor a given upstream Mach number. After the shock wave, the upstream flow will be locally deflected byan angle θ that corresponds to a local inclination angle β of the detached shock wave, were the theoreticalθ − β −M relation is given by Eq. (83). This relation allows the comparison of the local oblique-shockproperties given by the later equation with the numerical results given by the immersed boundary method.

Figure 6 corresponds to a free-flow Mach number of 2,5 and to a Reynolds number of 5.5 × 107. Thelevel of refinement of the Cartesian grid is 1,000 control volumes over the diameter, resulting in 25,000,000volumes overall, being 16,000,000 volumes in the 4D× 4D uniform region, while the surface grid has 10,000points that define the geometry.

Only the regular 4D × 4D region of the Cartesian grid is shown if figure 6, that presents the levels ofvariable βT , given by

βT = |∇T |120 . (84)

This variable has been chosen because it allows a simultaneous visualization of the detached shock wave, theexpansion waves, the solid surface, the boundary-layer separation points, the shear layers, the recirculatingregion and the shock waves formed after the shear layers reattachment point. In this figure it is possible tovisualize that the internal flow has no influence on the external flow, since the boundary condition imposedusing a discrete forcing approach acts as solid wall for the external flow and as an external boundary conditionfor the internal flow.

Figure 7 presents the comparison between the theoretical solution represented by the θ− β−M relationalong the detached shock wave and the corresponding values calculated using the immersed boundary method.

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Figure 6. Visualization of the supersonic flow over a circular cylinder using variable βT .

Figure 7. Local oblique shock properties along the detached shock wave. The continuous line represents the theoreticallocal oblique-shock relations given by the θ − β −M diagram for M∞ = 2, 5, and the squares represents the numericalvalues obtained using the immersed boundary method.

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V.D. Subsonic Laminar Flow over a Circular Cylinder

This case was selected following the work of Inoue17 and Hatekayama, where the sound generated by alaminar flow around a two dimensional circular cylinder is studied numerically. For this case, the Machnumber is 0.2 and the Reynolds number is 150. The cartesian grid has 25, 000, 000 control volumes overall,with 100, 000, 000 degrees of freedom. The 16D × 16D regular region is compound of 16, 000, 000 controlvolumes. The grid is large enough to capture acoustic wave propagation but has enough resolution (250control volumes over the diameter) to physically characterize the sound generation phenomena.

Figure 8 presents the vorticity field were the solid lines represent positive vorticity and the dashed linesrepresent negative values. Figure 9 presents the visualization of the aeroacoustic variable βT , defined byEq. (84), that in this case is also capable of showing simultaneously the acoustic wave propagation and VonKarman’s vortex street.

Figure 8. Vorticity field. Solid lines represent positive vorticity and dashed lines represent negative values.

Figure 9. Visualization of the aeroacoustic variable βT , defined by Eq. (84).

The overall sound pressure level (OASPL) field is presented in figure 10. This figure illustrates a directnoise computation in the regular region of the cartesian grid using a high-order immersed boundary methodthat results in a sharp definition of the solid boundary. The quasi-symmetric nature of the OASPL fieldshows the level of time development achieved for the aeroacoustic field.

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Figure 10. Contour levels of the overall sound pressure level (OASPL).

Figure 11 presents a semi-logarithmic plot of the overall sound pressure level as a function of the ra-dius along the positive vertical direction above the cylinder. The continuous line represents a polynomialapproximation of the numerical results given by

SPL(r) = 140.53− 16.69log(r), (85)

and this type of logarithmic relation is consistent with the results obtained by Inoue17 and Hatekayama.

Figure 11. Semi-logarithmic plot of the overall sound pressure level as a funcion of the radius along the positive verticaldirection above the cylinder.

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V.E. Subsonic Turbulent Flow over Two Circular Cylinders in Tandem Configuration

The flow around two circular tandem cylinders is very interesting case for an aeroacoustic evaluation of theimmersed boundary approach, since it can be treated as a first two-dimensional approximation of a flowaround an aeronautic landing gear. In addition, this case becomes an intermediate step towards a flowover high-lift devices, specially the flow over slated profiles. The case simulated corresponds to a subsoniccompressible flow, with a Reynolds number of 3,000,000, based on the cylinders diameter, and a free-flowMach number of 0.166. The cylinders are 4 diameters spaced, from center to center.

The surface grid has 2000 points defining the geometry of the cylinders, being 1000 points for eachcylinder. The Cartesian grid has 2,310,000 volumes overall, being 2,000,000 only at the uniform region ofthe domain (see figure 12). This resolution of the grid corresponds to a discretization of 125 volumes overthe cylinder diameter (see figure 13).

Figure 12. Cartesian grid (gray) and surface gird (black) for the two cylinders in tandem.

Figure 13. Detail of the Cartesian grid (gray) and surface grid (black) for the first cylinder in tandem.

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Figure 14 presents the instantaneous countour levels of βT , given by Eq. (84), over the region where theCartesian grid is regular. Using this variable, it is possible to see the strong destabilizing effect that thevortex wake of the first cylinder has over the second one, associated to the spreading of at least 8 diametersalong the orthogonal direction of the vortex wake of the second cylinder. This spreading effect is also presentin figure 15.

Figure 14. Instantaneous levels of variable βT , given by Eq. (84).

Figure 15 presents the overall sound pressure levels (OASPL) calculated using the immersed boundarymethod proposed in this work. It is important to note that this approach produces a very sharp definition ofthe boundary, even for an aeroacoustic variable as the OASPL, since there is a strong discontinuity betweenthe OASPL values in the interior and the exterior of the two cylinders.

Figure 15. Contour levels of the overall sound pressure level (OASPL).

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Figure 16 presents the effects of an increment of the spatial resolution, from 125 to 400 control volumesover the diameter. The instantaneous levels of βT , given by Eq. (84), are presented. A comparison withfigure 14, with resolution of 125 control volumes over the diameter, shows a more detailed simulation of theaeroacoustic field.

Figure 16. Instantaneous levels of variable βT , given by Eq. (84).

VI. Conclusion

A finite-volume fourth-order immersed boundary method is proposed in this work. The numerical resultsobtained for the problems of the subsonic laminar boundary layer over a flat plate, the oblique shock in thesupersonic laminar flow over a double-wedge airfoil and the detached shock in the supersonic turbulent flowover a circular cylinder show, for all cases, an excellent agreement with the available analytical results. Afterthis validation, this immersed boundary method is applied to the direct computation of the noise generatedby the laminar subsonic flow over a circular cylinder and by the turbulent subsonic flow over two cylindersin tandem configuration, where a very sharp definition of the solid boundary can be observed in the OASPLfields.

Acknowledgments

The first author wants to thanks Prof. Aristeu da Silveira Neto, from the Universidade Federal deUberlandia, Brazil, for the fruitfull scientific discussions over the immersed boundary methodology thatwhere the cornerstone of this work. The authors also want to thank Silicon Graphics, Inc. of Brazil for thelending of a computer server that was used to develop part of this work.

References

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