AIAA-23130-885

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Technical NotesTECHNICAL NOTES are short manuscripts describing new developments or important results of a preliminary nature. These Notes should not exceed 2500words (where a figure or table counts as 200 words). Following informal review by the Editors, they may be published within a few months of the date of receipt.Style requirements are the same as for regular contributions (see inside back cover).

Characteristic Ghost-Cell

Boundary Condition

A. Gross∗ and H. F. Fasel†

University of Arizona, Tucson, Arizona 85721

DOI: 10.2514/1.23130

I. Introduction

T HE modeling and improvement of boundary conditions hasalways drawn the attention of researchers and engineersinterested in simulations of time-dependent compressible flows.Reflections of acoustic waves at the boundaries and distortions of outgoingflow structures are undesired and have to be minimized. At the same time, boundary conditions must also enforce mean inflowand out flow conditions. Care has to be taken that numerical couplingbetween the inflow and out flow boundaries, which can potentiallyresult in nonphysical oscillations in the computational domain, isavoided. Review papers that summarize some of the current approaches for designing boundary conditions for compressibleflows were published by Poinsot and Lele [1], Nicoud and Poinsot [2], and Colonius [3]. Nonreflective boundary conditions derivedfrom the linearized Euler equations [4] were shown to work well for problems in which the disturbance amplitudes are small.

Characteristic boundary conditions can also be derived for thenonlinear case (large disturbance amplitudes, for example, vorticespassing through the boundary) [5–8]. Combinations of gridstretching and filtering near the boundaries may reduce wavereflections, depending on the grid-stretching parameters, the filter properties, and the extent of the buffer layer (or sponge layer).However, the success of these and similar approaches dependsheavily on trial and error. Regardless of the approach chosen, theintuitive approach of prescribing all flow quantities at the inflowboundary and extrapolating all flow quantities at the out flowboundary does not work. So-called soft boundary conditions arerequired. The method presented here is based on the boundaryconditions developed by Thompson [5,6] and further extended byPoinsot and Lele and Kim and Lee [7,8]. The present approachdiffers from previous approaches in that, different from before,instead of modifying the flux differences at the boundary, flowvariables in ghost cells are described such that the resulting fluxdifferences satisfy the nonreflecting conditions.

II. Derivation of Boundary Condition

The compressible Navier –Stokes equations in curvilinear coordinates at a boundary (without loss of generalization) can bewritten as

J @Q

@t @E

@ S (1)

where S contains all flux derivatives other than @E=@ . Differences@Q can be expressed as differences of the characteristic variables,@W R

1@Q, where R is the matrix of right eigenvectors of theJacobian A @E=@QRR1, and the eigenvalue matrix is

diag U U U U A c U A c

(2)

with A 2 x 2y 2zq and U xu yv zw. Equation (1)

can be transformed into a characteristic form in the direction normalto the boundary:

J @W

@t

@W

@ R

1S (3)

The second term on the left-hand-side can be rewritten as

L @W

@q

@q

@ (4)

where differences are computed for a state vector q, which isdefinedhere as

q u v w p

T (5)

The backward transformation is

@q

@

@W

@q

1

1L (6)

The transformation matrices are

@W

@qR1

@Q

@q

~ x 0

c~ z

c~ y 1

c2~ x

~ y c

~ z 0 c

~ x 1c2

~ y~ z

c~ y

c~ x 0 1

c2~ z

0 2c

~ x

2c~ y

2c

~ z1

2c2

0 2c

~ x 2c

~ y 2c

~ z1

2c2

26666664

37777775

(7)

and

@W

@q

1

~ x~ y

~ z 1 1

0 c

~ zc

~ yc

~ x c

~ xc

~ z 0 c

~ xc

~ y c

~ y

c

~ yc

~ x 0 c

~ z c

~ z0 0 0 c2 c2

26666664

37777775

(8)

with ~ x;y;z x;y;z=A . The generalized version of the characteristicboundary conditions was first proposed by Kim and Lee [7,8].

The boundary conditions were implemented into a compressiblemixed finite volume/ finite difference code in which the convectiveterms are discretized with a fifth-order-accurate scheme based on a weighted essentially nonoscillatory (WENO) extrapolation of thecharacteristic variables in combination with the Roe scheme [9]. Theviscous terms are discretized with fourth-order-accurate finitedifferences. At the boundaries, the state variables need to be

Received 10 February 2006; revision received 4 October 2006; acceptedforpublication12 October 2006. Copyright © 2006 by theauthors.Publishedby the American Institute of Aeronautics and Astronautics, Inc., withpermission. Copiesof this paper maybe made for personal or internal use, oncondition that the copier pay the $10.00 per-copy fee to the Copyright

Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code $10.00 in correspondence with the CCC.∗Research Assistant Professor. Member AIAA.†Professor. Member AIAA.

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prescribed in ghost cells (Fig. 1) before the governing equations areadvanced in time. The flow variables at position i are employed for computing all matrices. Differences @q=@ at i are computed asqi qi1, and differences @q=@ a at i are computed as qa qi,where qa defines a known ambient state. With @q=@ and @q=@ a,vectors L and La can be computed from Eq. (4). A new L0 isassembled according to [1,5–8]: supersonic inflow (L0 ( La),

subsonic infl

ow (L0 ( La;1 La;2 La;3 La;4 L5 T

),subsonic out flow (L0 ( L1 L2 L3 L4 La;5

T if a down-

stream pressure is imposed or L0 ( L1 L2 L3 L4 L4

T if

the condition @p=@t 0 is enforced), and supersonic out flow(L0 ( L).

For a “perfectly nonreflecting” boundary, the parameter is set tozero. This, however, will not guarantee that the mean static pressurein the computational domain is maintained over time. For the resultsshown here, the parameter was set to 0.25 [1], resulting in a “partially nonreflecting” boundary. Finally, differences@q=@ 0 arecomputed from the modified L0 vector via Eq. (6) and the ghost cellsare updated:

q i1 qi @q

@ 0

(9)

The characteristic boundary condition, which will be referredto asBC1, is compared with two simpler boundary conditions, BC2 andBC3 (Table 1). For BC2, velocities and temperature are enforced at the inflow boundary, whereasthe static pressure is extrapolated from within the computational domain, assuming zerofirst derivatives. At the out flow boundary, velocities and temperature are extrapolatedassuming zero second derivatives and the static pressure isprescribed. For BC3, all flow quantities are extrapolated assumingzero second derivatives.

The high order accuracy of the Navier –Stokes code appears to beconflicting with the first-order accuracy of the finite difference

stencils employed at the boundaries. The approach is, however,consistent with the general layout and philosophy of the code. Low-order-accurate boundary conditions were deliberately chosen for making the code more robust and because comparable approaches(such as the mirror image boundary condition at walls) are commonpractice infinite volume codes.It has tobe kept inmindthat the order of accuracy of the extrapolation acts as an additional physicalcondition.

III. Results

Twoinviscidtest cases are being considered. Forthese test cases, a fourth-order-accurate explicit Runge–Kutta scheme was employedfor time integration. As an example of a low Reynolds number application, simulations of a low-pressure turbine cascade wereconducted using a second-order-accurate implicit Adams–Moultonmethod in time. For all simulations, the gas was assumed to beperfect, with a ratio of specific heats 1:4, reference Mach number M

0:1, and Prandtl number Pr

0:72.

A. Circular Explosion

A circular explosion was generated by prescribing a pressureperturbation in the center of a computational domain withdimensions x 0:05; . . . ; 0:05 and y 0:15; . . . ; 0:15. Theinitial condition is given by

p p1

1 if r < 0:0015

0 otherwise(10)

where r2 x2 y2, u v 0, p1 1=M 2, and T 1. Thegrid was equidistant, with 129 387 cells. The evolutionof the blast wave was computed over a time interval of T 0:048, witha timestep of t

3

105. Instantaneous visualizations of the pressure

perturbation p p1 at t 0:012 are shown in Fig. 2.Theblastwavetravels with the speed of sound, c 1=M 10, and reaches the sideboundaries at t 0:005. The intensity of the wave reflection isdependenton the boundary condition. Judging from Fig. 2, boundaryconditions BC1 and BC3 yield similar results, although, with 0:25, BC1 is only partially nonreflective. BC2 leads to strongwave reflections. To allow for a more quantitative comparison, theroot mean square of the pressure perturbation in the region x; y 0:05; . . . ; 0:05 was plotted over time (Fig. 3). The pressuredeviation changes sign at t 0:005, the time instant when the wave

reaches the side boundaries. At t 0:005

2p

0:0071, the wavereachesthe corners of theregion of interest andthe pressure deviationbegins to decay. For BC2, the reflected wave is stronger than theincident wave and a second reflection can be observed at t 0:015.

ambient

q /d ξ(d )a

d q / d ξ

ξ

b o u n d a r y

g h o s t c e l

l s

d o m a

i n

i

i−1

i+1

Fig. 1 Sketch of computational domain at boundary.

Table 1 Boundary conditions employed in the present study

Inflow Out flow

BC1 Characteristic boundary condition

BC2 u;v;T u;v;T a @2u;v;T =@x2 0@p=@x 0 p pa

BC3 @2p;u;v;T =@x2 0 @2p;u;v;T =@x2 0

BC1 BC2 BC3

Fig. 2 Circular explosion; pressure perturbation; contour levels from1 to 1, with increment 0.2 (zero contour level removed).

0 0.01 0.02 0.03 0.04t

-0.05

0

0.05

p - p

∞

BC1BC2

BC3

Fig. 3 Circular explosion; root mean square of pressure perturbation.

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Also, the pressure does not converge to the ambient pressure for t > 0:01, indicating that additional damping (such as a sponge layer)may be required to stabilize the computation. Because BC3 does not enforce the ambient conditions, the pressure does not approach theambient value for t ! 0:05.

B. Vortex Convection

A vortex that is convected through the out flow boundary is a problem that was considered by Poinsot and Lele [1]. As initial

condition, an incompressible inviscid vortex was prescribed, with

u u1 Cy

R2c

expr2

2R2c

(11)

v Cx

R2c

expr2

2R2c

(12)

p p1 1C2

2R2c

expr2

R2c

(13)

T

1, u

1 1,

1 1, p

1 1=

M 2

, r2

x2

y2, Rc

1:5,

and C 1. The Navier –Stokes equations were integrated in timeover a time interval of T 20, with a time step of t 0:01. Thedimensions of the computational domain were x10; . . . ; 10 andy60; . . . ; 60, with 129 199 cells and equidistant grid linespacing in the region x; y10; . . . ; 10. In the regions y60; . . . ;10 and y 10; . . . ; 60, a cubic polynomial grid-stretching function was employed for distributing 35 cells in the ydirection on either end, such that the grid line spacing in the ydirection over the entire domain was smooth. Instantaneousvisualizations of spanwise vorticity,!z r v ,areshowninFig. 4.Disturbances are generated at the inflow boundary as the vortexpasses through theout flow boundary at t 8. These disturbances arealso convected in the downstream direction with u1 1 and reachthe out flow boundary at t 32. At t 40, for the chosen contour levels, the domain appears to be free of disturbances for BC1. For BC2, disturbancesare generated continuouslyat theinflow boundaryfor t > 8, which may becausedby wavereflections at the boundaries,leading to a coupling between the inflow and out flow boundaries [1].To allow for a more quantitative comparison, theroot mean square of the pressure perturbation, spanwise vorticity, and divergence r v for the region x; y10; . . . ; 10 were plotted over time in Fig. 5.For t > 8, the fastest recovery to the ambient pressure is achievedwith BC1. Self-sustained pressure fluctuations can be observed for

BC2. The spanwise vorticity is not a good indicator for quantifyingthe quality of the boundary conditions. The divergence, whichshould theoretically be identical to zero, is the lowest for BC1.

C. Low-Pressure Turbine

A linear low-pressure turbine (LPT) cascade that was studied

intensively, both experimentally (Sondergaard et al. [10]) andnumerically (Rizzetta andVisbal [11] and Gross and Fasel [12]),waschosen as an example for a low Reynolds number application.Details about the setup of the current simulations and the operatingpoint can be found in [12]. For the results shown here, the Reynoldsnumberbased on axial chord lengthC x was25,000, thebladespacingwas 0:88C x, and the inflow angle was in 35deg. The out flowconditions were out 60deg, vout=vin 1:64 [10] (the same

t = 8 t = 24 t = 40

BC1

BC2

BC3

Fig. 4 Vortex convection; instantaneous visualizations of spanwise

vorticity;contour levelsfrom0:0002to 0.0002, withincrement0.00005,and from 0.2 to 0.8, with increment 0.2 (zero contour level removed).

a)

0 10 20 30 40t

-0.02

0.00

0.02

p - p

∞

BC1BC2 BC3

b)

0 10 20 30 40t

10-5

10-4

10-3

10-2

10-1

∇ × v

BC1BC2 BC3

c)

10 20 30t

10-6

10-5

10-4

10-3

∇ ⋅ v

BC1BC2 BC3

Fig. 5 Vortex convection; root mean square of a) pressure perturba-

tion, b) spanwise vorticity, and c) divergence.

Table 2 Grid block resolutions

Block Short Long

1 10 30 20 302 20

20 20

20

3 500 100 500 1004 260 100 260 1005 105 110 130 110

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follows from mass conservation when incompressibility is assumed,vout=vin cos in= cos out), pin=pout in=out 1:012 (approx-imate values taken from earlier Navier –Stokes simulations), andT out=T in 1. These inflow and out flow conditions were needed for

BC1. The equations were advanced in time over a time interval of T 40, with a time step of t 0:001. Both a short and a longcomputational domain (Table 2, Fig. 6) were generated, withcomparable resolution around the blade and in the near wake region.For the long domain, strong grid line stretching in the axial directionwas applied near the inflow and out flow boundaries to dissipatedisturbances. An instantaneous visualization of spanwise vorticityfor the short grid with BC1 is shown in Fig. 7. At the low Reynoldsnumber conditions considered here, the laminar boundary layer separates from the suction side of the blade around x 0:6. Theseparated boundary-layer (free shear layer) velocity profile has aninflection point that gives rise to an inviscid instability mechanism,resultingin a strongamplificationof disturbances. As the disturbanceamplitudes reach large nonlinear levels, spanwise vortices emergethat greatly enhance the wall-normal momentum exchange and lead

to boundary-layer reattachment upstream of the trailing edge of theblade.

When the time-averaged spanwise vorticity is plotted near theout flow boundary (Fig. 8), theresultobtainedwith BC2 and theshort computational domain exhibits a distortion of the isocontours near the out flow boundary. With BC1, almost identical results wereobtained for both grids. Distributions of spanwise vorticity !z,

averaged in the y direction and in time (over the interval T 40),are shown in Fig. 9. With BC1, the predicted downstream distributions of !z are almost identical. With BC2, a noticeableupstream influence of the out flow boundary can be observed for theshort grid. Frequency spectra of the aerodynamic coef ficients

c x;y F x;y

12

1v21(14)

that were computed from the aerodynamic forces in the x and ydirection, F x and F y, are shown in Fig. 10. With BC2 and the short grid, a very broad spectrum that is considerably different from allother results is obtained. It was concluded that this combination wasinadequate for making predictions of the dynamic behavior of thefl

ow. Results obtained with BC1 for both grids and with BC2 for thelarge computational domain are qualitatively similar: the main peak (fundamental) in the spectra is at f 3:3, with higher harmonics at

a)

2

5

1

3 4

b)

2

5

1 3 4

Fig. 6 Block boundaries of a) short andb) long computational domain.

Fig. 7 Instantaneous visualization of spanwise vorticity.

BC1 , short BC1 , long BC2 , short BC2 , long

Fig. 8 LPT simulations; isocontour lines of time-averaged spanwisevorticity; contour levels from 10 to 10, with increment 1 (zero contourlevel removed).

1 1.2 1.4 1.6x

0

10

20

30

ω z

BC1, shortBC1, longBC2 , shortBC2 , long

Fig. 9 LPT simulations; spanwise vorticity averaged in y and in time.

a)

0

0.2

0.2

0

0.2

0 5 10 15 20

f

0

0.2

A ( c

x )

BC 1, short

BC 1, long

BC 2 , short

BC 2 , long

max=0.65

b)

0

0.2

0.2

0

0.2

0 5 10 15 20

f

0

0.2

A ( c

y )

BC 1, short

BC 1, long

BC 2 , short

BC 2 , long

max=0.97

Fig. 10 LPT simulations; frequency spectra of aerodynamic coef ficients in a) x direction and b) y direction.

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n f (n 2; 3; . . .). The noise between the peaks in the frequencyspectra is larger when BC2 is employed. It cannot be excluded that amplitude and frequency of the observed self-sustained oscillationswere also influenced by a numerical coupling between the separationbubble oscillation and the out flow boundary. With the distance from the blade trailing edge to the out flow boundary (in the streamwisedirection) being L 1:23, the average convection velocity beingv 1:64, and the speed of sound being c 10, the Rossiter modefrequencies are [13]

n

f L

v L

c v(15)

or f 1:11n, and the dominant peak in the frequency spectra canbeexplained as Rossiter mode n 3.

IV. Conclusions

The generalized characteristic boundary condition by Kim andLee [7,8] was modified for a finite volume method with ghost-cellapproach and tested for two inviscid problems, a circular explosionand a vortex convection problem, and for a low-Reynolds number problem, the flow through a linear LPT cascade. In comparison withtwo simpler noncharacteristic boundary conditions, the character-

istic boundary condition was found to be more transparent for outgoing waves and/or vortical structures and resulted in a faster relaxation of the flow to the ambient conditions, after thedisturbances had leftthe domain. With respect to the time-dependent LPT flow, the low background noise level achieved by thecharacteristic boundary condition appeared to be highly beneficial,because a significantly smaller computational domain could be usedfor accurately predicting the flow dynamics.

Acknowledgments

This work was funded by the Air Force Of fice of ScientificResearch (AFOSR) under grant number F9550-05-1-0166, by theOf fice of Naval Research (ONR) under grant number N00014-01-1-0932, and by the Deutsche Forschungsgemeinschaft (DFG) under grant number GR 2117/1-1.

References

[1] Poinsot, T. J., and Lele, S. K., “Boundary Conditions for Direct Simulations of Compressible Viscous Flows,” Journal of Computa-tional Physics, Vol. 101, No. 1, 1992, pp. 104–129.

[2] Nicoud, F., and Poinsot, T., “Boundary Conditions for CompressibleUnsteady Flows,” in Arti ficial Boundary Conditions at Interfaces,edited by L. Halpern, F. Natef, and L. Tourette, Nova Science,Hauppauge, NY, 2000.

[3] Colonius, T., “Modeling Artificial Boundary Conditions for Compressible Flow,” Annual Review of Fluid Mechanics, Vol. 36,

Jan. 2004, pp. 315–345.[4] Giles, M. B., “Nonreflecting Boundary Conditions for Euler Equation

Calculations,” AIAA Journal , Vol. 28, No. 12, 1990, pp. 2050–2058.[5] Thompson, K. W., “Time Dependent Boundary Conditions for

Hyperbolic Systems,” Journal of Computational Physics, Vol. 68,No. 1, 1987, pp. 1–24.

[6] Thompson, K. W., “Time-Dependent Boundary Conditions for Hyperbolic Systems, II,” Journal of Computational Physics, Vol. 89,No. 2, 1990, pp. 439–461.

[7] Kim, J. W., and Lee, D. J., “Generalized Characteristic BoundaryConditions for Computational Aeroacoustics,” AIAA Journal , Vol. 38,No. 11, 2000, pp. 2040–2049.

[8] Kim, J. W., and Lee, D. J., “Generalized Characteristic BoundaryConditions for Computational Aeroacoustics, Part 2,” AIAA Journal ,Vol. 42, No. 1, 2004, pp. 47–55.

[9] Gross, A.,and Fasel,H. F.,“High-Order WENO Schemes Based on the

Roe Approximate Riemann Solver,” AIAA Paper 2002-2735,June 2002.

[10] Sondergaard,R.,Rivir, R.B., andBons, J.P., “Control of Low-PressureTurbine Separation Using Vortex-Generator Jets,” Journal of

Propulsion and Power , Vol. 18, No. 4, 2002, pp. 889–895.[11] Rizzetta,D. P.,and Visbal, M.R., “NumericalSimulation of Separation

Control for Transitional Highly Loaded Low-Pressure Turbines,” AIAA

Journal , Vol. 43, No. 9, 2005, pp. 1958–1967.[12] Gross, A., and Fasel, H. F., “Numerical Investigation of Low-Pressure

Turbine Blade Separation Control,” AIAA Journal , Vol. 43, No. 12,2005, pp. 2514–2525.

[13] Rossiter, J. E., “Wind-Tunnel Experiments on the Flow over Rectangular Cavities at Subsonic and Supersonic Speeds,”Aeronautical Research Council, Reports and Memoranda No. 3438,1966.

C. Kaplan Associate Editor

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