AIAA-95-0163 Using High-Order Accurate Essentially Non-Oscillatory Schemes for Aeroacoustic Applications Jay Casper ViGYAN, Inc. Hampton, VA

Kristine R. Meadows NASA Langley Research Center Hampton, VA

33rd Aerospace Sciences Meeting & Exhibit January 9-1 2,1995 / Reno, NV

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L’Enfant Promenade, S.W.. Washington, D.C. 20024

-~ ~

AIM-95-0163

USING HIGH-ORDER ACCURATE ESSENTIALLY NON-OSCILLATORY SCHEMES

FOR AEROACOUSTIC APPLICATIONS

lay Caspcr * ViGYAN. Inc.

Kristine R. Meadows NASA Langley Research Center

ABSTRACT

The numerical study of aeroacoustic problems places stringent dem'ands on the choice of a computarional algo- rithm. particularly when shock waves are involved. Be- cause of their dual capacity for high-order accuracy and high-resolution shock caplitring. the recently developed class of essentially non-oscillatory (FNO) schemes has generated considerable interest in regard to such problems. Because FNO schemes w e adaptive stenciling, their ap- plication requires careful consideration. The use of such schemes for aeroacoustic applications is investigated, with particular attenlion to the control of the adaptive stenciling procedure. A modification ofpreviouslydeveloped rtencil- biasing procedures is proposed. This nonlinear stencil- biasing approach allows a freer adaptation near disconti- nuities lhan is allowed by the previous biasing methods, wilhout disturbing the biased. stable stencils that are de- sired in smooth regions. These new methods arc tested on a sound-shock inleraclion in a converging-divergingnozzle and an axi-symmetric shock-vortex interaction. Numcri- cal resulrs indicate a reduction in error for both problems when compared with rcsulls in which other stencil-biasing "-"""A,.<"- .,l.n,,

The desire to obtain acoustic information from a nu- merical solution that involves shock waves is a demanding proposition for a computational algorithm. High-order ac- curacy is required for the propagation of high-frequency, low-amplitude waves. In addition, high resolution of dis- continuities is desired, without spurious oscillations that can degrade the solution. The class of essentially non- oscillatory (ENO) shock-capturing schemes',' has been designed to have such properties. In this work, it is in- tended to demonstrate the usefulness of these schemes and to suggest some modificalions that will make them more robust with regard to CAA problems that involve shock waves.

The dual capacity of EN0 schemes for high-order ac- curacy and non-oscillatory shock-capturing is achieved through the use of adaptive stenciling. That is, the local polynomial approximation operator adapts its interpolation set to the smoothest available part of the solution. In this way, FflO schemes can approximate the smooth regions of a piecewise continuous function to high-order accuracy without the oscillatory behavior that is associated with in- terpolation across steep gradients. Furthermore, adaptive stenciling enables high-resolution shock-capturing. Al-

p'."""'cJ '"L UJW. though these properties are desirable, previous research has shown that the accuracy of these schemes can degener- ate when the stencils are allowed to freely adapt3 Further research indicates that this accuracy problem can be reme-

MTRODUCTION

'!his work IS motivated by the desire 10 develop IIumCri- cal methods that will be USCfUl i l l the study O f aerOaCOuStiC

d i d by biasing the stencils toward those that are linearly stab~e.4,5,6 Ifowever, osci~lations can occur when this bi.

phcnomcna that occur i n shocked flows. For example. the presence of shocks in jet flows. on airfoils. and in super- sonic combustion inlets contributes significantly to sound in to the adanlive nrocedure.

is implemented in a problem with moving disconti- nuities, which suggests then- for further consideration

generation. Problems such as these represent some of the more challenging aspects of the ongoing research in the dewinping area o f computational aeroacoustics (CAA).

'Research I'ngineer. Advanced Technology Group. XGYAN. Inc.. 30 Rescarch Dnvc. Ilampton. VA 23666

I Acmrpacc Engineer. Fluid Mechanics and Acoustics Division. NASA I.&('. Ilampton. VA 23681 (.'yynghr 0 1 1 9 4 hy the American Instilureof Aemnauricsand As- tronautics. Inc. No copyfight is asscncd in the IJnited Stares undcr 'Etle 17. 1I.S. Code. The US. (iovemmcnl has a royally-frce li- ccnsc to ercrcisc dl rights undcr the copyright claimed herein for Gwrmrnent purposes NI othcr rights are reservcd by the copyright owner.

- -. The following two Seetions briefly describe the

schemes employed in this work, with particular attention paid to the adaptive stenciling algorithm. The methods for constraining this adaptation through the use of bias- ing parameters are discussed. The numerical method is then applied to a sound-shock interaction i n a quasi-one- dimensional converging-diverging nozzle. The effccts of stencil biasing on the solution are examined with regard to the entropy wave generated by the interaction. A modifica- tion to this stencil biasing is proposed and tested. The i n - teraction of a vortex and a shock wave in an axi-symmetric

flow is then considered. Numerical results for this prob- lem, obtained from various modifications of the numerical scheme, are compared. Concluding remarks are made i n the final section.

NUMERICAL mmoD

For the sake of brevity, the necessary details ofthe EN0 schemes to be used in this work are presented within the context of a one-dimensional scalar equation.

a a -u + -f(u) = 0 at az

A control-volume formulation is obtained by integraling Eq. 1 on an interval [ z ~ - , / ~ , Z , + ~ / Z ] with center zi and "volume" Azi , The one-dimensional scalar conservation law can then be written

is the cell average of u on the i-th interval at time 1 . Tem- poral integration of Eq. 2 can be accomplished by treating Eq. 2 as a system of ordinary differential equations. via a method-of-lines approach. In particular, the Runge-Kutla methods of Shu and Osher' will he used. ?hese mea- ods are high-order accurate and total-variationdirrunishing (TVD) in the sense that the temporal operator does not in- crease the solution's total variation in lime. .The right-hand side of Q. 2 is approximated in a manner similar to that inuoduced by Harten, et ai.'; a brief description hollows.

To approximatetheright-handside ofEq. 2 to high-order accuracy, the spatial operator must include a high-order pointwise approximation to u(z, f) . However, at a given time f , only the cell averages in Eq. 3 are available. 'There- fore, a pointwise "reconstruction" of the solution from its cell averages is required. To this end, let R be an op- erator which reconstructs the cell averages and yields a piecewise polynomial R ( z ; U ( t ) ) of degree r - 1 which approximates u(z, t ) to high order, wherever u ( r , t ) is sufficiently smooth. This operator R acts in a piecewise manner in that the solution is locally reconstructed within each cell. Let P; denote the polynomial of degree r - 1 which approximates U(T, 1 ) in the i-th cell, at lime 1 . ; .e.

P;(z) 5 R ( z ; t i ( t ) ) , zi-i/z 5 z 5 ~ i + 1 / 2

= u(r,t) + O(h') (4)

The specific method used in this work is the "reconstruction by primitive" proposed by Ifarten el al.' and is not detailed here.

This piecewise reconstruction can cause jumps in the approximate solution at the cell intCIfdCCS that are O( h ' ) in smooth regions and O( 1 ) near discontinuities. The fluxes in rq. 2arethen approximated by solving thelocal Riemann problems a1 the cell in1crf:ices. 'Thus, the right-hand sidc of F4.2 is replaced by ils high-order approximation. which yields

- where

j i t 1 / 2 ( t ) = fR"'( Pt ( z i+r /? ) 1 P t t l ( z t t i / ? ) (5b)

and fR"'(u~, UR) denotes the flux that is associated with IhesolutionoftheRiemann problem whose iniiialstaies are UL and UR . Upon temporal integration ofw. 2 wilh an ap- propriately high-order Kunge-Kutta method? the scheme in FZ. 5 is locally r-th-order accurale i n the L1 sense. (See Ref. 1 for details.) The extensions of these schemes to hy- perbolic systems and muliiplc spatial dimensions thal are used in this paper can be found i n Refs. I and 7.

ADAM'IVIS STENCILING

The most unique aspect of the reconstniction operalor R is its use of adaptive stenciling. That is. the interpolation set used for the approximation of U(Z, 1 ) within a given cell is allowed to shift in an attempt to use the smoothest possible information. This adaptive stenciling makes FNO schemes highly nonlinearand is the focus of their applicaiioii in this paper.

Todete~ineaIocalpoIynomialP, ofdegreer- 1 within the i-th cell, informalion from I' cell averages is required. including ti, itself and r - 1 of iis neighbors. Because any stencil is assumed l o he contiguous. a particular stencil can be identified by its left-most index. Let j'(i) denole the left-most index of the slencil that is used to locally deter- mine the polynomial of degree r - 1 in [ x , - ~ / ~ , x i t l / ? ] .

The index Y(i) is derermincd in a hierarchical manner. Cells are annexed to the interpolation set based on local smoolhness criteria. lnforniation in regard to the smooih- ness of the solution on a given stencil is obtained lrom the local differences ofthe cell averages on Ilia1 stencil. let A: denote the operator that yields lhe k-lh forward diffcrencc on a stencil of k + I cells with left-most index i. which IS

defined recursively by

d

A f u = - 0;

A?u = A:;,'- A ; - ' u , k = 2 . 3 ,

The algorilhm begins by setting j i ( i ) = i . Alone, this one-cell stencil results in a piecewise constant reconsuuc- lion and a numerical scheme that is spatially first-order ~ ~2

2

accurate. In order to choose jk+’(i) , k = I , . . . , r-1, the two stencils considered as candidates are those obtained by annexing a cell lo the left or right of the previously deter- mined stencil. The selected stencil is the one i n which the k-th difference is smaller i n magnitude:

where Akfi and A i < are the k-th differences obtained, re- spectively, by annexing a cell to the left or right of the pre- viously determined stencil. Because this algorithm allows the reconstruction stencil to shift freely with the detection of any numerical gradient. i t will be referred to as “freely adaptive.”

Rogerson and Meiberg3 have reported a degeneration of accuracy that can occur when the freely adaptive stencil algorithm in Q. 6 is used. Shu‘ has suggested that the algorithm in Fq, 6 be modified 10 bias the reconslruction stencil toward one that results in a linearly stable scheme. For prescnt purposes. the desired reconstruction stencils are centered i f r is odd and one cell upwind if 1‘ is even. In this manner, the resulting schemes have an upwind biased flux. as shown for thecases r = 3 and r = 4 in Fig. I .

This biasing can be accomplished by implementing a factor n i n the stencil search i n F4. 6:

otherwise

( 7 4 where (o,,.nn) = ( 1 , t ) or (h , 1) . forbiasingtotheleftor right. respectively, with 6 > I , For Ax sufficiently small, the hiascd stencil algorithm i n Q. 7a will yield the desired lincarly stable stencil i n any smooth region where all the derivatives of the solution are n o n - ~ e r o . ~

‘Yo affect biasing toward linearly stable stencils near points where the k-th derivative of the solution might van- ish. A t k i d has suggested another type of biasing. The stencil is biased toward the linearly stable one wherever the local differences are small. This philosophy is imple- mcntcd for present purposes as

i f IALul < 6 and lA!,UI < 6

(7‘)) then j’+’(i) = j , k+l ( I ) ,

where c is a small parameter and j : ( i) identifies the stencil ohtained hy annexing the k-th cell i n the linearly stable dircctiun. ?lis additional hiasing is particularly important when solving problems that develop regions i n which the solution is Note that when h = I and c = 0 thc hiased stencil algorithm i n Qs. 7a.b becomes the frceiy adaptive algorithm in 13. 6. Because thcse hiasing parametcrs are fixed relative to the solution. the stencil .d

algorithm in Eq. 7a.b will bereferred to as “linear biasing.” Reasons for modifying this biasing will become apparent.

SOUND-SHOCK INTERACTION I N A NOZZLE

me interaction of a sound wave with a shock in a quasi- one-dimensional converging-diverging nozzle is numeri- cally investigated. The governing equations are the quasi- one-dimensional Euler equations:

( 84 a a - (Au) + z ( ~ ~ ) = rf at

where

The variables p, u, P, E , and A are the density, velocity, pressure, total specific energy, and nozzle area, respec- tively. The equation of state is

1 2

P = ( y - 1 ) p ( E - - TI2 ) ,

where y is the ratio of specific heats which is assumed to have a constant value of 1.4.

me flow variables are normalized with respect tostagna- tion conditions and the area with respect to the value at the throat. The spatial domain ofthe nozzle is 0 5 x 5 1 . The nozzle shape is determined by requiring a linear distribu- tionofMachnumberfromM = OSat the in le t toM = 2.8 at the exit. when the flow is isentropic and fully expanded. The resulting area distribution A(z) is illustrated in Fig. 2.

Given the prescribed area distribution, the Mach 0.8 inflow state is retained at z = 0, and the outflow condition at z = 1 is determined such that a shock forms at z = 0.78, which corresponds to a pre-shock Mach number of M = 2.36. A steady-state solution is obtained by implementing a third-order (r = 3) EN0 scheme with the biased stencil algorithm in Qs. 7a.b until residuals are driven to machine zero. The biasing parameters used are u = 2 and c = 0.001 . It should he noted that this numerically converged initial conditioncannot be obtained with the freely adaptive algorithm in Eq. 6. After the steady state is achieved, an acoustic disturbance is introduced at x = 0:

P ( 0 , t ) = P , ( 1 + / 3 s i n w t ) p(0,f) = P(0,t)”’

u(0,f) = u, + -[ c ( 0 , t ) - c, ] 2

7-1

where the subscript i denotes the steady inlet state. w is the circular frequency, /3 is the amplitude. and c = is the local sound speed. Fig. 3 depicts the pressure perturha- tion 6 P ( t ) = P ( x , t ) - P(x, 0) at 40 equally spaced time

3

iotcrvals during one period of the incoming acoustic wave. The calculation was performed on a uniform mesh of 256 cells, with w = 30 and /3 = 0.001.

'lhis problem is taken from Ref. 8 and is re-examined for the effect of stencil biasing on the entropy wave down- stream of the shock. This entropy wave is expected as it arises from the movement of the shock. However, previoiis research' has shown that thecalculation of this wave is hin- dered by the numerical production of spurious entropy thal is generated by slowly moving shock waves. Furthermore, the research indicated that less of this spurious entropy is produced when the stencil is allowed to shift more freely. However, if all the stencils are allowed to freely adapt, then the convergence of the initial steady solulion will be prohibited, Therefore, to retain the stencil biasing where it is desired but allow more freedom near a discontinuity, the linear biasing parameters 0 and f in Qs. 7a.b are replaced by

0' = 0(l - 8' ) + Bk , 8 = r ( l - 0') (Sa)

where 8' is a simple switch given by

8' = min( 1, a I A $ i I ) ~ a > 0 (9b)

where AtU denotes the k-th difference obtained by annex- ing a cell in the linearly stable direction. For a given value o f a , thevalue I/aistheminimumvalueof lA*ul at which full switching is desired. By setting a = 0, the biasing pa- rameters revert to theirconstant values. The values ofthese nonlinear biasing parameters are not significantly different from the linear parameters except near a discontinuity in the solution's k-th derivative.

The steady-state and perturbed solutions were recalcu- lated using the nonlinear biasing discussed above with 3 = 2 , r = 0.001, and a = 2. In Fig. 4, the quantity 6S(1) = S(z, t ) - S(z,O) is plotted for one time value. where S = P/p'. The two solutionsrepresent results with both linear and nonlinear stencil biasings. In this measure, the amplitude of the entropy wave computed with nonlin- ear biasing is two-thirds less than the solulion computed with linear biasing.

This result suggests that the nonlinearly biased stencils have a slightlywider bandwidth for free adaptation near the shock than do the linearly biased stencils. Fig. 5 supports this conclusion with examples of typical stencils near the shock location. The stencils represent those used for the reconstruction ofthecharacteristic variable associated with X = u - c . (The reconstruction is based on characleristic variables. See Refs. I and 7 for details.) The variable plotted in Fig. 5 is the stencil offset j 3 ( i ) - ji(i). where j : ( i ) = i - 1. as previously shown in Fig. 1. ? l e values -1.0, or 1 represent a left-shifted,centered, orrighl-shifted reconstruction stencil, respectively. With the linear biasing. the stencil is centered on lhe shocked cell, whereas the adaptive biasing of Eqs. 7 and 9 allows a shift to either

side of the shock, while the desired centered stencils are retained in the smooth regions.

An approximation 10 the correct value for the amplitude of the downstream entropy perturbation can be obtained by the solution of lhe Riemann problem pictured in Fig. 6. The s l a t s Li, and lJq are those of the initially steady shock. Upon perturbing the state IJ, by an amount 6U1. the S l a t s U2 and U3 are readily delermined. The solution of the Riemann problem is well documented (e.g., Ref. 9) and is not detailed here.

This analytical solution is approximate in two ways. First. the value for 6U1 is measured from the numerical solution jus1 upstream of the shock. Secondly. this analy- sis d o s not take into account the small amount by which total entropy changes downslream of the moving shock. IJsing this approach, the value for the change in S between states (12 and U3 is approximately 1 . 1 3 ~ The com- puled amplitudes are 2 . 9 7 ~ 1 0 ~ ~ for (he linearly biased algorithm and 9.20~ for the nonlinearly biased algo- rithm. l l s e values represent errors of 162.8% and 18.6%. respectively.

AXI-SYMMETRIC SHOCK-VORTEX Ih'TISRACTION

n e previous example motivates the study of a more complicated problem. that of an interaction of a voriex with a shock in an axi-symmetric flow. The governing equalions are the axi-symmetric Ihlcr qualions: I

where

r p i r I)u 1

The variables p , P, E , and I' are the density, pressure. total specific energy, and axial radius. respectively. The vari- ables u and v are the Cartcsian componenls of the velocity vector P. The equation Lif state is

I L P = ( 7 - l ) p [ f 2 - ; ( u ? +.?)I

where y = 1.4. The physical problem is described with respect to the

(z, r, 8) coordinate syslcm. where the z-axis coincides with the axis of symmetry. I I e initial conditions are im- plemented in two steps. First. a steady normal shock is ~_,

positioned on the surface defined by r = x, , with a flow parallel to the axis of SylIUIEUY and a pre-shock Mach number Af = 1.3 . At I = 0. a toroidal vortex is imposed upsueam of the shock. The vortex is taken from L.amblo and augmented with a solid-body filanient core ofradius 6 . Outside the core radius. Ihe velocity induced by the vortex at a point Q in the z-r plane is derived from the sucam function

where

E ( d l , d ? , m ) = d : c o s 2 - + d : s i i i ' ~ 4 2 2

ti is the circulation, and the distances dl and d z arc mea- sured from Q to the core center points that lie i n the z-r plane, as pictured in Fig. 7. At any point within the core. the velocity is obtained from the value of when dl = 6. The suength of the velocity field induced by this vortex is determined by setting ti = 0.001 U L , where 7 1 ~ is the steady pre-shock llow speed.

Q s . loa-c aresolvedin theupper-halfx-r planeon ado- main { a < r < 6 ) x {O < r < R ) . All flow variables are normalized with respect to the static conditions upstream of the steady shock. The values 6 - a and R are made large enough relative to the vortex suength to neglect the effccts of the interaction at the boundaries. Non-reflecting conditionsii are applied at the inflow (I = a) and out- ilow ( r = 6 ) boundaries, fluxes are zeroed on the axis of symmetry. and tangency is enforced along the boundary r = 12.

The vortex is i n i t i d l y positioned such that the center of the filament core is 7.56 upsueanl o f r , and 1256 from the axis of symmetry. At this distance from the axis, the fila- ment core can be assumed to have a circular cross section. An interaction zone is predetermined, within which all in- teresting details of the solution are resolved on a fine coni- putational mesh, as shown in Fig. 8. The con~putational mesh is 256x236 and is composed of a uniform interior region that is smoothly connected to an outer suctched region. The interior interaction zone has mesh spacing A I = A r = 6 1 5 . Every fourth point of this mesh is plotted in Fig, 9.

For 1 > 0 , the vortex convects downsueam and passes through the shock, at which point the most significant part of the interaction occurs. The shape of the shock is dis- torted. the vortex strength is altered by thepost-shockstate. and an acoustic wave is produced. This interaction is dc- picted in Figs. loa-c. The numerical scheme implemawd here is a third-orderENO scheme with the frecly adaptive stenciling algorithm in F4. 6. The range of 50 equally spaced pressure contours is resuicted to enhance the visu- alization. The unit of time for these figures is T, = 6 1 7 1 ~ .

-

4

As in Ihe previous examplc, the focns of the present dis- cussion is the numerical a lg~i r i t l~n~. A discussion of this problem with respect to the physics of sound generation will appear in a future paper.

To test the effects of stcncil biasing oil this interaction. the solution is examined at 1 = I O Ta , after the entire iil- ament core has emerged downstream of the shock. Fig. 1 1 a shows a restricted range of pressure contours on a con- fined area within the interaction zone. Again, the numer- ical scheme is implemented with frecly adaptive stencils. The sueamwise recnnslruction stencils for the characteris- tic variable associated with the eigenvalue A = 71 + c are plotted in Fig. I Ib. llie shading in this ligure represents the three possible values ofthestencil offset j 3 ( i ) - j i ( z ) . me slcncils vary greatly when allowed to adapt frwly, par- ticularly in a case such as this where only small gradients are present in an otherwise uniform region. Of particular concern are the white regions in which the stencil offset is I . which represents a downwind-shifted stencil for this characteristic variable. Although the calculation for this particular test case does not appear to be adversely af- fected. previous r e ~ e a r c t i ~ , ~ , ~ indicates that. when linearly unstablestencils are continuously used over the duration of a calculation, accuracy can be severely compromised. The objective here is the development of a robust numerical shock-capturing scheme that can be used for a variety of CAA problems that involve shocks. Therefore, an attempt to bias the stencils is examined.

I o Fig. 12a,spuriouswavcs aboveand belowthefilament core appear to the right of the shock when the scheme is applied with linear stencil biasing in F!. 7 with u = 2, c = 0.001, and a = 0. Ilowever. the reconsLructicin stencils in Fig. 12harecenteredin thesmoothregionsasdesired. Figs. 13a and 13h depict the pressure field and stencils obtained with the nonlinearly biased stencil algorithm in Q s . 7 and 9 with u = 2, c = 0.001, and a = 2. The spurious wave lhat appears i n Fig, I2a is only slightly visible beneath the vortex in Fig. l3a. Also. the reconstruction stencils are centered (ix., stable) everywhere except near the shock and the vortex filament core. 'Illis result indicates that the nonlinear biasing of stencils makes E80 schemes more robust with regard to aeroacoustic applications that involve shocks.

CONCI.UDING KBMAKKS

The control of the adaptive stenciling employed in high- order accurate 1;NO schemes is essential for aeroacoiistic applications. The modifications ]hat have been suggested by other authors',s in regard to the biasing of stencils to- ward l h o x that are linearly stable have been demonstrated to Serve their purpose i n smooth regions of a flow. IIow- ever, thccases tested in this work have shown that this type of linear biasing can cause errors near moving shocks. A proposed nonlinear hiasinglias been shown to reduce some

5

of the spurious entropy generated by a sound-shock inter- action in a quasi-one-dimensional nozzle. Such a nonlinear biasing allows a freer stencil adaptation near a movingdis- continuity than is allowed by the use of constant biasing parameters. This nonlinear stencil biasing was also tested on a shock-vortex interaction and shown to reduce spuri- ous errors in a manner comparable to results ohrained with a freely adaptive algorithm, while biased. stable stencils were retained in smooth regions.

FIGURES

ut + a u x = 0, a > O

\ a) - 3 :

X

REFERENCES

I . Ilarten, A., Engquist. B. , Osher, S.. and Chakravarthy, S., “Uniformly lligh Order Accurate Fsentially Non- oscillatory Schemes 111,” Journal of Computational Physics, Vol. 71. No. 2, 1987. pp. 231-323.

2. Shu, C. and Osher. S., “Efficient Implementation of Es- sentially Non-Oscillatory Shock-Capturing Schemes.” Journal of Computalional Physics. Vol. 77. No. 2, 1988,pp. 439-471.

3. Rogerson, A. and Meiberg. E. “ANumerical Study oflhe Convergence Properties of EN0 Schemes,’’ Journal of Scientijc Compuling, Vol. 5. No. 2, 1990, pp. 151-167.

4. Shu, C.. “Numerical Experiments on the Accuracy of I340 and Modified EN0 Schemes,” Journal o/Srienri>C Computing,Vol. 5 , No. 2, 1990. pp. 127-150.

5. Atkins. 11.. “IIigh-Order ENOMethods for the {Jnsteady Navicr-Stokes FAuations,” AlM 91-1557. AIAA 10th Compritational Fluid Dynamics Conference. Ilonolulu, Hawaii, June, 1991.

6. Casper. J. . Shu, C. W., and Atkins, 11.. “A Comparison of Two Formulations for Iiigh-Order Accurate hsen- tially Non-Oscillatory Schemes.” A I A A Journal. Vol. 32,No. 10, pp. 1970-1977,Ocrober. 1994.

7. Casper. I. and Atkins. 11, “A ~iinite-VolumcIligh-Order F i O Scheme for -0-Dimensional llyperholic Sys- tems.” Journalo/CotrtpirrarionalPh~sirs. Vol. 106, May, 1993. pp. 62-76.

8. Meadows. K.. Caughey, I>.. nnd (:asper, J., “Compiiting Unsteady Shock Waves for Aeroacoustic Applications,” A I A A Journnl. Vohinie 32. No. 7, pp. 1160-1366. July. 1994.

9. Liepniann. 1I.W. and Roshko, A,, Elcrnents of Gasdvnamics. John Wiley & Sons. Inc. New York. 1957.

10. Iamb. 11.. lIydrodynamic$. (’nmbridgeiJniversity I’ress. 1932.

I I . Atkins. 11. and C q v x . J.. “Non-Reflective Boundary Condilions for Iligh-Order Mcthods.” AIAA Journal . Volume 32. No. 3. pp. 512-51R.March. 1994.

I f;+l,2

Figure 1: Preferred reconstruction stencils t ha t result in fluxes tha t are a) one-half cell upwind and b) one cell upwind.

A

0.0 0.2 0.4 0.6 0.8 I .0

Figure 2: Nozzle area distribution

6pX lo3 1.0 r I

o n

-1.0 I I. I X 0.0 0.2 0.4 0.6 0.8 1.0

Figure 3: Envelope of the pressure perturbation. -

?

4

3.0

1.5

0.0

-1.5

-3.0

I1 II

- I I I1

- - Linear Biasing I \ I 1 - Nonlinear Biasing I I !I

-

- 1 i,; \; I 1 I1 . ’ ‘ I X

Figure 4 : Entropy perturbation, comparing stencil biasing types.

Nonlinear Biasing

I

01, = = 3 - 0 -

- 1 I I 0.76 0.77 0.78 0.79 0.80

f ~ / f y

/y u4 U , + 6 U I I

I ............................. ....................................................................

t

r

t A

Figure 7 ; Toroidal vortex core

A T

-d n b U *I

(-2’36 *I

Figure 8: scale). Shading denotes fine-grid interaction zone.

Diagram for initial conditions (not to

92 142 -8 42 -108 -58

Figure 9: Cornputatiorlal grid (<:wry fourtl, point).

147.

99.

I

@

- 1.8 41

Figure 10% Pressure, t = 0, 0.994 < P < 1.000.

r/6 147.9

99.1 -10.8 41.9

Figure IOb: Pressure, t = 12T6, 1.792 < 1’ < 1.813

147.

99. 8 41.

a

135.3

112.7 4.1

s

19.9

Figrirr I l a : Pressure, t = 1OT6, 1.792 < P < 1.813, freely adaptive stenciling, r? = 1,c = 0, a = 0.

135

1 1 7 . 7 4.1 19.9

z/a

d

Figure I l h : Rcronstruction stencils associated w i t h = 11 + c for the solution in Fig. I l a . The values

of js(i) - j;(i) are - 1 for black, 0 for grey, and 1 for wliitr. -

9

d

135.2

112.7 4.1 19

Figure 12a: Pressure, 1 = IOTa, 1.792 < P < 1.813, linear-biased stenciling, u = 2 , ~ = 0.001, a = 0.

.r / A 135

117

1 , -

X / 4.1 19.9

Figure 1% Reconstruction stencils associated witli X = u + c for the solution in Fig. 12a. ‘The values of j n ( i ) - jz(i) are - 1 for black, 0 for grey, and 1 for wliite.

,J

135.3

112.7

‘Fi

4.1 19.9

Figure 13a: Pressure, 1 = IOT,, 1.792 < P < 1.813, nonlinear-biased stenciling, d = 2, c = 0.001, a = 2.

7 /A 135

1 1 7 ~

. , -

X / 6 . 4.1 19.9

Figure 1311: Reconstruction stencils associated with X = u + c for the solution in Fig. 13a. The values of ja(i) - jz(i) are - 1 for black, 0 for grey, arid 1 for white.