ADAPTIVEEAGLE N 9 3 " 2:7 4 GDYNAMIC SOLUTION ADAPTATION AND GRID

QUALITY ENHANCEMENT S...3

Phu Vinh Luong

Naval Oceanographic OfficeStennis Space Center, MS 39529

J. F. Thompson, B. Gatlin, C. W. Mastin, H. J. KimNSF Engineering Research Center for Computational Field Simulation

Mississippi State UniversityMississippi State, MS 39762

In the effort described here, the elliptic grid generation proce.dure in the EAGLE grid code has been separated from the maincode into a subroutine, and a new subroutine which evaluatel _v-

eral grid quality measm'_ at each lgid point hal been added. Theelliptic grid muline can now be called, either by a CFD code to sen-eraie a new adaptive grid based on flowvariables and quality inca-rares through multiple adaptation, o¢ by the EAGLE main codeto generate a grid based on quality measure variablea through litai,,ic adaptation. Arrays of flow vatinblea can be read into theEAGLE grid code for use in static adaptation u welL These majorchanges in the EAGLE adaptive grid system make it easier to con-vert any CFD code that operates on a block-slructured grid (orsingle-block grid) into a multiple adaptive code.

IHI_,DD1/ClIDB

The requirements of accuracy and efficiency for obtainingsolutions to PDE's have alway1 been a conflic_ in numerical meth-ods for solving field problems. On the one hand, it k well knownthat increasing the number of grid pointsimplies de_em_ng the lo-

cal truncation error. This, however, results in bng computationtime due to large numbers of grid points. On the other hand, short-er computation time can be achieved by decreau i the number ofgrid points, but the result is a le_ accurate solution.

Adaptive grid generation techniques are a means for resolvingthis conflict. For many practical problenm, the initial grid may notbe the best mired for a particular phydcal problem. Fc_ example,the location of flow features, such m shocks, botmdmy and shearlayenk and wake regions, are not known before the grid is gener-ate& In multiple adaptive grid generation, grid points are movedcontinually to respond to these featurel in the flow field m they de.velop. Thin adaptation can reduce the oscillatiom umciated withinadequate resolution of large gradients, allowing sharper shocksand better repreaentation of bouada_ iayenL Th,,- it ia postholeto achieve both efficiency and high acx:uracy for numerical mlu.tiom of PDE'L Several basic technique, involved in adaptive gridgeneration are dik-'u_d in Ref. (1).

In the earlier form of the adaptive EAGLE _em (2, 3), thecoupling of the adaptive grid syutem with a CFD code required theencaplmlation of both the entire grid code and the CFD flow codeinto separate subroutines, and the construction of a driver to calleach. _ wu inefficient that it included mine mmeceua_ partaof the grid code and required significant umdification, and per-Imp, remmmutng, of the CI:D code. In particular, the flow codeazr_ and/o¢ the grid code mTays had to be modified to be compat.l_le in mucture.

The convemion procedure il accomplished by adding the el-liplic grid generation subroutine, and certain other mbroutinel

from the EAGLE grid ,ystem that are invoh,ed in the elliptic gridgeneration protest, to the flow code. The CFD code may then callthe eRiptic grid generatiou routine at each lime step when a new

grid is desired. The CFD code passe, its current solution to thisEAGLE routine via a scratch Rle. This structure eliminates theneed for compatibility between CFD and grid arrays. One restric.tions is that the initial grid must be generated by the EAGLE sys.tern, or be processed through that system. This provides the neces-saryparameters and structural information to be read from flies bythe adaptive EAGLE routine.

In the present work, the control function approach is tuml asthe basic mechanism for the adaptive grid generation. The staticand multiple adaptive grid generation technique_ are investigatedby formudafin 8 the control functions in termJ of either grid qualitymeamr_ the flow mlution, or both.

Previomwork (2, 3) with the adaptive EAGLE system allowedthe grid to only adapt to the gradient of a variable. The work de-scribed here has extended thk adaptive mechanism to also allowadaptation to the curvature of a variable or to the variable itselLThe aystem provides for different weight funcdom in each coordi-nate direction. In addition, the mechanism now includes the abil-ity to calculate the weight functions mweighted averages of weightfunctions fi'om several flow variables and/or quality measures.This allows the adaptation to take into account the effect of manyof the flow variable, instead of just one. The construction of theweighted average of flow variables and quality measures, and thechoice of adaptation to gradient, curvature, orvariable, are allcon-trolled iu each coordinate directions through input parameters.The quality measures now available in the EAGLE grid system areskewness, aspect ratio, arc length, and smoothnem of the grid.These grid quality measures, and the resulting control and weightfunction values, can be output for graphical contouring.

ADAPTIVE MECHANISM

Control Function A nnroaeh

The control funcdon approach to adaptation is developed bynoting the correqxmdence between the 1D form of the system,

xw + ex, - 0 (1)

and the differentia] form of the equidism'bution principle,Wxe = ¢omtant,

wxw + w,z, = 0 (2)

where P k the function to control the coordinate line sparing, andWi, the weight function.

https://ntrs.nasa.gov/search.jsp?R=19930018275 2020-04-30T10:30:04+00:00Z

From(I)and(2) the control function can be defined in termsof the weight function and its derivative as

r w, (3)

This equation can be extended in a general 3D form as

w,, (4)

This approach was suggested byAnderson (Ref. 4, 5), and has beenapplied with _ for 2D configurations by Johnson and

Thompson (ReL 6) and for3D configurations by Kim andThomp-sun (Re£ 2) and by 'Ih and Thompson (Ref. 3).

The complete generalization of (4) was proposed by Eiseman(_f.7) as =

where W_is the weight function chosen for the _t direction. Thisdefinition of the control functions provides a convenient means tospecify three separate control functions, with one in e_ coordi-nate direction.

In order to preserve the geome_ characteristics of the ex-isting grid, it is practical to construct the control functions in mucha manner that the control functionsdefined by (5) are added to theinitial set of control functionsobtained from the geometry, i.e.,

P_ -(P_; + C_P_. _ = 1,2"3 (6)

where

(P,). : control function based on geometry

(P_). : control function based On weight function

In these equations the weight function Wcan be computed bydifferent formulas for different adaptive mechanisms:

Adaptation to

Var_b/¢ : W = 1 + IV!

Gradien: : W - ! + IVVI

Curvature : W = (I +plgl)Jl + alVVI 2

(7)

where Vcan be either a flow solution variable or a grid quality mea-

sure. Here a and flare on the range 0-1, and

. T'V (s)X" (1 + IVVP)_

isthecurvatureofthevariableV,

Using these definitions of the control functions, the ellipticgeneration system becomes an adaptive grid generation system.This system is then solved iteretlvely in adaptive EAGLE by thepoint SOR method to generate the adaptive grid.

Grid Oualitv Measures

The objective of tiffs part of the investigation was to developa means of evaluating grids through the computation of certaingrid properties that are related to grid quality and to develop tech.niques for estimating the truncation error. Following Kerlick and

Klopfer (Re£ 8), and Gatiin, et. al. (Ref. 9), the grid quality mea-sures are taken as skew angle, aspect ratio, grid Laplacian, and _'clength. Techmques for estimating the truncation error due to thework of Mastin (ReL I0) are also included. At each grid point in

a general 3D grid, each property can have three values associatedwith the three directions. The approach taken in this investigation

is to treat each surface of constant _ separately for ease in graphi-cal interpretation.

The minimum skew angle between intersecting grid lines isone of the most important measurable grid properties. This anglecan be expressed in terms of the covariant metric elements as

Sincegn - &=, 8n = 8nandsn = 8_, the three skew anglesassociated with each grid point in a 3D grid are 0t_ 03 and 031'

asimaxatlo

Since aspect ratio is the ratio of the length of the sides of a gridcell, it can be defined in two different ways. For example, on _:

face of constant _, this ratio can be expressed in terms of me_c

elements 8u and 8_ as

A_, l _ (lOa)

or

Large changes in aspect ratio of grids from one part of the field toanother may inhibit the convergence of viscous flow solutions toa steady state.

A nsefid measure of the smoothness of a grid is the Laplacianof the curvilinear system, _, / l 1, 2, 3, which is simply therate of change of grid point density in the grid. For a perfectly uni-form grid, the grid Lap]acian would vanish everywhe_, but ex-ceedingly large values may arise in highly stretched grids. Themathematical representation of the gi'id I..aplacian is defined in

terms of the contravariant metric elements 80, the contravariant

base vectors a t and the position vector • as

Another important inc.*mare of the grid quality is the local rateat which grid spacing changes. On acoordinate surface of constant

_3, and along a coordinate line of constant _z the grid spacing canbe defined as

E

F

d, - [(x,.i - x__ + (y,+_- y_ + (-,÷_ - .0q½ (12a)

The normalized rate at which grid spacing changes (ARCL) is then

d, - dH

(ARCL), = ½ (d, + d,-O (12b)

The objective of this section is to present heuristic error esti.mates which give order of magnitude appro_dmations for thetruncation error and the solution error in the numerical solution

of the Elder equations for compres_le flow and other systems ofconservation laws. Any conservation law can be written in a gener-al formu

u,+f.+&, + h. = 0 (13a)

The transformation of this system to an arbitrary curvilinear coor-dinate system is

U, + F,_ + G,z + HO = 0 (13b)

where 4"gis the Jacobian of the transformation and

G -/_ _ + _ + _:h)

Let h be the spacing of the fine grid, and nh be the spacing of thecoarse grid. Let L, be the difference approximation operator onthe fine grid, and L., be the difference approximation operator onthe coarse grid. Then the finite difference approximation of thePDE can be represented on the fine grid as

u, + f, + g, + h, = LdF, G,H) + T(h)" (13c)

and on the coane grid as

u, + f= + &, + h, = L.,(F,G,I'/) + T(nhy (13d)

where a is an integer.

From (13¢:) and (13d), the estimate of the mmcation error onthe fine grid can be computed as

T(hF - _ (14)

A aimilar procedure can be used to compute the error in thenumerical solution. Such a procedure has long been used in thenumerical solution of ordinary different_l equations and is re.ferred to as Richardson extrapolation. Even though numericalsolutions must be computed on both fine and coarse grids, the er-ror estimates which result do not have the large peaks at solutions/nguiarities which can be encountered with the truncation errorcomputed from difference approximation of higher derivatives.Thus the solutionerror estimates may sometimes be more useful

in the construction of adaptive grids.

Assume that there are two numerical solutions of order p ac-curacy for (13b) that have been computed on a fine grid and on acoarse grid, with grid spacing h and nh, respectively, in each coor-

dinate direction. Assuming that the samepth order method is usedin both cases, then the relation between the two numerical solu-tions and the actual solution u of the PDE can be established as

u = U. ÷ R(8)' (15a)

and

- = u,= + R(nhy (15b)

From these equations, an extrapolated value ofu can be computedas

n,u. - u_ (15c)u = (w,- 1)

Thus the estimate of the error inthenumerical solution computed

on the h grid is:

. - u, _ (15d)= (n*- I)

RESULTS AND DISCUSSION

The adaptive grid generation system based on the controlfunction approach as described in the previo_ chapters has beenused to generate static and multiple adaptive grids for several ge-ometries(gel 11). Some of these results are presented in this sec-tion. The static adaptive grids were obtained by adapting the initialgrids to either grid quality m_ variables or to existing flowsolution variables. The multiple adaptive procedure was tested onseveral different configurations with the adaptive MISSE Eulerflow code (Re£ 12) for transoni-" and supersonic flow cases, andwith the adaptive INS3D flow code (Reg. 13) for incompressibleflow.

Ada_ntaflon to Oualit v Measures

Some examples of the grid quality adaptation are shown inFigure 1 for adaptation to various quality measures. (In ReL 11,

color contour plots of the quality measures and the other adaptivefeatures are given.) Figures 2a-d shows the difference of the av-erage skew angle between the initial and adaptive grids. The samenumber of total adaptive iterations were run in each case. Thecontrol functJom were updated based on the geomeUy of the pre-vious grid, rather than the initial grid" at each adaptation.

Comparison of Figure lb with Figure la shows that adapta-tion to the skewness is effective in reducing the skewness in one re-gion, while increasing the skewness in other regions of the grid. Asmall improvement inaspectratiooccurs,but the smoothness ofthe grid is decreased.

Comparison of Figure lc with Figure la indicates that adapta-tion to aspect ratio does improve both aspect ratio and smoothnessof the grid; the skewnem is increased, however. Comparison ofFigure ld with Figure ia shows that adaptation to smoothness ira.

prove= the skewness and aspect ratio of the grid effectively, but theadaptive grid is not as smooth as the initial elliptic grid.

Figure le shows the beneficial effect of including adaptationto aspect ratio, arc length, aad smoothnea, with adaptation toskewness: the skewness is reduced more by the combination thanwith skewness adaptation alone. A little improvement oco.n's inaspect ratio; the =moothness of the grid does, however, decrease.

Resultsfromtheseexamplesshowthat the adaptation to thecombination of all grid quality measures, or to each individually,can improve some grid properties while damaging others. For ex-ample the adaptation to the Laplacian of this particular grid can:educe the skewness, but the resulting adaptive grid is not assmooth as the initial grid. The choice of the adaptive variable forthe adaptation very much depends on what property of the gridneeds to be improved and the configurations of the grids.

Results of multiple adaptation performed with the adaptiveMISSE Euler flow code are shown in Figures 3-8. In all these

plots, NIT is the total number of time steps, INT indicates thenumber of time steps at which the tint adaptation is performed,NCL is the number of time steps between each adaptation, andMAXINT indicates the number of time steps at which the lastadaptation is performed. Values of weight functions(AWT, AW'/'_ weight coefficients (C1, C_O,adaptive variablesdensity (RH01, RttO_ pressure (PRE$_, PRE$_) ate given forthe _ and the _= directions, respectively. For example,AWT = GRAD, CURV, C, = 0.5, C2 - 0.3, ,t_/O - t,0,PRE5 = O,I and a - I, ,8 = I, can be interpreted as adapta-

tion to the density gradient in the _ direction with C_ = 0.5,and to the curvature of the pressure in the _= direction withC= = 0.3 with coefficients of gradient and curvature

a = 1,/3 = l, respectively.

double w_lm_ (mnerxanic Euler)

Results obtained from a supersonic flow at Mach- 2 over fine(121 x 41) and coarse (81 x 31) double-wedge grids are shown in

Figures 3-7. Figure 3 shows the pressure contours obtained from300 time steps on the initial and adaptive grids (121x 41). The gridwas adapted to the density gradient in the flow direction (R/-/O =

1,0) with C1 - 0.7 and to the pressure gradient in the normaldirection with C2 = 0.5. A total of 4 adaptations was used forthis case, with control functions updated from the previous grid.

Figure 4 shows the pressure coefficients on the lowerwall, andconvergence histories of the two solutions are shown in Figure 5.

In Figure 5, the high peaks at each adaptation are due to the useof the previous solution on the new adapted grid without integra-tion onto the new grid. From these figurer, clearly the adaptivegrid gives a much better representation of the shock regions as wellas the expansion region=. Shocks are much sharper for the solutionobtained on the adaptive grid. A record of the CPU time on anIRIS 4D/440VGX machine shows that the total CPU time for theinitial grid (121x 41) without adaptation was 1481.51 CPU secondsand for the adaptive grid (121 x 41) was 1599.02 CPU seconds, an8% increase.

Contour plots on the pressure of the initial fine grid (121 x4i),the initial coarse grid (81 x 31), and the adaptive grid (81 x 31) areshown in Figure 6. The coarse grid was adapted to the combina-tion of density and preumre in _t direction, with weight coefficientC t - 0.5, and to the gradient of this combination in _=directionwithweightcoeRicient C= - O. 5,(AWT - VAK GP,,4D,RHO =1,t, PRES = t,1).

Different adaptive mechanisn_ applied to the coarse grid inthe multiple adaptation processare shown in Figure 7. Figure Toshows the pressurecontours obtained on the adaptive grid of Fig-tare 7a. The initial grid was adapted to the curvature of the com-bination of density and pressure in both directions (A W/" = CURE',CURV, RHO = 1,1,PRES = 1,1). The total numberof adaptationswas4withC| = 0.7. C= " 0.7. Thecoeffldantsofthegradi-ent and curvature were a = I and fl = 0.5, respectively, and

the updates were from the original control functions.

Figure 7¢1show= the pressure contours obtainedon the adap-tive grid of Figure 7c. The adaptative mechanism for this case waspresmm= gradient in both directions with Ct " 0.7, C= - 0.7end total number of adaptations wes 4, (AWT = GRA_ GRAD,PJ_O= 0,0,pm_s = t,l).

The initial grid, adapted to the gradient of the combination ofdensity and pressure in the _ direction only is shown in Figure 7e.TotalnumberofadaptationswasSwith CI = 0, C2 = 0.9,andupdates were applied to the previous control functions. Pressurecontours obtained from this adaptive grid are shown in Figure 7£

From these figures, the representation of the shocks on the

adapted coarse grid is much sharper and closer to the fine gridsolution than the nonadaptive coarse grid. The total C'PU time forobtalning 300 time steps solution for the adaptive grid was approx-imately800 seconds for each adaptive mechanism, nearly 50% sav-ing time compared to that of the fine grid.

The adaptation to the combination of density and pressure in

Idirection and to the gradient of this combination in _=directionof Figure 6 gives a smoother behavior of the pressure coeff'tcientbehind the shock than the adaptation to the gradient of pressurealone of Figures 7c and 7d. The adaptation to the curvature of Fig-ures 7a and "Togives a better result, however with a little over pre-diction of the pressure coefficient right behind the shock. Theadaptation to the gradient of the combination of the density and

pressure in ]_ direction only in Figures 7e and 7f gives the closestsolution to the fine grid solution.

From the= results, clearly multiple adaptive grids produce abetter representation of the shock regions, aswell as the expansion

regions, than that of the same nonadaptive grid. Among theseadaptive mechanisms, the use of the better results than the use ofsingle variable. Another advantage that shouldbe mentioned hereis the controlling of the direction in which adaptation is applied.As shown above, the adaptation in only one direction (_2) gives theclosest solution to the fine grid solution. Moreover, the grid in this

adaptive mechanism is not being disturbed as much as by theadaptation in both directions. The minimum skew angle for thiscase is higher compared to those of adaptation in both directions.Ofcoune, this is true only for a certain number of adaptations and

a particular vahm of weight coefficients.

wind tunnel (m_rr_nic Euler_

Results from the supersonic flow at Mach= 2 in awind tunnelare shown in Figure 8. These resultswere alsoobtalned in 300 timesteps. Figure 8a is the initial grid, Figure 8b is the adaptive gridadapted to the error estimation in both directions, and Figure 8cis the adaptive grid adapted to gradient of the combination of den-sity and pressure in both directions. The number of adaptationwa= 5 for both cases, with Ct = 0.6, C: = 0.55 for the

adaptation to gradient of the combination. Shocks are muchsharper for solutions obtained on the adaptive grids than on thenonadaptive grids for this configuration in supersonic flow aswell.

Results from these examples show that multiple adaptive grids

captured very well major features of the flow field in supersonicflow for these particular configurations. The adaptations to thecombination of the grid quality measures, such as skewness of the

grid and the flow solution, for these particular grids not only makethe _ more skewed but alsoresulted in poor resolution of the

major features of the fiow field. On the other hand the adaptationto the error estimation and the use of the weighted average in

weight functions computed from several flow variables does, infact, improve the solutions.

The computation of the weight functions and the choice of theadaptive solution variable are independent from one direction toanother thus enabling the users to have more freedom in choosingsuitable adaptive mechanism for each kind of flow. For example,in the case of boundary layers and shocks occurring in the sameflow field, the usen may choose to adapt the grid to the velocitymagnitude gradient in the normal direction to capture the bound-an/layer regions and to the pressure gradient in the flow directionto capture the shocks.

l_.t, mtrJL_tdatm_0neomnrtsslble Navinr-Stoke_

Remtts of multiple adaptation performed with the adaptiveINS3D incompressible flow code are du3wu in Figures 9-12.These results are obtained for incompress_le laminar flow for a

__=

m

|

=

!

im

2-block backward facing step, (grid size for the first block is (21x 35) and (81 x 41) for the second block). The Reynolds numberused in this investigation for the backward facing step was 183.32for comparison with experimental data.

The grid constructed for the backward facing step consideredin this case is the same as the geometry of the experiment. Howev-er, the step length downstream of the grid is only 30 times thelength of the step height, while the step length for the experimentwas much larger. Figure 9¢ shows the velocity magnitude contoursobtained from 5000 time steps on the initial grid of Figure 9a, Ve-]ocityvectors are shown in Figure 9b.. Figure 10 shows the velocitymagnitude, vorticity contours end velocity vectors obtained from5000 time steps on the adaptive grid. The initial grid was adaptedat 500 end I000 time steps to the vorticity magnitude in the direc-tinn normaito the walls with C_ = 0, Cz - I,(AWT-VAR,FAR, FORR ="0,I). Total number of adaptations was 2 for thiscase, and updates were applied to the initial control functions.

Skin friction coeffic/ents on the lower and upper wails (begin-ning at the step) obtained from initial end adaptive grids areplotted in Figure 1I. Velocity profiles at the step and several Ioca-tions downstream (nearest to the experimental data) along withdigitized experimental data are shown in Figure12.

Results from these figures show that the velocity profiles ob-tained from the adaptive old are closer to the experimental data

than for the nonadapfive grid. However, there are some wigglesof the skin _ction coefficient obtained from the adaptive grid oc-curring at the separation region of the lower wall. This may be dueto the redistn_oution of gr/d spacings in this region. Digitized val-ues of the reattachment length from Figure ll are appro_nately7.67 for both solution& while the experimental value was 7.9 for

this particular Reynolds number. The difference of these valuesmay be due to the difference of the step length of the experimentand thegriddownstream.

A record of the CPU time on a Cray 2 machine shows that the

total CPU time for the initial gridwithout adaptation was.25956.26CPU seconds and for the adaptive grid was 26363.74 CPU seconds.Since there is only 2 adaptive iterations the increase in time for thiscase is 12%.

180 de_m-ee turn around duet(incompressible Navim'.-Stokes)

Most flow solvers for incompreuffole flow require grid lineswhich are packed closely to the walls in order to resolve the botmd-ary layer regions. This results in a large number of grid points andhence long computer times. The multiple adaptation can be usedto reduce the cost of computer time by allowing the use of acoarsergrid. In the present investigation, a fine grid (111 x 51) with spacingoff the walls of 0.002 and a coarse grid (111 x 31) with spacing 0.004off the walls are considered for the turn around duct. The result

of the adaptation on the coarse grid is compared with the nonadap-tlve fine grid solution, while the Reynolds number for the turnaround duct was 500. Results obtained from 6000 time steps onfine, coarse and adaptive grids for turn around duct are shown in

Figures 9-18.

Figure 13 shows the velocity magnitude contours obtained onthe initial and adaptive grids. The initial coarse grid was adaptedto the velocity magnitude gradient at 1000, 1500, 2000 and 2500time steps, in the direction normal to the flow direction, (AWT -GRAD, GRAD, VOMA = 0,1). Total number of adaptations was4 with CI = O. 1, Cz = 0.5, and the updates were applied tothe initial control functions. Figures 14 and 15 show the skin frlc-tion and pressure coeffi_ents of the inner and outer wailaobtalnedfrommane, fine and adaptive coarse grids.

Figure 14 shows that the behavior of the skin friction coe_-ctents for the adaptive grid are much closer to the fine grid soin_onthan the nunadapttve coarse grid. Figure 15 showu that the adapta-tion for this case did aot help significentlyin the improvement ofthe preumra coefficients, however.

Figure 16 shows the velocity magnitude contours obtained onthe initial and another adaptive grid. The initial coarse grid wasadapted to the combination ofvorticity and quality measure aspectratio of the grid in the direction normal to the flow direction, (AWT= VAR, VAR, VORR ,, 0,1, ASPE = 1). HereCl " 0.3, C2 = 0.5. Figures 17and 18 show the skin frictionand pressure coefficients of the inner end outer wails obtainedfrom'coarse, fine end adaptive coarse grids.

Figure 17 shows that the behavior of the skin friction coe_-cient of the outer wall is almost identical to that of the fine grid.The representation of the skin friction of the innerwall is smootherthan that of the nonadaptlve grid but with a large change after theseparation region toward the outlet of the duct. Figure 18, againindicates that the adaptation did not help in the improvement ofthe pressure coefficients for this case either.

A record of the CPU time on an IRIS 4D/440VGX machine

shows that the total CPU time for the initial grid (111 x 51) was23870.61 CPU seconds and for the adaptive grid (111 x 31) wasapproximately 13800 CPU seconds for each adaptive mechanism.From FtIDtres 13 and i6, it can be seen that in both adaptations thegrids get finer at the turn. Correspondingly the skin friction coeffi-cinnts obtained from adaptive grids have higher pick at the turnand capture separation region well, as shown in Figures 14 and 17.Moreover, the reattachment point obtained from adaptive grid ofFigure 14 is closer to that of the fine grid than the adaptation ofFigure 17 end the non-adaptive grid.

The widely-used EAGLE grid generation system (Ref. 14)has been extended and enhenced so that it can be readily coupledwith existing PDE solvers which operate on structured grids to pro-vide a flexible adaptive grid capability. The adaptive EAGLE gridcode can be used for generating not only algebraic grids and ellip-tic grids but static adaptive grids as well. In the s_atic adaptation,the grid can be adapted to an e_stlng PDE solution or to grid qual-itymeasuresor to acombination of both. The test cases show that

some grid properties can be improved by the static adaptation togrid quality measures.

In this study, the weight functions can be formulated asweighted average of weight functions from several flow variables

or several quality measures or the combination of both. Differentweight functions and adaptive variables can be applied in eachdirection. These operations are controlled through the input pa-rameters in static as well as multiple adaptation mode.

There are several mccesd_ incoxporationsof the adaptiveEAGLE packed subroutines into flow codes, including INS3Dfrom NASA Ames and the MISSE Euler solver developed at Mls-siutippi State University. Several configurations are consideredfor each of these adaptive flow codes for the investigation of thenew weight fun_om formulations and gridquality measures in

the multiple adaptation.

Results obtained from the adaptive MISSE Euler flow codeshow considerable succe_ as measured by improvements in shockresolution on coante grids in the eompress_le flows. Some successhas been made in capturing separation regions on coarse grids ofthe adaptive INS3D flow rode in incompreuible flows. For furtherstudy, the interpolation of the previous solutions to the newadapted grids would be recommended, especially for the adaptiveINS3D flow code and the implementation of arbitrary blockadaptation in multi-block eonfigurations.

AC_O_E_E_NT

This work was supported in part by GrantF08635-89-C-0209 from the Air Force Armament Directorate,Egfin AFB, (Dr. Lawrence IAjewski, monitor) end in part by Con-tract NA_-36949 from NASA Marshall Space Flight Center(Dr. Paul M_mnaughey, monitor).

[lllptl¢ _ld

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Figure la_ Elliptic Grid

_t tm grSd ...........

_| It4 Jrid

i11ttUbt t, ILLo1

6¢ _,,l', _. S,. S, o# I_m,

Figure le. Adaptation to _ aspect ratio, and Laplacia.n.

N

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ia

84

Im

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Figure lb. Adaptation to skewness.

Is, i) _14

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ll_Bt |w Irid

ITtt_ll • it, _ • a

it_er._, |..So 0# Immm

F_,ttre ld. Adaptation to I.,apladsn.

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its

#41

44

AVERAGE SKEW ANGLE

• _ (Ulm_i)

. . . . . .............._I • 6 4 ? I O I0 II lJl IS 14 16 i4 IT 18 |g i

Figure 2a. The difference of *ve_e _anitle betweeninitial and adaptive grids.

i

AVERAGE _ ANGLE_yL_

m

u

M

m

m

, | .... | .............

II I 4 $ • T II • 10il Ul_ll141111tll'f|lllll*NILt

F_ttre2b. The _1[ of average skew a_le ia theadapt,doe to ,q)ea ratio.

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A_GE SK]_Y A_GLF

mu_ va. _

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adaptationtoLaphc/an ofthegrid.

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Figure 2d. Average skew angle in the adaptation to all grid

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HITs'mn

I_lmr41

3a

HITs3Qil

121m41

PI_SSUm

3b

NITs_QQ

121N41

_f_HJt

3C

N I "l'nl"m='% Intstan, ©¥1s20o 0mMtnt :2m 3d

m, ¢ u l.lead, cualll. ?, ft. S, r_ea I. m# i tqtmn_

Figure 3. Contour plots of pressure on initialandadap_veSr_

U

-4LI

-4.4

-4k4

-4.?

-4.4

°lg

pressure Coefficient (Lower wall)

I_m-adatptaU_.. Vs. Ji4apt,ll_

• _uzta4n)-- &4Sl_lll_4l )

i i , I n t-4 • l IO II i

Figure 4. Adaptation with mw- _ _ PRE$ - 1,1,cw - 0.7, 0.7.

|.0

O.6

0.0

"-4.6

-l.0

*l J5

-8.41

-ILil

-11.0

-SJI

-4.0

Convergence History

_et.mmr ap._IL_lO_. I _HZ_O, I .ev_..|

-4A

4J

\\_,

bdl_d (!111x41)

N IN 150 H4

Fmeathm _b_"

m

Figure 5. Convergence history of the initiLl and

adaptive grid mlutionL

[|1 igt IC grid

............... NITiqnn121141

F_(SSmK

NITx30o (_121N41

(lllptic 0rid

llill#llul grid

01x3i 6C

N|T=3OOoinlzll_l,cvl: "wl

PRFHliRF

IIII :200. ill Ixp:rlOo prllss I, 1 _f

il #_ir. 8_o _0. S, O. S. _ I, I

Figure 6. Contour plots of the pressure on fine, coarse

/_4g! IH grill

Oix31, i|_ |, I_l|lll. 5 78NITz_. Intsl_.cgl:_

PA(SSO_

mu_ Int s I_. mf Iupu_Im;, pi-ag81, I_lEcur_. cur_. CUSO, 7.0.7. _I • I

i

w

NI I #_ll/ _i_81x31

PR(|m

Nlt=_ao 6d1111131

U1x31 7CN IT8300. I_l# Inn. ©vl:2g

Pfl(IIORf

Nil lists liio, Ol ilqesnO, P¢'INIIII I • I 7dm,I =ormd0 grid, ©_ii. 7. O. 7

RdIptl_Qrid _l[|p|le Irld

llll W_lt _CN iTli_ma, inl | Inn, i:ilil _9

PI_FS_IW

nimlntsZnn, attlq_lnlI.O rlls=O.1 7fi,¢sgclil.e¢_4hcl_O.O.O.g, r hosa. !

Figure 7. Contour plots o(pz_l;u_ on ilc_t.iva grids.

NT_C,_Oe 9a0tl.0S0iletiIS.le:l$3.)2

IRLOC IW

aE,

r.

?

(lllptl¢ irld

Ida 8a

Ill/pI ll..IJ Irlld

NTWiamSII_ 9b

. O|t. O_, I_t_5, R_ 113.32

V(L_ I_' n_._ 11Folk

NTIIIIC= 5000

Die, _,ktt.mS. I_tm 113, 32

i 8b

Figure 9. Contour plots of velocity magnitude, and velocity

vectorson initialgrids.

idlpt hm Irld Ildept li.I w'id

i sc

Figure 8. Contour plota of _ on initial andw'nvmsam, rm. tON lOa

Dtm IS Iml_S li_tlm 3_• 0 °o •

U[I.OC I1_

vVrr r r JlilJ!iiilll i i i.... i

Y*le_lty pruflle

zJ

radium-, uer. c_ll..O.S lob u

_I_ Ntll41 _

-_--U =u= u u u L. u, u, u

.,.l _,-. w_-. m.ll..II.S |01_ u

Fiju:e 10. Contour plots ofvelacity mai_itude and velocity I

8k_ F-rleUon CoeHl_4mt in BF8z

UE

m A, Wen,

-4.1m

-4LIII• • 4 41 Ill 10 *n W 14 tl N

cut

Ill

U • ! a_

m.eu..c_m.4J)

-.4dl _ u u u IJI LII IJ I.II u

12b

ldl

t_

t.4

t-u

Figure 11. Sign friction of the lower and upper walls obtainedfrom initial and adaptive pidL

12c

IA

1.4

1.1

L4

Velocity profile

ltt._L_Jk_t_lUtm. I_

•,, .., u u L,, ,_, z., ,J u _,_,.,&&,u

F"q;ure 12. Veloc/ty profde at the stepx = 2.'/'/,x = 6.27,andx - 8.128.

Elliptic erid

IIix31

deub im simc intl 13a

RChl_t lue erld

mtux=_, I_ , tun, 2_0,

dtm_O, n9 , lie,. 511. Imt_ 10 . 13b

UELOC ITV fl_GN I _lB[

n t mixltilltl

4tu|. I: . IImL_DO . betma l| .

UI[L/_C I TV _GN I TIJOE

met zgrml, orsm. c_ s . I, .S

_..,m. i..! i,.,-w,. 13d

F'q_re 13. Contour plots of velocity magnitude on initial coarsegrid and adaptive coarse grid with awl = 8rod, 8rod,

VOMA - 0,I, ¢w - 0.1, 0.5.

Skin Friction CoefficientDt .,G.OJLPNI4k I .l14t_= l O,JNtm@0QO,IUe-500

0.34

0JS

O.ll

_3 0.10

O.O6

0.00

-O.O6

-e.fG

_ (llleil)

htkthd (111"_'1)

idlapU_ (Iildl)

I 4 • | to I_. 14 IS 18 tO

Figure 14. Skin h'iction coefficients of the inner and outer wallsobtained from _ fine, coarse and adaptive grids with,,w =

_ VOM.A = 0,1, cw = 0.1, 0.5.

Pressure Coefficient

Dt=,_o_. Lj_t, , j oJet ,,,,#,o_P.qe-eeoI..o

_'4.s "

-L, t I ! _ _--, (ll_dl)

0" t v _ jbialt4_(Ill|_ _ hdt_d (tlLl_)I)

/-4.0 0 t • • 8 tO It 14 IS i8

Figure 15. Pressure coef_ene, o/the inner and outer walkobtained from initial fine, coarse and adaptive coarse srids with

,swf- y,_,_ VOMA = O,l,¢w - 0.1,0-5.

F|llptic grid

111_31

ntmzSOml, laOfl. 1580.2000.2506

:,_m, dtm-o, g2, ne-Soa , bet*a m. 16"0

ULrLOC ITV _SN I TU_

nt m=GtLq]g

dt _fl. 02. NosSd_. bereft ira. 1_

I&'LOC I TV _q_ ITU_

o_

O_

OJO

O.IO

o.to

0,06

0.00

-4.10

0

Skin Friction CoefficientDt=O.O_Prs=o+i Jh_t= l oJ4t=eOOo,hffi_oo

lm[t._.l (111z61)

-- I_/,+-_ (inx,,_1)

-- _l+qm,, ())sz:)1)

, | , , , , l -- -

| • 6 8 10 12 14 14 18 zo

t.m

Figure 17. Sldn fr/ction coeffic/ents of the inner and outer wailsobtained f_om initial fine, coarse and adaptive coanz grids with

a_v - m_ v_ I/ORR = 0,1,.4_vE - I, cw = 0.1, 0.5.

+

1.0

0 a

0.0

-0.4S

-l.O

-I,ll

-3.0

-_L6

-4+O

Pressure CoefficientDt =O.O_hPn_O.l Jls_= l O_ItmlOOO.Ib=600

4 I • 10 I_ l• 11 18 ZO

L/!

Figure 18. Pressure coefflc/ents of the ismer and outer waitsobudned from initial fine, coarse and adaptive coarse grids with

a_t = _ m_, VORR = O,I, ASPE = l,¢w = 0.I, 0.5.

i

z

•lll lqmr, l_llr, 1:1410.3, n ..16d

Figu_ 16. Contour plots of velocity magnin.sde ou initial coarsepid and adaptive com'se grid with mw - _._ _ t,'ORR = 0,1,

.,,/SP_" - 1, cw - 0.I, 0.5.

1. J. E Thompson UASurvey of Dynamically- Adaptive Grids in theNumerical Solution of PDE",Journa/ofAppliedNumericaiMathe-ma_s, Vol. 1, pp. 3-27, North-Holland, 1985.

2. H.J. Kim and J. E Thompson" 'Whree Dimensional Adaptive GridGeneration on a Composite Block Grid",A/AA 88-0311,A/AA26th Aerospace Sciences Meeting. Reno, Nevada, 1988.

3. J. Tu and J. F. Thompson, "Three Dimensional Solution-Adap-tive Grid Generation on Composite Configurations", A/AA90-0329,A/AA 28th Aerospace Sciences Meeting. Reno, Nevada,1990.

4. D.A. Anderson,"Equidistribution Schemes, Polsson Generators,and Adaptive Grids",Applied Ma_ and Computa:/o_ VOl.24, p. 211, 1987.

5. D.A. Anderson, "Generating Adaptive Grids with ConventionalGrid Scheme", A/AA 86-0427, A/AA 24th Aerospace ScienceMeeting. Reno, Nevada, 1986.

6. B. E Johnson and J. F. Thompson, "Discussion of a Depth-De-

pendant Adaptive Grid Generator for Use in Computational Hy-draufics", N_ Grid Genzvution in Computarional Fluid Me-

chan/cs, J. Hauler and C. "l_ylor, Ed., Pineridge Press, 1986.

7. E R. Eiseman, "Adaptive Grid Generation', Computer MethodJ inAppliedMechanicJand_, Vol. 64, p. 321, 1987.

8. G.D. Kerlick and G. H. IGopfer, "Assessing the Quality of Cutvili-near Coordinate Meshes by Decomposing the Jacobian Matrix",N_I Gr/d Genenu/m, J. E Thompson, Ed., pp. 787-796,North-Holland, 1982.

9. B. Gatlin, et al., "Extensions to the EAGLE Grid Code for QualityControl and E_iency', A/AA 90-0148, A/AA 29th AerospaceSciencee Meeting. Reno, Nevada, 1991.

10. C. W. MMlin, "Error Estimates and Adaptive Grids for the Nu-merical Solution of Conservation Laws", _ o[_ Fw_International Coherence on Computation _ pp. 73-76,Boolder, CO, June, 1990.

11. Phu Laong. "Analy_ and Control of Grid Quality in Computa.tional Simulation", PhD Dia6ertation" IVAui_ppi State Universi.ty, December, 1991.

12. D.L. Whiffieid, "Implicit Upwind Finite Vohtme Scheme for the3D Euler Equntiont _, Mississippi State Univez_ty, MSSU-EIRS-A,SE-85-1, September, 1985.

13. S.E. Rogers, D. Kwak and J.L. Chang. "INS3D - An Incompress-ible Navier-Stokes Code in Generalized 3D Coordinates",NASA TechnicalMemorandum 100012,NASA Ames ResearchCenter, November, 1987.

14. J. E Thompson,'A Composite Grid Generation Code for General3D Region - the EAGLE Code",A/AA Jouma/, VOl.26, No. 3, p.271. 1988.

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