Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1996A9618485, NCC2-374, AIAA Paper 96-0524

A feasibility study of a cell average based multi-dimensional ENO scheme foruse in supersonic shear layers

Oshin PeroomianCalifornia Univ., Los Angeles

Sukumar ChakravarthyCalifornia Univ., Los Angeles

AIAA 34th Aerospace Sciences Meeting and Exhibit, Reno, NV Jan 15-18, 1996

The performance of a multidimensional cell average based ENO scheme is evaluated for use in supersonic confinedshear layers. Different ENO recipes for limiting the polynomial coefficients are investigated and their resultsdocumented. For shear layers with strong shocks a coefficient-by-coefficient technique must be utilized in order toavoid strong oscillations near discontinuities. Also, ENO polynomial interpolation on primitive variables, calculatedfrom the conservation variables, was studied. Oscillations in pressure and species concentration were greatly reducedby this process. (Author)

Page 1

AIAA-96-0524

A Feasibility Study of a Cell Average Based Multi-DimensionalENO Scheme for use in Supersonic Shear Layers

Oshin Peroomian and Sukumar ChakravarthyUniversity of California Los Angeles

Abstract

The performance of a multi-dimensional cell averagebased ENO scheme is evaluated for use in supersonicconfined shear layers. Different ENO recipes for limiting thepolynomial coefficients are investigated and their resultsdocumented. For shear layers with strong shocks acoefficient-by-coefficient technique must be utilized in orderto avoid strong oscillations near discontinuities. Also, ENOpolynomial interpolation on primitive variables, calculatedfrom the conservation variables was studied. Oscillations inpressure and species concentration were greatly reduced bythis process.

Introduction

The use of cell average based multi-dimensional ENOschemes described in Harten and Chakravarthyfl],Chakravarthy[2], and Casper[3] have been mostly restricted toblunt body and oblique shock type calculations. Pointwiseschemes such as Shu and Osher[4] have been extensivelytested in 1-D problems and have been applied to onlyCartesian grids in multiple dimensions. Shu et al. [5] usedthis type of scheme for temporal simulations of a highsubsonic shear layer. Others such as Atkins[6] have extendedthese ideas into curvilinear coordinates and have also appliedthem to shear layers[7]. Of course, many researchers haveinvestigated the compressible shear layer numerically and asmall survey of these was done by Peroomian[8]; however,this overview focuses on the numerical schemes used ratherthan the physical results obtained. The need for high orderaccurate methods in numerical simulations of shear layer orany other shear flow instability is evident in the work ofAtkins[7] who points out that second order methods used withcoarse grids can lead to spurious energy transfer betweenmodes. Most of the simulations of shear layers using ENOschemes have been done for moderate or high subsonic freeshear layers, where at most shoddets appear within thefiowfield. In most cases these shocklets are stationary withinthe computational box (temporal simulations). For the caseof the confined shear layer as shown by the spatialsimulations by Sigalla et al. [9], Lu and WuflO] andPeroomian[8], strong shock-expansion structures developwithin the fiowfield which are convected downstream. Allthree analyses were done using second order TVD schemes.These TVD schemes require a lot of grid points for accuracyThe feasibility of a higher-order multi-dimensional ENOscheme given in [2] is studied for the use in supersonic

confined shear layers. The reason for choosing such a testproblem comes from the fact that the fiowfield is rich withstrong 2-D shocks and expansions which are convected in theflow direction and the initial shear layer profiles themselvespossess large gradients within a small layer near thecenterline. These features pose problems for some schemeswhich work well for 1-D and 2-D blunt body calculations.Many engineering fluid flow applications involve complexgeometries which can have embedded shear layers and otherhigh gradient structures. The need for a method that canhandle these complex geometries and maintain ENOcharacteristics is still a continuing subject of research. In thispaper, the scheme will be presented first followed byimprovements and enhancements which can improve thesolution.

Governing Equations and Numerical Scheme

The Euler equations used to model a compressibletwo-species non-reacting mixing layer can be written in anon-dimensional conservation form:

.dt dx dy

or

dtwhere

ppupv

puut + ppuv

pvpuvv2 + p

y — 1 2

The scheme discussed below is an ENO based multi-dimensional shock capturing finite volume scheme in twodimensions. The scheme is discussed in Harten andChakravarthyfl] and Chakravarthy[2]. The equations areintegrated over a control area^4,

and by using Leibniz's and Gauss's rules one can obtain,

where n=nxi+nyj' is the outward unit normal to thecontrol surface, fit = -xnx - yny is the movement of the

qdxdy is the average of thecontrol surface, and - J_A

conservation variable over the control surface. For the presentpurposes, let us assume a stationary grid; however, the codewritten for this analysis can handle moving grids. Also, it isassumed that the discretization of the domain is done by theuse of quadrilaterals with nodes ABCD (i.e. structured grids),Based on these assumptions, the equations can now bewritten as the following:

where q , ,, is the cell average in cell j,k; A/k is the area ofj » K

cell j,k; ABCD are the nodes of the quadrilateral and n arethe cell face unit normals. The heart of the numerical methodis calculating the integral of the flux, F , across each cellwall. The integral in the equation above is evaluated usingnumerical formulae. Depending on the order of the method,one can choose the numerical integration technique. Forsecond order methods one can use a midpoint rule, and for afourth order method a two point Gaussian quadrature can beused.

Since only the cell averages are updated at each timestep, one needs a way to evaluate the fluxes at the cell walls.This is done by fitting a polynomial for each conservationvariable to the cell averages. The polynomial is one order lessthan the order of the method. Therefore for a fourth ordermethod one can write the following multi-dimensionalpolynomial:

central cell is used to solve for bo in such a way that the cellaverage matches the average of the polynomial in the centralcell. Examples of two stencils are shown in Figure 1. Oncethe polynomial coefficients are known for each cell, achecking routine is run to see if any of the interpolations havebeen done across a discontinuity. It is well known that thiswill give rise to Gibb's type phenomenon which manifestsitself within the numerical solution and appears as theoscillations within the flow field near shocks and reduces thefidelity of the method. The checking routine is a hierarchicalmethod of comparing derivatives of polynomial at the cellcentroid (i.e. the coefficients) to the ones of the neighboringcells. This is the heart of the ENO interpolation method.

The initial recipe for this process was taken fromHarten and Chakravarthy [1] and Chakravarthy [2]. Forexample, in a third order method, the LI norm of the firstderivatives of the polynomial for the central cell (j.k).< / = b , | + &, , is calculated initially. Then this norm1 l>j,k < L'j,k

is compared to e = a n / + K*2 / . where a is a weight

factor to bias the central stencil (1 £ a £ 2) and / representsthe cells to the top, bottom, left and right of the central cell. If^el ^min ^ ^ ' ^en ̂ e ^ocus shifts to cell /. Now, the next

level of derivatives is compared, i.e pJ

If there was a shift at the first level, then the norm for cell /*,,„is compared with that of its neighbors' (top, bottom, left andright) without placing any weight on the norms; however, ifthere was no shift in the first level, then a weight is placed onthe norms in order to bias the centered stencil coefficients.After all the levels, if a shift away from the central cell hastaken place, a "shift" in the coefficient bo is done in thefollowing way:

OQ = u, t ~"eno J'K Clmbm.lm=\

are the cell averaged geometry properties, i.e.etc. This again insures the fact that the

where aQI = *„ - x,

;,* /cell average is still the average of the polynomial used in theevaluation of pointwise values within the central cell. Thepolynomial used to evaluate pointwise values at theintegration nodes of the central cell now becomes:

where xc, yc is the location of the centroid of the cell. Theaverage of the polynomial in each cell is fitted to the cellaverage values of the dependent variable in a neighborhood ofcells. The number of cells in such a neighborhood couldexceed the number of polynomial coefficients that must becomputed. The coefficients (bi-b?) are solved using a leastsquares approach and a cell centered stencil, i.e. first, theequation for the central cell is first subtracted from the othersand then least squares is applied. Finally, the equation for the

As will be alluded to later, this is somewhatinconsistent since the comparison of the derivatives was doneby coefficients only but the above shifting of the polynomial(coefficient bo) is essentially equivalent to the following:

eno+2b4 (x

* c -xj , k

(y -ycj.k ci

=4.5 = 1.6

enob4 = b4 be = be

eno I eno I

and

bn = ii . i. — A CtmOm.eno"eno J'K _iim=l

X-eno cj,k "eno 7,*

The inconsistency arises from the fact that, byguaranteeing is the "smallest" one does not

necessarily guarantee the same foreno

(It seems

that a comparison of the latter quantities is more consistent,although more expensive computationally, and so will bediscussed later). Another problem with this-scheme lies inpicking the coefficients if a weight is placed on the centralstencil. The scheme picks the central coefficients if their LInorm is less then a times that of the neighbors; however, ifthe coefficients from the neighbor are selected, they are notmultiplied by a.

Once the "best" polynomials (polynomials whosecoefficients are not found by interpolating acrossdiscontinuities) are found, then one can go about finding thefluxes at the cell faces. At each integration point we have avalue from the left hand side (evaluated using that cell'spolynomial) and a value from the right (evaluated using theneighbor's polynomial). The "correct" values of the fluxes,F(q) • n , at the integration points are found by a Riemannsolver.

Once the fluxes are known, the right hand side ofequation (6) is known and therefore the equation can beintegrated using an explicit scheme such as a Runge-Kuttamethod.

Spatial Simulations of a Supersonic Shear Layer

A non-dimensional computational box of 16x1 (lengthby height) was chosen for the simulations. In the x-direction,the grid was taken to be uniform; however in the y-direction,a special clustering was used near the centerline to resolvethe initial profiles and eigenfunctions. The entire flowfield issupersonic; therefore at the inflow boundary, all conditionsare enforced and at the outflow boundary, where no boundaryconditions are necessary, the values from the polynomials areused. Wall boundary conditions are employed at the top andbottom boundaries. The profiles used to initialize thecomputational box are those from Peroomian[8]. Theseprofiles are the solution of the steady compressible boundarylayer equations for the parameters by Tarn andHufl 1].

• = 1.29 = 1.40(7/2)

These profiles are much different than the hyperbolictangent profile. Based on the linear analysis from [8], all thevariables where forced by a sinusoidal forcing at thefundamental frequency.

/'

where the qr and q^ are the real and imaginary parts of theeigenfunctions and e is the forcing amplitude taken to be 5%for these simulations. Finally a third order Runge-Kutta timeintegration with a CFL=0.5 was used to advance the solutionin time.

Performance of the TVD and ENO schemes

Many problems were encountered when the multi-dimensional ENO scheme was used in the investigation ofthis supersonic shear layer. These difficulties orshortcomings of the scheme in question can be attributed tothe type of physical phenomena which are initially present inthe flow field or develop as the shear layer evolves and theinability of the ENO recipe to prevent the oscillations arisingdue to these phenomena. First, these problems will beexplained and then some improvements will be given.

The initial conditions in the computational box areonly functions of y. These are the basic flow profilesobtained from the solution of the steady compressibleboundary layer equations. These profiles, although smooth,have derivatives with large magnitudes, especially the higherderivatives (i.e. q , , etc.) In fact, these profiles havemuch larger derivative magnitudes than those of hyperbolictangent approximations, especially at non-unity densityratios. This causes the ENO scheme to switch to a one-sidedstencil near these large gradients. As the solution is updatedin time, the scheme at a particular point keeps switching todifferent stencils instead of calculating the solution using acentral stencil. This type of overswitching in the type of ENOrecipe given in the previous section occurs even if the centralstencil is given more weight than neighboring stencils andcan affect the accuracy and stability of the scheme.

Another challenging phenomenon for the schemeoccurs in the nonlinear stage of the shear layer evolution. Thegrowth of the instability wave gives rise to strong shocks andexpansions which are located next to one another. Tocomplicate matters further, these structures move downstream

causing all sorts of problems for this implementation of ENO,resulting in the numerically computed pressure becomingnegative. The ENO recipe described above has been testedextensively for blunt body, oblique shock and other similartypes of calculations and has not shown any problems. Anexplanation for the failure of the scheme might be the factthat in blunt body or oblique shock calculations the samephenomenon gives rise to high derivatives in both x and y,and therefore these derivatives are similar in magnitude.However, in the shear layer, the derivatives in y are due tothe basic profiles and can be of a different order ofmagnitude than the derivatives near shocks, i.e., in the centerof the shear layer the same phenomenon is not responsible forthe large gradients in both x and y. Due to these phenomena,as the shear layer develops, the initial ENO recipe leads to anegative pressure (see Sections 6 and 7).

The numerical problems which lead to negativepressure are given below.

1. Guaranteeing ENO behavior on the conservationvariables does not necessarily guarantee ENO behavioron the primitive variables. This is especially a problemfor the pressure and the species concentration. In theshear layer, the pressure is taken to be constant. Whenthe numerical scheme calculates the ENO polynomialsand decodes the pressure from these polynomials,unwanted "wiggles" and oscillations are introduced inthe pressure which can lead to negative pressure duringthe shear layer evolution in certain cases. In order toovercome this problem, a new recipe is given in the nextsection which performs ENO directly on the pressure.The species concentration, Ca can physically take onvalues between 0 and 1 ; however, due to the fact that theconservation variable p is limited more severely in someregion of the flow than pCa> the value of the speciesconcentration can go above 1 (or below zero due to otherfactors). This is compounded by the problem describedin statement W3 and, in the regions of the flow where thestrong shocks and expansions exist, Ca can take onvalues as high as 1.14. This can also cause problems inthe solution procedure unless the value is artificially setto 1 in situations where the values goes above unity.This, of course, is not a desirable situation, and someremedies will be presented in the coming section. In fact,one should note that any high order ENO scheme cannotguarantee 0 £ C because it is not TVD (i.e. itdoes not set slopes to zero at local maxima and minima);however, the scheme must be such that the overshootand undershoot at these location be small (say aboutO(h"\ as alluded to byShu andOsher[4].

Unlike the TVD scheme, the current ENO schemes mayallow growth at local maxima or minima. The TVDscheme sets the slopes to zero at these location andbecomes locally first order. In the TVD scheme, this waseasy since in each cell two slopes (one from the right andone from the left) were computed. This is one of thereasons why the ENO scheme in question is TotalVariation Bounded (TVB) instead of being TVD.

Another fact to remember, which can affect the decisionof which set of variables (conservation, primitive, orcharacteristic) to use in the interpolation process, is thatan extrema in say the conservation variable pCa does notimply an extrema in p and Ca or visa-versa.

3. The current recipe for ENO takes a hierarchicalapproach in comparing polynomial derivatives in a.certain neighborhood. Here is a specific example whichcan occur which will fool the current ENO recipe.

Let the 2-nd order polynomial in a cell be:

where xc,yc are the coordinates of the centroid of the celland b; and bi are the x and y derivatives at the cellcentroid, respectively. Suppose least squares resulted inbi = 12 and bi = 8 and the neighbor with the lowest LInorm had b{ = 4 and b} = 10. Now depending of theweight placed on the least squares coefficients, ENOmay not switch to the lower norm. In fact, currentstudies show that placing weight on the norm of thecentral stencil (the least squares coefficients) can causeoscillations near discontinuities in certain cases (i.e. nearstrong shocks). This might be due to the fact that theconcept of giving weight to the centered stencil comesfrom a one-dimensional or a truly direction-by-directionalgorithm and needs to be more carefully implemented inthe multi-dimensional general geometry scheme. Whenno weighting is placed on the central stencil, a secondorder scheme will only pick the central stencil atmaxima or minima and at other locations will pick aone-sided stencil in one of the directions. This results ina less accurate scheme. A better solution to this problemwill be given later (sec Section 7).

4. The full implications of grid stretching on the ENOproperties of the polynomials are not well understood. Inregions of large grid stretching, some calculations showthat ENO can fail and oscillations can occur within thesolution. This was not investigated fully in this analysisand is left for consideration at a later time.

Some of these issues remain yet unresolved and arestill being investigated. However, some modification andimprovements to the multi-dimensional ENO scheme wereinvestigated in order to be able to carry out a solution of theshear layer to a time-periodic state. These modifications arediscussed in the following sections.

Pressure Decoding and ENO for Shear Layers

In schemes where the equations are solved inconservation form, the pressure needed for the Riemann

solver and fluxes is decoded from the conservation variablesusing an equation of state which is a nonlinear relation:

~

where y is a function of the conservation variable pCa. Thisequation is used after an ENO step in the spatialdiscretization is carried out. Unfortunately, ENO interpolationof the conservation variables does not guarantee ENO on thepressure field. This could lead to problems especially inboundary layer or shear layer calculations which have highgradients in the smooth basic flow profiles. In this type ofcalculation, the initial basic flow pressure is constant but theconservation variables e, p, pu, pv and pCB vary within theshear layer. The pressure is decoded from these variablesafter the ENO interpolation which yields a non-constantpressure. To take care of this problem by grid refinementwill lead to a grid which is much finer than necessaryresulting in very large computational times. An example ofthe oscillations produced in the pressure profile can be seenfor the parameters considered in Section 4. If a regular leastsquare interpolation in the y direction is implemented on theshear layer profiles without an ENO step, one can seenonoscillatory profiles for all variables including a nearlyconstant pressure. However if an ENO routine is also calledat the end of the interpolation, one sees that ENO still givessmooth profiles for the conservation variables but thepressure is no longer constant and "unwanted" oscillationshave been created. This is shown in Figure 2 where pressurealong the y axis is plotted for interpolations on theconservation variables with and without an ENO step. Thisclearly illustrates the statement that guaranteeing ENO for theconservation variables does not automatically mean the samefor the primitive variables. For the current case the velocitiesand species concentration behave smoothly after the ENOstep, and therefore only the pressure presents a problem;however, as the solution progresses the species concentrationcan also become a problem, as mentioned above. Thisproblem with the "wiggles" in the pressure persists as thesolution is advanced in time and can lead to a negativepressure (N.P.); as a result, a solution becomes impossible.This happens well before any shocks have been developed inthe flow. If ENO is turned off, the solution behaves nicely andno wiggles are seen in the pressure field.

In order to overcome these difficulties, a new recipe isused which increases the polynomial evaluations by one.First, the polynomials are found for the conservation variablesusing the usual least squared approximation. For the form ofthe polynomial used, the coefficient, bo, corresponds to thevalue of the conservation quantity at the centroid of the cell.From these centroidal quantities, the pressure at the centroidof each cell is calculated. Then a least squares interpolationis run on the pointwise values of the pressure. Finally, anENO step is performed on/;, p, pu, pv and pC0, and energy isdecoded from these quantities. An alternative to this is toperform an ENO step before finding the centroidal values.From some preliminary numerical tests, it seems that this isnot necessary and the two variations of this concept give

similar results. When this method is used on the test case, thewiggles in the pressure are no longer present. Figure 3a and bshow the pressure contours at times 18 and 21 (time-periodicstate not reached) for a case with ENO on the energy andENO on pressure respectively. Figure 4 is similar to Figure 2,with the results of the current method added to the figure.Again, it is clear that even at t = 0, the wiggles are no longerpresent. However, this fix on the pressure by itself is notsufficient to reach a time-periodic state^ This is due to thefacts given in statement #3 of the previous section.

As a final test, an oblique shock problem is run usingM^ = 2.0 and a deflection angle of 9 = 10° . The results

for the pressure and the conservation quantities are shown inFigures 5a and b. It can be seen that ENO on the pressuregives satisfactory results. In fact, they are as good as thesolutions obtained when ENO is run on the conservationvariables.

Improvements to the Multi-Dimensional ENO Recipe

As stated in an earlier portion of this chapter, theoriginal ENO recipe does not truly limit the derivatives.Examples were shown in which the ENO recipe can be"fooled" into picking the wrong stencil. This occurs simplybecause the derivatives in the two directions for the currentproblem are not similar in magnitude. It is conjectured that atruly coefficient-by-coefficient recipe is needed. In thissection a first attempt is made at such a recipe. The goal is toget a general method without paying attention to efficiencyand speed of the algorithm. Those problems will beaddressed later and are not in the scope of the currentanalysis.

As before, least squares are used to calculate thecoefficients of the polynomials in each cell. For example, forthe third order method a stencil of thirteen cells is used to fita polynomial with six coefficients. The new recipe thencompares the derivatives of the polynomial in the central cell(j, k) to the ones from the other twelve cells. The firstderivatives of the polynomials in the neighboring cells can beevaluated up to second order at the centroid of the centralcell as the following:

& ,—— = 0-ta* i i/ (x -x 3 CVC^ °j,k

-y.

6P— = b- +2b5 (y -y \fy i Ll 5/ cj,k cl

(x -x )cj,k cl

The second derivatives and the cross derivative areonly known up to first order and are proportional to the

coefficients of the polynomial, i.e. — — = 2i>4/ , etc. Nowdx*-

let:

kl= min— / = 1.....12

SPmm —|

= mm

<f>. = min

= mn

Then let:

/ = !,...,12

= 1,.. .,12

/ = 1.....12

/ = !,...,12

b, = xsignQmin b,. ,a<j>, I = 1,...,5*ovtt-* 'Ji* * \

was the inadequacy of these minor changes which led to thecoefficient-by-coefficient recipe given above. Table 1 givenbelow shows all the runs made using the original ENO recipeand the results obtained.

Table 1—Runs made with different parameters.

Run

123

456

78

Order

333

333

22

Description ofMethod

ENO on energyENO on pressureSame as run 2, butforce 0 Z Ca 51Same as 3,ct=1.5Same as 3,a=1.0Same as 3, with

adhoc TVDlimiters

Same as 3Same as 3, a= 1.0

Results

Fail(N.P.,t=21)Fail (N.P., t = 27)Fail (N.P., t = 29)

Fail (N.P., t = 32)Fail (N.P., t = 33)

Reaches timeperiodic state, butdestroys sinusoids

see Figure 6Fail(N.P.,t=37)Highly diffusive,

30% error in growthrate

bo. . ="/. - Z a, b.Results from the 3-rd Order ENO Scheme with

Comparisons to results from a TVD Scheme

The factor of two in front of the coefficients in the lasttwo equations comes from taking the derivative of thepolynomial. Of course in practice, one would just comparecoefficients of the polynomial at the last level and notmultiply them with these factors. The factor a is the sametype of weighting factor as given in a previous section and thefunction xsign restores the correct sign of the derivative (orcoefficient) chosen. Several improvements achieved over theold algorithm are stated below.

1. The weight factor is applied in such a way that a suddendrop in the coefficient magnitudes due to the switchingdoes not occur.

2. Each coefficient can by weighed differently.

3. A coefficient-by-coefficient selection process isachieved.

4. The inconsistency in evaluating b* is avoided.eno

The results of calculations with this algorithm will begiven in the next section. The algorithm presented above wasa final product of many different permutations of the initialalgorithm. The table below shows the results of calculationswith the different changes made to the initial algorithm. It

The improvements on the multi-dimensional ENOscheme proposed in Section 7 in order to overcome all thenumerical problems encountered when using the initialscheme were tested by running a case from Section 4 andcomparing the results to those found using a second orderTVD scheme given in Chakravarthy[12], As was conjecturedin previous sections, it was the ENO recipe which wascausing all the problems in the numerical simulation, and thisnew recipe avoids most of those problems and reaches a time-periodic state. This is something that was impossible innearly all the simulations conducted with the old recipe asshown in Table 1.

The ENO scheme simulation was run using a 241x145grid with a minimum by = 0.003. The TVD simulation wasrun using a 241x171 grid with a minimum Ay = 0.002, Theobject of the ENO scheme simulation was to see if theproposed recipe in Section 7 was an answer to some of theproblems described and not to get detailed accurate results.After further modification to the current recipe, the grid usedin the TVD simulations will be used to get more accurateresults. The third order Runge-Kutta time integrationemployed guarantees the stability of the centered stencil. Theweight factor for the central stencil coefficients, a, waschosen to be 2, and the species concentration was not forcedto be between zero and one for the cell averages in the timeintegration. However, the pointwise values used at the

integration points were reset to one or zero when the valueswere outside this range.

The following is an outline of the results compared tothe original scheme, besides the fact that solutions are nowpossible.

1. The minimum pressure at any time within thecomputational box is 0.0034 which is only 3% lowerthan that of the minimum pressure found using the TVDscheme. Similar trends were found for the maximumpressure.

2. The maximum species concentration was about 1.12observed in the grid stretching area. This is not betterthan that of 1.14 during the failed attempts using theoriginal scheme. It seems that in order to do a better jobon the species concentration, ENO should be applieddirectly on the species concentration as opposed to theconservation variable pCa.

3. The statement in #2 can be attributed to other quantitiesas well. The current scheme is based on limiting theconservation variables as opposed to the characteristicvariable limiting used in the TVD scheme. The problemwhich is still not solved by the current improvement isthe fact that limiting p«, pv and pCa does not guaranteethat H, v and Ca are limited as well. In essence, in certainparts of the flowfield p is limited more severely thanthese other conservation quantities and when theprimitive variables are decoded from these conservationvariables new maxima and minima can occur. Asubsonic pocket was observed in the simulations withENO which did not appear in the simulations conductedusing the TVD scheme which might be attributed to thispoint.

4. Better treatment of sinusoids occurs with a relativelyaccurate magnitude for the growth rate.

5. Almost all of the oscillations near shock-expansionstructures are removed for the pressure (some still existfor the other primitive variables).

In order to substantiate these statements, severalfigures are presented from the current simulation. Figure 7a,bshow the centerline pressure histories at different x locationsfor the ENO and TVD schemes respectively. One canobserve the kinks hi the maximums and minimums of thesinusoids in the last three plots for the ENO scheme. FigureThe instantaneous centerline pressure for t=56.0 are given infigures 8a,b for the ENO and TVD schemes respectively.Again differences near the sinusoid peaks are observable. Thegrowth rate can be found by fitting a line through the first fewpressure peaks. This is done in figures 9a and b where\n(p - p) vs. x are plotted and a least squares fit isemployed. The comparison of the growth rates found in thesimulation to that from linear analysis[8] is very good. Thecoarser grid used for this simulation along with the problemsdiscussed above lead to the type of kinks seen in the pressure

plots. The pressure, Mach number and species concentrationcontours are given in Figures 10a,b, 1 la,b, and 12a,b. In thepressure contours, the rotation of the structures occurs morerapidly for the ENO scheme than for that seen in the TVDscheme simulations also a rarefaction of the rotated structureis observed towards the end of the box in the ENO schemecalculations. The entire flowfield simulated using the TVDscheme is supersonic Mm^ « 1.25); however, in the ENOscheme simulations subsonic pockets form away from theoutflow boundary M • » 0.9 . Currently tests are beingdone to see whether ENO all of the primitive variables cangive better results. It is obvious that more improvements areneeded on this cell average based multi-dimensional ENOscheme in order to use it for accurate modeling of shear layerswith strong shocks and expansions. These improvementsrange from the stencils and neighborhoods used forpolynomial evaluations to the type of variables used for thepolynomials themselves. These improvements are a subject ofan ongoing study which is beyond the scope of the currentanalysis. It will try to unify certain aspects of the differenttypes of schemes used for shock capturing.

Acknowledgments

This work was supported by NASA DrydenResearch Center under Grant NCC2-374.

References

[1] Harten, A. and Chakravarthy, S. (1991) "Multi -Dimensional ENO Schemes for General Geometries", ICASEReport No. 91-76.

[2] Chakravarthy, S. (1994) Rockwell Science Center ReportNo. SC71039TR.

[3] Casper, J. (1991) "Finite-Volume Application of HighOrder ENO Schemes to Two-Dimensional Boundary-ValueProblems," AIAA Paper No.91-0631.

[4] Shu, C.W. and Osher, S. (1989) "EfficientImplementation of Essentially Non-Oscillatory ShockCapturing Schemes IT, J. Comp.Phys., 83 , 32-78.

[5] Shu, C.W., Erlebacher, G., Zang, T.A., Whitaker, D., andOsher, S. (1991) "High-Order ENO Schemes Applied toTwo- and Three-Dimensional Compressible Flow", ICASEReport No. 91-38.

[6] Atkins, H. L., (1991) "High-Order ENO Methods for theUnsteady Compressible Navier-Stokes Equations", AIAAPaper No. 91-1557.

[7] Atkins H.L. (1992) "Evaluation of a Finite VolumeMethod for Compressible Shear Layers", AIAA Journal, 30,1214-1219.

[8] Peroomian, O. (1995) "Effect of Density gradients onConfined Compressible Shear Layers", Ph.D. Dissertation,University of California Los Angeles.

[9] Sigalla, L., Eberhardt, D., Greenough J. and Riley, J.(1991) "Numerical simulation of confined, spatiallydeveloping shear layers: Nonlinear interaction betweeninstability modes", AIAA Paper No.91-1643.

[10] Lu, P.J. and Wu, K.C. (1991) "Numerical investigationon the structure of a confined supersonic mixing layer", Phys.Fluids A,3, 3063-3079.

[11] Tarn, C.K.W. and Hu, F.Q. (1989) "The instability andacoustic wave modes of supersonic mixing layers inside arectangular channel", J. Fluid Mech.^03, 51-76.

[12] Chakravarthy, S. (1986) "The Versatility and Reliabilityof Euler Solvers Based on High-Accuracy TVDFormulations", AIAA Paper No.86-0243.

2nd order 3r(j ordar

Fig. 1 Stencil of cells for least squares.

o 5 10 15Fig. 3a Pressure contours at time =18.0 with ENO on

energy for the test case parameters.

Fig. 3b Pressure contours at time = 21.0 with ENO onpressure for the test case parameters.

2.9620E-2

2.96106-2

2 96006-2 '——•0 10

Without ENOENOonenwgy

Fig. 2 Decoded pressure as a function of y forinterpolation with and without ENO.

2.95405-2

2.9S30E-2

2.9620E-2

2.9610E-2

2.9600E-2•0.10

Fig. 4

ENO on PressureWithout ENOEno on Energy

-0.05 0.00 0.05 0.10

Same as Figure 7 but with results from ENOinterpolation on pressure added to the figure.

0.35 r

0.30

0.25

0.20

Exact Solutionp,/p2.1.717for M.. 2.0 and 8-10°

0.150.0 0.2 0.4 0.6 0.8 1.0

Fig. 5a Pressure as a function of x across the obliqueshock using ENO interpolation on pressure.

1.5 r

1.3

1.1

pu

0.90.0 0.2 0.4 0.6 0.8

Fig. 5b Some conservation variables as a function of xacross the oblique shock using ENOinterpolation on pressure.

1.0

0.070 -

0.060 -

0.040

0.020

Fig. 6 Centerline pressure at time = 56.0 for test caseparameters using a 3rd order ENO with adhocTYD limiters.

EUU

,Ttm«

Fig. 7a Centerline pressure histories at different xlocations from simulations using the TVDscheme.

Fig. 7b Centerline pressure histories from 3rd ordersimulations using the coefficient-by-coefficientENO scheme.

0105

O.OS5

0.005oo 2.0 40 6.0 a.o 10.0 t:.o u.o i«.oX

Fig. 8a Instantaneous centerline pressure at time =56.0from simulations using the TVD scheme.

0.10

Fig. 8b Instantaneous centerline pressure at time =56.0from 3rd order simulations using the coefficient-by-coefficient ENO scheme.

J& -6.0

Growth Rale Irom Line Fil - 0.389Growth Rate from Linear An • 0.395

Fig. 9a

4.5X

First five pressure peaks vs. x from simulationsusing the TVD scheme.

10

•55

40

Fig. 9b

Slope ol line . 0.407Growth rue (lin analysis) . 0.39S

First four pressure peaks vs. x from 3rd ordersimulations using the coefficient-by-coefficientENO scheme.

C1735080 16224901509890 13973012847101172120 1059530 09469370 083434500721753006091620 0496575 038397900271387C 0158 795

p

0 1482560 1388010 1293450 115890 1104350 100979009152390 08206860072613330631579OOS3702600442473003479150 025336600158813

16.0

Fig. lOa Pressure contours at time =56.0 fromsimulations using the TVD scheme.

15

Fig. lOb Pressure contours at time =56.0 from 3rd ordersimulations using the coefficient-by-coefficientENO scheme.

M

ra

5 737195 «8025 1388518396845405'4 2413439421736433 343833044662 745492446322 147151 S47981 54881 Fig. 1 la Mach number contours at time =56.0 from

simulations using the TVD scheme.

5565665 2575749494846413943333402523 717113409023100932792842484742176651 86856: 56047y 25236

0.5

12.0

Fig. l ib Mach number contours at time =56.0 from 3rdorder simulations using the coefficient-by-coefficient ENO scheme.

16.0

11

8.0 12.0 16.0

Fig. 12a Species concentration contours at time =56.0from simulations using the TVD scheme.

4.0 8.0 12.0 16.0

Fig. 12b Species concentration contours at time =56.0from 3rd order simulations using the coefficient-by-coefFicient ENO scheme.

12