[American Institute of Aeronautics and Astronautics 17th AIAA/CEAS Aeroacoustics Conference (32nd...

A Moving-Body High-Order Immersed Boundary

Method for Computational Aeroacoustics

Roberto F. Bobenrieth Miserda �

Ana L. Maldonado y

Braulio Gutierrez z

Universidade de Bras��lia, Bras��lia, DF, 70910-900, Brasil

The objective of this work is the development of a moving-body fourth-order immersedboundary method for compressible and viscous ows and to apply it to directly compute therotor-stator interaction noise in turbofans. The unsteady and compressible Navier-Stokesequations are numerically solved using a �nite volume discretization where the uxes 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 theimmersed boundary points, resulting in a sharp representation of the solid boundary sincethe prescribed surface velocity is directly imposed in the boundary volumes. At thesevolumes, pressure and temperature are obtained by imposing a null value for the spatialderivatives of these variables in the outward normal direction from the solid wall. Thespatial derivatives at the boundary volumes are calculated with a fourth-order accuracy,and thus preserving the overall spatial accuracy of the numerical scheme. In order toavoid the numerical oscillations resulting from the discrete forcing approach applied atinitial conditions, a pseudo-force and its associated pseudo-work are introduced in theright-hand side of the momentum and energy equations in order to gradually accelerate,using a non-inertial frame of refence, the entire ow �eld from the stagnation condition tothe free- ow condition. The numerical results obtained for the rotor cascades show a verysharp de�nition of the moving surface for both the forced and induced ows. In the forcedcase the pressure gradient is favorable, since the ow through the cascade is forced by anexternal body force. On the other hand, the pressure gradient is adverse in the inducedcase, since the ow is induced solely by the rotor motion and the pressure rises due to thework done over the uid by the moving surfaces. In both cases, the pressure signal spectrashows clearly the blade-passing frequency and its harmonics.

�Associate Professor, Departamento de Engenharia Mecanica, [email protected], AIAA Member.yGraduate Student, Departamento de Engenharia Mecanica, [email protected], AIAA Member.zGraduate Student, Departamento de Engenharia Mecanica., [email protected], AIAA Member

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American Institute of Aeronautics and Astronautics

17th AIAA/CEAS Aeroacoustics Conference (32nd AIAA Aeroacoustics Conference)05 - 08 June 2011, Portland, Oregon

AIAA 2011-2753

Copyright © 2011 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Nomenclature

� Inclination angle of an oblique shock�M Visualization variable based on the Mach number gradient�T Visualization variable based on the temperature gradient� Inclination angle of an oblique shockC2 Second gas constant in Sutherland’s formulacp Constant-pressure speci�c heatcv Constant-volume speci�c heatD Diameter of the circular cylinder�t Time step�ij Kronecker’s delta functione Internal energy per unit massek Kinetic energy per unit masseT Total energy per unit mass� Jameson’s sensor Ratio of speci�c heatsi Unit vector in the x-directionj Unit vector in the y-directionk Unit vector in the z-directionk Thermal conductivityL Characteristic length� Dynamic viscosityM Mach numbern Unit vector in the normal directionPr Prandtl numberp Pressureqs Volumetric uxqi Heat- ux density vector� 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)�ij Stress tensorT Temperaturet Timeta Acceleration time from stagnation to free- ow 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)x First spatial coordinatey Second spatial coordinatez Third spatial coordinate

Subscript1 Free- ow properties

Superscript� Dimensional property

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I. Introduction

The term immersed boundary method was �rst used in reference to a method developed by Peskin1 tosimulate cardiac mechanics and associated blood ow. The distinguishing feature of this method was thatthe entire simulation was carried out on a Cartesian grid, which did not conform to the geometry of theheart, and a novel procedure was formulated for imposing the e�ect of the immersed boundary on the ow.Since Peskin introduced this method, numerous modi�cations and re�nemets have been proposed and anumber of variants of this approach now exist. Initially, the immersed boundary methods were developed forincompressible viscous ows. Only recently, immersed boundary methods for compressible viscous ows hasbeen developed, and examples this type of application are the works of de Tullio2 et al., Cho3 et al., Liu4 andVasilyev, and Ghias5 et al. In all these work the resulting numerical schemes are second-order accurate inspace and time. Bobenrieth6 et al. proposed an immersed boundary method that is fourth-order accurate inspace and third-order accurate in time and applied this methodology to aeroacoustic problems that involvecomplex geometries. The objective of this work is to extend the later work for moving boundaries with theintent of directly compute the rotor-stator interaction noise in turbofans.

II. Governing Equations

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

@�

@t+

@

@xi(�ui) = 0; (1)

@

@t(�ui) +

@

@xj(�uiuj) = � @p

@xi+@�ij@xj

+ fi; (2)

@

@t(�eT ) +

@

@xi(�eTui) = � @

@xi(pui) +

@

@xi(�ijuj)�

@qi@xi

+ fiui: (3)

In the same way, the unsteady nonlinear Euler equations can be written as:

@�

@t+

@

@xi(�ui) = 0; (4)

@

@t(�ui) +

@

@xj(�uiuj) = � @p

@xi+ fi; (5)

@

@t(�eT ) +

@

@xi(�eTui) = � @

@xi(pui) + fiui: (6)

where the nondimensional variables used in the above equations are de�ned as:

x =x�

L�; y =

y�

L�; z =

z�

L�; t =

t�

L�=U�1; u =

u�

U�1; v =

v�

U�1; w =

w�

U�1;

p =p�

��1 (U�1)2 ; � =

��

��1; T =

T �

T �1; e =

e�

(U�1)2 ; � =

��

��1; f =

f�

��1 (U�1)2=L�

: (7)

The viscous stress tensor is given by

�ij =1

Re1(�Sij) =

1

Re1

��

��@ui@xj

+@uj@xi

�� 2

3�ij@uk@xk

��; (8)

where the Reynolds number is de�ned as

Re1 =��1U

�1L�

��1: (9)

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

eT = e+ ek = cvT +ui ui

2(10)

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and the heat- ux density vector is

qi = � �

( � 1) M21Re1 Pr

�@T

@xi

�; (11)

where the Mach and the Prandtl numbers are respectively de�ned as

M1 =U�1a�1

=U�1p R� T �1

; Pr =c�pk�1

��1: (12)

In this work, the Prandtl number is considered a constant with the value 0.71. For a thermally andcalorically perfect gas, the equation of state can be written as

p = ( � 1) �e (13)

and

T = M2

1 p

�: (14)

The molecular viscosity is obtained using Sutherland’s formula

� = C1T 3=2

T + C2; C1 =

"(T �1)

1=2

��1

#C�1 ; C2 =

C�2T �1

; (15)

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 ow �eld from the stagnation condition to thefree- ow condition during during the acceleration time, ta. For a free- ow velocity U1, the components ofthe pseudo-force f are given by Eq.(16), Eq.(17) and Eq.(18).

fx =f�x

��1 (U�1)2=L�

=�� (U�1 sin(�=2� �) cos(�)=t�a)

��1 (U�1)2=L�

=

��

��1

��L�=U�1t�a

�sin(�=2� �) cos(�)

=� sin(�=2� �) cos(�)

ta(16)

fy =f�y

��1 (U�1)2=L�

=�� (U�1 sin(�=2� �) sin(�)=t�a)

��1 (U�1)2=L�

=

��

��1

��L�=U�1t�a

�sin(�=2� �) sin(�)

=� sin(�=2� �) sin(�)

ta(17)

fz =f�z

��1 (U�1)2=L�

=�� (U�1 cos(�=2� �)=t�a)

��1 (U�1)2=L�

=

��

��1

��L�=U�1t�a

�cos(�=2� �)

=� sin(�)

ta(18)

For t 6 ta, where � is the angle between the projection of U1 in the z-plane and the x-axis (attack angle)and � is the angle between the projection of U1 in the y-plane and the x-axis (side slip angle) After this

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acceleration time, the value of the pseudo-force fi must be zero, since the free- ow conditions are achieved,resulting, for t > ta in

fx = 0; fy = 0; fz = 0: (19)

III. Numerical Method

In order to be numerically solved, the governing equations are written in the following vector form:

@U

@t+@E

@x+@F

@y+@G

@z= R; (20)

De�ning tensor � as� = E i + F j + G k; (21)

Eq. (20) is rewritten as@U

@t+r �� = R: (22)

For the Navier-Stokes equations, the vectors U, E, F, G and R are given by

U =

2666664�

�u

�v

�w

�eT

3777775 ; (23)

E =

2666664�u

�u2 + p� �xx�uv � �xy�uw � �xz

(�eT + p)u� u�xx � v�xy � w�xz + qx

3777775 ; (24)

F =

2666664�v

�vu� �xy�v2 + p� �yy�vw � �yz

(�eT + p) v � u�xy � v�yy � w�yz + qy

3777775 ; (25)

G =

2666664�w

�wu� �xz�wv � �yz

�w2 + p� �zz(�eT + p)w � u�xz � v�yz � w�zz + qz

3777775 ; (26)

and

R =

26666640

fx

fy

fz

fxu+ fyv + fzw

3777775 : (27)

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For the Euler equations,the vectors U, E, F, G and R are given by

U =

2666664�

�u

�v

�w

�eT

3777775 ; (28)

E =

2666664�u

�u2 + p

�uv

�uw

(�eT + p)u

3777775 ; (29)

F =

2666664�v

�vu

�v2 + p

�vw

(�eT + p) v

3777775 ; (30)

G =

2666664�w

�wu

�wv

�w2 + p

(�eT + p)w

3777775 ; (31)

and

R =

26666640

fx

fy

fz

fxu+ fyv + fzw

3777775 : (32)

Integrating Eq.(22) over the control volume V , and applying the divergence theorem to the �rst term ofthe right-hand side results

@

@t

ZV

UdV = �ZV

(r ��) dV +

ZV

RdV = �ZS

(� � n) dS +

ZV

RdV: (33)

De�ning the volumetric mean of vectors U and R in the control volume V as

U � 1

V

ZV

UdV (34)

and

R � 1

V

ZV

RdV; (35)

respectively, Eq. (33) is written as@U

@t= � 1

V

ZS

(� � n)dS + R: (36)

Evaluating Eq. (36) for a hexahedral control volume, Eq. (37) is obtained�@U

@t

�i;j;k

= � 1

Vi;j;k

"ZSi+1=2

(� � n) dS +

ZSi�1=2

(� � n) dS+

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ZSj+1=2

(� � n) dS +

ZSj�1=2

(� � n) dS+

ZSk+1=2

(� � n) dS +

ZSk�1=2

(� � n) dS

#+ �tR; (37)

where Si+1=2; Si�1=2; Sj+1=2; Sj�1=2; Sk+1=2 e Sk�1=2 are the surfaces that de�ne 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 � is constant over the control surfaces, it is possible to de�neF(U)i;j;k as a function of the ux of tensor � over the control surfaces as

F(U)i;j;k = � 1

Vi;j;k

h(� � S)i+1=2 + (� � S)i�1=2+

(� � S)j+1=2 + (� � S)j�1=2+

(� � S)k+1=2 + (� � S)k�1=2

i+ R; (38)

and the resulting spatial approximation of Eq. (37) is�@U

@t

�i;j;k

= F�U�i;j;k

+D�U�i;j;k

(39)

where D(U)i;j;k is an explicit arti�cial dissipation.In order to advance Eq. (39) in time, a third-order Runge-Kutta is used as proposed by Shu and reported

by Yee.9 This yield to the following three steps:

U1

= Un �

hF�Un��D

�Un�i; (40)

U2

=3

4Un

+1

4U

1 � 1

4

hF�U

1��D

�U

1�i; (41)

Un+1

=1

3Un

+2

3U

2 � 2

3

hF�U

2��D

�U

2�i: (42)

In order to calculate F(U)i;j;k, the ux of tensor � over the control surfaces must be calculated. For thecontrol surface Si+1=2, this ux is given by

(� � S)i+1=2 =

2666664(� � S)1(� � S)2(� � S)3(� � S)4(� � S)5

3777775i+1=2

: (43)

The �rst component of the vector de�ned by the above equation is associated to the continuity equationand given by

(� � S)1 = �i+1=2 (qs)i+1=2 ; (44)

where the volumetric ux 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 : (45)

The second, third, and fourth components are associated to the three components of the momentumequation and the �fth component is associated with the energy equation. For the Navier-Stokes equations,those components and given by Eq.(46) to Eq.(49)

(� � S)2 = (�u)i+1=2 (qs)i+1=2 + pi+1=2 (sx)i+1=2 �h�i+1=2 (Sxx)i+1=2

i(sx)i+1=2 �

h�i+1=2 (Sxy)i+1=2

i(sy)i+1=2 �h

�i+1=2 (Sxz)i+1=2

i(sz)i+1=2 ; (46)

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(� � S)3 = (�v)i+1=2 (qs)i+1=2 + pi+1=2 (sy)i+1=2 �h�i+1=2 (Sxy)i+1=2

i(sx)i+1=2 �

h�i+1=2 (Syy)i+1=2

i(sy)i+1=2 �h

�i+1=2 (Syz)i+1=2

i(sz)i+1=2 ; (47)

(� � S)4 = (�w)i+1=2 (qs)i+1=2 + pi+1=2 (sz)i+1=2 �h�i+1=2 (Sxz)i+1=2

i(sx)i+1=2 �

h�i+1=2 (Syz)i+1=2

i(sy)i+1=2 �h

�i+1=2 (Szz)i+1=2

i(sz)i+1=2 : (48)

(� � S)5 = (�eT )i+1=2 (qs)i+1=2 + pi+1=2 (qs)i+1=2 � ui+1=2 (sx)i+1=2

h�i+1=2 (Sxx)i+1=2

i�

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

h�i+1=2 (Syy)i+1=2

i� wi+1=2 (sz)i+1=2

h�i+1=2 (Szz)i+1=2

i�h

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

i h�i+1=2 (Sxy)i+1=2

i�h

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

i h�i+1=2 (Syz)i+1=2

i�h

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

i h�i+1=2 (Sxz)i+1=2

i�h

ki+1=2 (@T=@x)i+1=2

i(sx)i+1=2 �

hki+1=2 (@T=@y)i+1=2

i(sy)i+1=2 �h

ki+1=2 (@T=@z)i+1=2

i(sz)i+1=2 : (49)

For the Euler equations, those components and given by Eq.(50) to Eq.(54)

(� � S)1 = �i+1=2 (qs)i+1=2 ; (50)

(� � S)2 = (�u)i+1=2 (qs)i+1=2 + pi+1=2 (sx)i+1=2 ; (51)

(� � S)3 = (�v)i+1=2 (qs)i+1=2 + pi+1=2 (sy)i+1=2 ; (52)

(� � S)4 = (�w)i+1=2 (qs)i+1=2 + pi+1=2 (sz)i+1=2 ; (53)

(� � S)5 = (�eT )i+1=2 (qs)i+1=2 + pi+1=2 (qs)i+1=2 ; (54)

In order to calculate the ux (� � S) according to Eqs. (44) to (49), it is necessary to approximate thevalues of the variables at the control surface Si+1=2 from the mean values of the conservative variables inthe control volumes, given by the vector

Ui;j;k =

2666664�

�u

�v

�w

�eT

3777775i;j;k

; (55)

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

eu =�u

�; ev =

�v

�; ew =

�w

�; feT =

�eT�: (56)

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The mean of the total energy is given by

feT = ee+ eek = ee+fuu+fvv +gww

2: (57)

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 = feT � ek = feT � eu eu+ ev ev + ew ew2

; (58)

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

p = ( � 1) � e; T = M2 p

�; � = C1

T 3=2

T + C2and k = �

�

( � 1) M2 Re1 Pr: (59)

It is important to note that the �rst terms in the right-hand side of Eqs. (44), (46), (47), (48), and (49)are the uxes of mass, momentum and total energy through surface Si+1=2 and the other terms are uxesthat 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 =2

3(eui + eui+1)� 1

12(eui�1 + eui + eui+1 + eui+2) (60)

for the primitive variables, exempli�ed in the above equation by the x�direction component of the velocity,and where

(�u)i+1=2 =1

3

��i + �i+1

�(eui + eui+1)�

1

24

��i�1eui�1 + �i+1eui�1 + �ieui + �i+2eui + �i+1eui+1+

�i�1eui+1 + �i+2eui+2 + �ieui+2

�+

1

3

�1

2

��i+1eui+1 + �ieui�� 1

4

��i+1 + �i

�(eui+1 + eui)� : (61)

for the conservative variables, also exempli�ed by the x�direction component of the speci�c momentum.The scheme proposed by Eqs. (60) and (61) is a centered one, and therefore, an explicit arti�cial viscosity

was previously included in Eq. (39). In this work the arti�cial dissipation model uses the basic idea proposedby Jameson8 et al. given by

D(U) =1

�tf[di+1=2(U)� di�1=2(U)] + [dj+1=2(U)� dj�1=2(U)]

+ [dk+1=2(U)� dk�1=2(U)]g; (62)

wheredi+1=2(U) = �

(2)i+1=2[Ui+1 �Ui]� �(4)i+1=2[Ui+2 � 3Ui+1 + 3Ui �Ui�1]: (63)

The �rst and second terms of the right-hand side of Eq. (63) are the second-order and fourth-order dissipationoperators, respectively, and the coe�cients for both operator are

�(2)i+1=2 = K (2)max (i;i+1) ; �

(4)i+1=2 = max

h0;�

K (4) � �(2)i+1=2

�i; (64)

where the recommended values for the constants are

K (2) = 1=4; K (4) = 1=256; (65)

and the pressure-based sensor i is given by

i =jpi+1� 2p

i+ p

i�1jjpi+1j+ j2p

ij+ jp

i�1j: (66)

9 of 25


In order to improve the numerical stability of the moving-body immersed-boundary methodology, a newvorticity-based sensor,

(rot)i+1=2 = K(2)rot � jO� uji+1=2; (67)

and a new divergence-based sensor,

(div)i+1=2 = K(2)div � jO � uji+1=2; (68)

are implemented in the arti�cial dissipation model by substituting the pressure-based sensor in the calculationof the coe�cient of the second-order dissipation, resulting in

�(2)i+1=2 = max

h(rot)i+1=2 ; (div)i+1=2

i: (69)

Since the calculation of the divergence and the vorticity of the velocity �eld is fourth-order accurate inspace, the resulting numerical method is also fourth-order accurate in space and third-order accurate in time.

IV. Moving-Body Immersed-Boundary Technique

The approach used in this work for imposing the boundary conditions at the boundary volumes, de�nedas 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 =

2666664�

� eu� ev� ew� feT

3777775i;j;k

: (70)

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

eu = uS ; ev = vS ; ew = wS ; (71)

where uS , vS , and wS are the components of the prescribed surface velocity of the moving body. Since thetotal energy is the sum of the internal and kinetic energy, the application of the no-slip condition results in

feT = e+1

2

�uS

2 + vS2 + wS

2�; (72)

bearing, for the boundary volumes

Ub

i;j;k =

2666664�

� uS

� vS

� wS

��e+ 1

2

�uS

2 + vS2 + wS

2��

3777775

b

i;j;k

; (73)

where the superscript b indicates that the �nite volume (i; j; k) is a boundary volume. It is important tonote that the (i; j; k) indexes of boundary volumes are constantly changing, since the surface points that liewithin the boundary volume are moving through the computational domain with the prescribed velocity ofthe moving body.

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

p =

�1

M21

�� T ; (74)

is derived in the normal outward direction from the solid wall. With this objective, it is de�ned n as a unitvector with a direction that is normal to the wall with outward sense, where the Cartesian components are

10 of 25


n = nxi + nyj + nzk and the magnitude is jnj =qn2x + n2y + 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 de�ned 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=

�1

M21

�@

@n(� T ) =

�1

M21

��@T

@n+ T

@�

@n

�: (75)

For an adiabatic wall, @T=@n = 0, and considering the boundary-layer approximation, @p=@n = 0, Eq. (75)yields

@�

@n= 0; (76)

and since

e =1

( � 1)M21T ; (77)

the adiabatic wall condition results in@e

@n= 0: (78)

De�ning 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

@�

@n=@�

@x

@x

@n+@�

@y

@y

@n+@�

@z

@z

@n= nx

@�

@x+ ny

@�

@y+ nz

@�

@z(79)

and@e

@n=@e

@x

@x

@n+@e

@y

@y

@n+@e

@z

@z

@n= nx

@e

@x+ ny

@e

@y+ nz

@e

@z: (80)

For the boundary volumes, Eqs. (76) and (78) apply and result in

0 = nx

�@�

@x

�bi;j;k

+ ny

�@�

@y

�bi;j;k

+ nz

�@�

@z

�bi;j;k

(81)

and

0 = nx

�@e

@x

�bi;j;k

+ ny

�@e

@y

�bi;j;k

+ nz

�@e

@z

�bi;j;k

: (82)

If nx > 0, in regular region of the Cartesian grid the derivative @�= @x in the boundary volumes can becalculated with fourth-order spatial precision using a forward �nite-di�erence approach as�

@�

@x

�bi;j;k

=1

12�x

��25� bi;j;k + 48�i+1;j;k � 36�i+2;j;k + 16�i+3;j;k � 3�i+4;j;k +O(�x)4

�: (83)

De�ning the di�erence operator

D+i � =

1

25

�48�i+1;j;k � 36�i+2;j;k + 16�i+3;j;k � 3�i+4;j;k

�; (84)

Eq. (83) is written as �@�

@x

�i;j;k

=25

12�x

�� bi;j;k +D+

i �+O(�x)4�: (85)

If n = i (nx = 1, ny = 0 and nz = 0), Eq. (81) gives

0 =

�@�

@x

�bi;j;k

; (86)

and introducing this result in Eq. (85) yields

� bi;j;k = D+i �+O(�x)4: (87)

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

� bi;j;k = D+j �+O(�y)4; (88)

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

� bi;j;k = D+k �+O(�z)4: (89)

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

� bi;j;k =jnxjDi�+ jnyjDj�+ jnzjDk�

jnxj+ jnyj+ jnzj: (90)

Following an analogous procedure, since @�=@n = @e=@n = 0, the averaged internal energy is calculated asthe weighted value

e bi;j;k =jnxjDie+ jnyjDje+ jnzjDke

jnxj+ jnyj+ jnzj; (91)

where the di�erence 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. (84), and the

other forward and backward di�erence operators are given by

D+i � =

1

25

�48�i+1;j;k � 36�i+2;j;k + 16�i+3;j;k � 3�i+4;j;k

�; (92)

D+j � =

1

25

�48�i;j+1;k � 36�i;j+2;k + 16�i;j+3;k � 3�i;j+4;k

�; (93)

D+k � =

1

25

�48�i;j;k+1 � 36�i;j;k+2 + 16�i;j;k+3 � 3�i;j;k+4

�; (94)

D�i � =1

25

�48�i�1;j;k � 36�i�2;j;k + 16�i�3;j;k � 3�i�4;j;k

�; (95)

D�j � =1

25

�48�i;j�1;k � 36�i;j�2;k + 16�i;j�3;k � 3�i;j�4;k

�; (96)

D�k � =1

25

�48�i;j;k�1 � 36�i;j;k�2 + 16�i;j;k�3 � 3�i;j;k�4

�: (97)

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

Ub

i;j;k =

26666664� bi;j;k

� bi;j;k uS

� bi;j;k vS

� bi;j;k wS

� bi;j;k

he bi;j;k + 1

2

�uS

2 + vS2 + wS

2�i

37777775 ; (98)

where the �rst and last components are given by Eqs. (90) and (91), respectively.

V. Numerical Results

The following numerical results are divided in two categories: forced ow and induced ow. In the forced- ow case a body force is applied in the entire ow�eld, given by Eq. (32), directly forcing the ow through therotor cascade whereas a prescribed velocity is imposed in the rotor cascade surface. In the induced- ow case,just the prescribed velocity is imposed in the rotor cascade surface and the ow through it is induced solelyby the movement of the solid surface, implemented in the numerical method by the boundary conditions atthe boundary volumes given by Eq. (98). The forced- ow case is non-physical, since in the real case onlythe movement of the solid surface must induce the ow through the cascade, but it was studied in orderto test the proposed moving-body immersed boundary technique in a favorable pressure-gradient conditionthrough the rotor cascade, where the moving boundary do not need to induce the ow. In the induced- owcase, the pressure gradient through the cascade is adverse, since work is performed by the moving rotor overthe uid. In both cases, only the Euler equations were solved.

12 of 25


V.A. Forced Flow

The rotor blade geometry used in this part is from the annular non-rotating cascade facility situated of theLaboratoire de thermique appliqu�ee et de turbomachines at EPF-Lausanne, Switzerland. The rig was builtfor aeroelastic analysis and �gure 1 shows the general setup.

Figure 1. General setup of the annular non-rotating cascade facility.

In the design condition, M = 0:31, the rotor blade has a stagger angle of 40:85o and the nominal inlet ow angle is � = 15:2o. For this case, the forced ow velocity is U�1 = 104:22 m/s and the downwardvelocity prescribed over the rotor surface is V � = �28:32 m/s, where � = atan (V=U1). The chord of therotor blade, c�, is 0.0778 m and the pitch between two consecutive blades is 0.05655 m. The geometry ofthe rotor blade is presented in �gure 2, where 104 equally-spaced surface points are used to discretize it.

Figure 2. Rotor blade geometry with a zoom over the trailing edge.

13 of 25


The regular region of the Cartesian grid has a size of 3c� 0:8c with a resolution of 103 control volumesover the chord, giving a total of 2:4� 106 control volumes. Figure 3 shows the resolution, at the leading andtrailing edges, of the rotor surface grid and regular the Cartesian grid.

Figure 3. Resolution of the surface grid and the Cartesian grid at the leading an trailing edges.

Figures (4) to (7) show the aeroacoustic visualization of the downward moving blade at four consecutivetimes. The downward movement is clearly seen, as well as the following vorticity wake. Figure (8) shows thepositions of the upstream (blue) and downstream (red) pressure probes and �gure (9) present the pressuresignal obtained in those probes, showing the favorable pressure gradient and the blade passing frequency(BPF). Figures (10) and (11) show the SPL spectra for the upstream and downstream probes, respectively.In both cases, the spectra shows a StBPF = 0:33, that corresponds to calculated value using the blade pitchand moving velocity. It is also noticeable that more than 20 harmonics peaks are present in both signals forfrequencies lower than St = 10. Figures (12) and (13) show the broadband content in the SPL spectra forthe upstream and downstream pressure probes. In both signals, there is a 70 dB decay in a span of St = 600.

14 of 25


Figure 4. Aeroacoustic visualization (�t) at t = 5:414.




15 of 25


Figure 8. Aeroacoustic visualization (�t) at t = 44:145, showing the positions of upstream (blue) and downstream (red)pressure probes.

Figure 9. Pressure signal at the upstream (blue) and dowstream (red) pressure probes, showing the favorable pressuregradient and the blade passing frequency (BPF).

16 of 25


Figure 10. Sound pressure level (SPL) spectrum as a function of the Strouhal number showing the blade passingfrequency (BPF) and its multiple harmonics for the upstream pressure probe.

Figure 11. Sound pressure level (SPL) spectrum as a function of the Strouhal number showing the blade passingfrequency (BPF) and its multiple harmonics for the downstream pressure probe.

17 of 25


Figure 12. Sound pressure level (SPL) spectrum as a function of the Strouhal number showing the broadband distri-bution for the upstream pressure probe.

Figure 13. Sound pressure level (SPL) spectrum as a function of the Strouhal number showing the broadband for thedownstream pressure probe.

18 of 25


V.B. Induced Flow

The rotor blade geometry used in this part is from the Advanced Noise Control Fan (ANCF) rig, locatedat the NASA Lewis Research Center. The geometry selected corresponds to the rotor pro�le at the meandiameter and �gure (14) shows the general setup. The regular region of the Cartesian grid has a size of17:5L�5:5L with a resolution of 300 control volumes over the reference length, L, giving a total of 7:875�106

control volumes with 8,908 surface grid points to de�ne the geometry. For this case, an upward velocity isprescribed over the rotor surface and equal to V � = 138:88 m/s and the freestream velocity is U�1 = 0:0m/s, meaning that the velocity �eld must be induced by the rotor movement. The Mach number associatedto the prescribed rotor velocity is 0.4. The reference length is 0.13335 m, the rotor chord is c = 3� L = 0:4m and the pitch between two consecutive blades is 0.66675 m.

Figure 14. ANCF rotor pro�le at the mean diameter (left) and zoom at the leading edge showing the resolution of theCartesian and surface grids (right).

Figures (15) to (18) show the aeroacoustic visualization of the upward moving blade at four consecutivetimes. The upward movement is clearly seen, as well as the generation and movement, fron left to right, ofthe induced vorticity wake.

19 of 25




20 of 25




Figure (19) shows the positions of the upstream (blue) and downstream (red) pressure probes and �gure(20) present the pressure signal obtained in those probes, showing the pressure rise due to the work done bythe blade over the uid.

21 of 25


Figure 19. Aeroacoustic visualization (�t) at t = 728:94, showing the positions of upstream (blue) and downstream (red)pressure probes.

Figure 20. Pressure signal at the upstream (blue) and dowstream (red) pressure probes, showing the pressure rise dueto the work done by the blade over the uid.

22 of 25


Figure 21. Sound pressure level (SPL) spectrum as a function of the Strouhal number showing the blade passingfrequency (BPF) at 0.2 and its second harmonic (2BPF) at 0.4 for the upstream pressure probe.

Figure 22. Sound pressure level (SPL) spectrum as a function of the Strouhal number.

23 of 25


Figure 23. Sound pressure level (SPL) spectrum as a function of the Strouhal number showing the broadband distri-bution for the upstream pressure probe.

Figure 24. Sound pressure level (SPL) spectrum as a function of the Strouhal number showing the broadband distri-bution for the downstream pressure probe.

24 of 25


VI. Conclusion

A �nite-volume fourth-order immersed boundary method for moving bodies is proposed in this work.The numerical results obtained for the rotor cascades show a very sharp de�nition of the moving surfacefor both the forced and induced ows. In the forced case the pressure gradient is favorable, since the owthrough the cascade is forced by an external body force. On the other hand, the pressure gradient is adversein the induced case, since the ow is induced solely by the rotor motion and the pressure rises due to thework done over the uid by the moving surfaces. In both cases, the pressure signal spectra shows clearly theblade-passing frequency and its harmonics.

Acknowledgments

The blade geometry of the Advanced Noise Control Fan used in this work was provided by Daniel L.Sutli�, Aero-Acoustic Propulsion Laboratory, NASA Glenn Research Center.

This work is sponsored by FAPESP (Funda�c~ao de Amparo �a Pesquisa do Estado de S~ao Paulo) andEmbraer (Empresa Brasileira de Aeron�autica S.A.) within the framework of the project Aeronave Silenciosa.

References

1Peskin, C. S., Flow patterns around heart valves: a numerical method, Journal of Computational Physics, Vol. 10, 1972,pp. 252-271.

2de Tullio, M. D., De Palma, P., Iaccarino, G., Pascazio, G., Napolitano, M., An immersed boundary method forcompressible ows using local grid re�nement, Journal of Computational Physics, Vol. 225, 2007, pp. 2098-2117.

3Cho, Y., Boluriaan, S., Morris, P. J. Immersed Boundary Method for Voiscous Flow Around Moving Bodies, 44th AIAAAerospace Sciences Meeting and Exhibit, AIAA 2006-1089, 2006.

4Liu, Q., Vasilyev, O. V., A Brinkman penalization method for compressible ows in complex geometries, Journal ofComputational Physics, Vol. 227, 2007, pp. 946-966.

5Ghias, R., Mittal, R., Dong, H., A sharp interface immersed boundary method for compressible viscous ows, Journalof Computational Physics, Vol. 227, 2007, pp. 946-966.

6Bobenrieth Miserda, R. F., Lauterjung Q., R., Maldonado, A. L. P., Ribeiro, I.D., Godoy, K., Neto, O.G., DirectComputation of Noise Generated by Complex Geometries Using a High-Order Immersed Boundary Method, AIAA, 2009.

7Ducros, F., Laporte, F., Soul�eres, T., Guinot, V., Moinat, P., Caruelle, B., High-Order Fluxes for Conservative Skew-Symmetric-like Schemes in Structured Meshes: Application to Compressible Flows, Journal of Computational Physics, Vol.161, 2000, pp. 114-139.

8Jameson, A., Schmidt, W., Turkel, E., Numerical Solutions of the Euler Equations by Finite Volume Methods UsingRunge-Kutta Time-Stepping Schemes, AIAA 14th Fluid and Plasma Dynamics Conference, AIAA-81-1259, 1981.

9Yee, H. C., Explicit and Implicit Multidimensional Compact High-Resolution Shock-Capturing Methods: Formulation,Journal of Computational Physics, Vol. 131, 1997, pp. 216-232.

10Bobenrieth Miserda, R. F., de Mendon�ca, A. F., Numerical Simulation of the Vortex-Shock Interactions in a Near-BaseLaminar Flow, 43rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA-2005-0316, 2005.

11Bobenrieth Miserda, R. F., Leal, R. G., Numerical Simulation of the Unsteady Aerodynamic Forces over a CircularCylinder in Transonic Flow, 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA-2006-1408, 2006.

12Bobenrieth Miserda, R. F., Jalowitzki, J. R., Lauterjung Q., R., On the E�ect of the Plunging and Pitching Motionsover the Dynamic Response of an Airfoil in Transonic Laminar Flow, 44th AIAA Aerospace Sciences Meeting and Exhibit,AIAA-2006-452, 2006.

13Bobenrieth Miserda, R. F., Carvalho, A. G. F., On the E�ect of the Plunging Velocity over the Aerodynamic Forces foran Airfoil in Subsonic Laminar Flow, 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA-2006-453, 2006.

14Bobenrieth Miserda, R. F., Carvalho, A. G. F., Direct Computation of the Noise Generated by Subsonic, Transonic,and Supersonic Cavity Flows, 46th AIAA Aerospace Sciences Meeting and Exhibit, AIAA-2009-0008, 2008.

15Bobenrieth Miserda, R. F., Maldonado, A. L. P., Gutierrez, B., Simulation of the Cascade-Gust Interaction ProblemUsing a High-Order Immersed Boundary Method, AIAA, 2009.

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