Adjustment of Dissipative Terms to Improve Two and Three- Dimensional Euler Flow Solutions

Transcription

1 Seyed Saed Bahranan Adjustment of Dsspatve Terms to Improve Two and Three- Dmensonal Euler Flow Solutons SEED SAIED BAHRAINIAN Department of Mechancal Engneerng Shahd Chamran Unversty AHWA IRAN Abstract: - The Euler equatons are a set of non-dsspatve hyperbolc conservaton laws that can become unstable near regons of severe pressure varaton. To prevent oscllatons near shockwaves these equatons requre artfcal dsspaton terms to be added to the dscretzed equatons. A combnaton of frst-order and thrd-order dsspatve terms control the stablty of the flow solutons. The assgned magntude of these dsspatve terms can have a drect effect on the qualty of the flow soluton. To examne these effects subsonc and transonc solutons of the Euler equatons for a flow passed a crcular cylnder has been nvestgated. Trangular and tetrahedral unstructured grds were employed to dscretze the computatonal doman. Unsteady Euler equatons are then marched through tme to reach a steady soluton usng a modfed Runge-utta scheme. Optmal values of the dsspatve terms were nvestgated for several flow condtons. For example at a free stream Mach number of 0.45 strong shock waves were captured on the cylnder by usng values of 0.5 and for the frst-order and thrd-order dsspatve terms. In addton to the shock capturng effect t has been shown that smooth pressure coeffcents can be obtaned wth the proper values for the dsspatve terms. ey-words: - Artfcal Dsspaton - Unstructured Grds - Fnte volume Euler Equatons - Shock capturng Introducton The conservatve form of the Euler equatons can descrbe flow condtons for any gven nvscd flow. But the dscretzed representatons of these equatons contan errors. These errors are due to numercal approxmaton and can cause soluton nstablty. Soluton nstablty of the dscretzed Euler equatons can occur near regons of hgh pressure gradents. These unstable regons nclude locatons of flow stagnaton or presence of shock waves. Stagnaton ponts or shock waves can ntroduce unwanted oscllatons nto the flow feld. To damp oscllatons for soluton stablty the Euler equatons should be modfed by addng extra dsspaton terms. The man objectve of ths study s to ntroduce proper nteger values to control the amount of dsspaton added for soluton stablty. In numercal vew pont dsspatve terms are mathematcal functons that are based on the flow varables. The proper representaton of these functons can have drect effect on capturng flow phenomena n regons of nstablty. For example to numercally capture a shock wave on a wng the correct amount of dsspaton must be added to the Euler equatons n the form of adjustable dsspaton coeffcents. An under value of added dsspaton can cause soluton nstabltes due to the presence of undamped shock oscllatons. On the other hand an over value of added dsspaton may over damp all the oscllatons and nstabltes causng the flow soluton to dverge from ther steady state condtons. The over dampng can at best gve the ncorrect locaton of the shocks. Numercally dsspaton functons are added to the Euler equatons n the form of dsspaton fluxes. These fluxes are descrbed based on the dervatves of flow varables. A thrd order artfcal vscosty has been appled by MacCormak and Baldwn [] whereby artfcal vscosty s made proportonal to second dervatve of the pressure feld n order to enhance the effect of dsspaton n the presence of strong pressure gradents and to reduce t n the smooth flow regons (c.f. Hrsch []). Another form of artfcal vscosty ntroduced by Steger [3] s based on the addton of hgher-order dervatves n numercal schemes. Jameson and others apply a blend of expressons wth excellent shock capturng propertes [4 5]. Here a pressure sensor adapted from Stolcs and Johnston [7] s used wth unstructured grds to solve nvscd flow around a crcular cylnder. The am of these solutons was to nvestgate and demonstrate the effect of dsspatve terms on the qualty of twoand three dmensonal Euler flow solutons. ISSN: Issue Volume 5 January 00

2 WSEAS TRANSACTIONS on FLUID MECHANICS Seyed Saed Bahranan Computatonal Grd Generaton The space dscretzaton of the examples presented n ths paper makes use of an n-house grd generator. The grd generator s based on a novel algorthm that can generate trangular or tetrahedral unstructured grds. For smplcty and ease of comparsons trangular meshes n -dmensonal space are generated for a crcular cylnder secton. To show consstency wth 3-dmensonal space sample tetrahedral grd and solutons are also gven for a crcular cylnder attached to a plane-ofsymmetry. Grd refnement s based on a combnaton of pont nserton and cell-subdvson methods. An edge-based and a cell-based connectvty matrx were employed smultaneously to prevent searches through the lst of edges. The art of ths grd generaton algorthm s n ts geometry movement capablty. The algorthm starts wth a very coarse ntal grd. Ths ntal grd whch can be constructed wth a few edges s shown n Fg. (a). The sold boundary whch s ntally represented wth four edges s refned and moved to ft a crcular geometry. The fnal computatonal grd output around a crcular secton s shown n Fg. (b). nteror faces and 0 outer-boundary faces gvng a total of 74 trangular faces []. Fnal computatonal tetrahedral grds are shown n Fgs (d) (e) and (f). Ths fnal grd has been employed to obtan all of the three dmensonal flow solutons presented n the proceedng sectons. (a) (c) (b) (d) (a) Intal grd (b) fnal grd (e) Fg. Computatonal grd output for a crcular cylnder D = L = 4 where: (a) and (b) show the ntal grd (c) and (d) show close vews of ntal and fnal surface faces (e) and (d) show fnal planeof-symmetry surface and outer-boundary faces (f) Fg. Grd generaton procedure for trangular cells around a cylnder secton for two-dmensonal cases The computatonal grd of Fg. has been used to obtan flow solutons for all of the twodmensonal cases presented n ths work. It should be noted that only a dagonal edge swappng procedure has been used to mprove the qualty of the two-dmensonal unstructured grds [0]. Smlar approach s taken for the generaton of three dmensonal tetrahedral cells around a crcular cylnder attached to a plane-of-symmetry and s detaled n Fg.. The ntal three dmensonal tetrahedral grds shown n Fgs (a) and (b) have been produced by decomposng hexahedral cells []. Ths ntal tetrahedral grd conssts of 8 planeof-symmetry faces 0 surface geometry faces 46 3 The Euler Equatons From a physcal-mathematcal pont of vew the Euler equatons can be vewed as the lmt of the Naver-Stokes equatons for vanshng dffuson effects. If we neglect vscosty and heat conducton terms n the Naver-Stokes equatons ther nvscd form known as the Euler equatons are obtaned. The three-dmensonal Euler equatons n Cartesan coordnates can be wrtten as: W E F G = 0 () t x y z Wth vectors W E F and G gven by: ISSN: Issue Volume 5 January 00

3 Seyed Saed Bahranan w ρ ρu w W= w = ρv 3 w 4 ρw w E 5 ρv ρuv F = ρv + ρuw ( E + ) v ρu ρu + E = ρuv ρuw ( E + ) u ρw ρuw G = ρvw. ρw + ( E + ) w The state equatons as a result of the nondmensonal parameters would become: p = ρt T = ( γ ) e = ( γ ) ρe (5) E = + u + v + w ( γ ) ( ρ ). (6) The next step n solvng the three-dmensonal Euler equaton requres the spatal dscretzaton of the governng flow equatons. The dscretzed equatons are then marched through tme to reach a steady state soluton. The procedures and formulaton of these steps are dscussed n the followng sectons. Where u v and w are the Cartesan velocty components E s the total energy per unt volume s the pressure and (w w w3 w4 w5) are the components of the unknown vector W. The frst row of the vector equaton (.e. Equaton ()) vectors W E F and G) correspond to the contnuty equaton the second thrd and fourth rows are the momentum equaton n x y and z drecton and the ffth row s the energy equaton. The vector W contans the tme dependent varables. If ths vector s neglected the remanng terms would be the steady state Euler equatons. The mathematcal character of the steady Euler equatons n subsonc flows s ellptc whle at supersonc flows ther character changes to that of hyperbolc. However by keepng the vector W the exstng unsteady Euler equatons n all of the flow regmes ncludng subsonc transonc and supersonc form a hyperbolc system of partal dfferental equatons. The unsteady Euler equatons can then be solved by a tme-marchng method to arrve at ther steady state soluton. To complete the above equatons the deal gas law equatons can be used n the followng manner. p = ρrt e= ct v γ = cp cv (3) ( ) E = ρe+ ρ u + v + w. Where c p s the specfc heat at constant pressure c v s the specfc heat at constant volume R s the unversal gas constant and γ s the rato of specfc heats. For ar γ =.4 and R = 87 (m/s). 4 Numercal Dscretzaton Havng derved the governng flow equatons the next stage s the development of a soluton procedure assocated wth the numercal dscretzaton of these equatons. In the present work the Euler equatons are frst dscretzed n space. Subsequently n a second step they wll be ntegrated n tme to a steady-state soluton (cf. Jameson et al [6]). Ths approach the so-called "method of lnes" leads to a system of non-lnear ordnary dfferental equatons. Thus a fntevolume spatal dscretzaton s used and the computatonal doman s dvded nto a fnte number of non-overlappng sub-domans or controlvolumes. 4. Fnte-Volume Spatal Dscretzaton Fnte-volume methods have become very popular durng the last few decades manly because they present some advantages of both the fntedfference and the fnte-element schemes (Hrsch []). In partcular they are very sutable for equatons n conservaton form and can be appled to the equatons drectly n the physcal space wthout the need to transform them nto a computatonal space. The advantage s an ncrease n flexblty of the method snce all of the problems assocated wth the evaluaton of the Jacobans of the transformaton matrx are removed. The fnte-volume method appled to the governng equatons can be obtaned by ntegratng Equatons () over the doman of nterest Ω. WdΩ + ( F) dω= 0 (7) t Ω Ω ISSN: Issue Volume 5 January 00

4 Seyed Saed Bahranan where s the dvergence operator n Cartesan coordnates. By applyng the Gauss theorem to the second term of Equaton (7) the system becomes: WdΩ+ F n ds = 0. (8) t Ω Ω Where Ω s the boundary of the doman Ω and n s the unt outward normal vector. The man property of the above form s that the evaluaton n tme of the conserved quanttes W depends only on the dstrbuton of the flux F on the boundary Ω. Therefore the problem of determnng W over Ω s reduced to the determnaton of the flux on the contour of the doman of nterest. In the fnte-volume method the computatonal doman s dvded n several sub-domans or control volumes and the conservaton laws () are appled to each of the resultng sub-domans. For each generc control volume of volume Ω Equaton (8) can be re-wrtten as: WdΩ+ F n ds = 0. (9) t Ω Ω Ths becomes a system of ntegral-dfferental equatons n tme. The fnte-volume method allows certan choces of control-volume to be used as the computatonal grd. In the present work the cell-centered fnte volume approach s used and therefore from now onwards the terms "cell" and "control-volume" wll refer to the grd cells only. As a consequence the unknown varables W are assocated wth the centers of the cells (See Fg. 3). Cell B A Cell (a) Numberng scheme for trangular cells A Cell C Cell (b) Numberng scheme for tetrahedral cells Fg. 3 Man and auxlary computatonal cells or control-volumes wth cell centers and for neghborng: (a) trangular and (b) tetrahedral cells B In order to use Equaton (9) for the components of W a further assumpton has to be ntroduced. In fact because the dstrbuton of W over the cell s not known the frst ntegral n Equaton (9) cannot be evaluated explctly. However by applyng the mean-value theorem to the control volume Ω expresson (0) for the average value W s obtaned. W = W dω= 0. (0) Ω Ω The Equaton (9) can now be wrtten as: Ω W t + F n ds = 0. () Ω If the cell volume s constant n tme the above equaton can be dvded by the control volume Ω whch results n: W + F n ds = 0. () t Ω Ω In ths manner Equaton () contans the cellaveraged values of the unknown conserved varables. Ths ndcates that only the averaged values of W and not the local values are known after the soluton process. An mportant consequence of the mean-value theorem s that f the values of the conserved varables W are assumed to be constant or vary lnearly over the control volume Ω the value at the center of the cell W C s equal to the mean-value W : W = W = W dω= 0. (3) C Ω Ω The coordnates of the cell centrods for a threedmensonal computatonal cell such as the one shown n Fg. 3(b) are: x C yc zc = xdω Ω Ω = ydω Ω Ω = zdω. Ω Ω ISSN: Issue Volume 5 January 00

5 Seyed Saed Bahranan Consderng the flux vectors for a threedmensonal case n terms of the Cartesan unt vectors F=F + Fyj + F zk where ( F = E F = F F = G ) equatons () and are combned and for a typcal control-volume fxed n tme can be approxmated by: faces W + ( Sx + Sy + Sz ) = 0 Ω E F G. t = (5) The summaton s over the n-faces formng the cell- E F and G are the Cartesan components of the flux at the th face and ( Sx Sy Sz ) are components of the area outward normal vector (n ds) or S assocated wth face-. Dependng on the geometry of the face area n ths case a trangular face of a tetrahedron the components of the area S are defned by ts three formng vertces (A B and C). These vertces are detaled n Fg. 3(b) for a trangular face- (shown n red) and are defned as follows. Sx = ( )( ) ( )( ) [ ] B A C A C A B A (6) Sy = ( B A )( C A ) ( C A )( B A ) [ ] Sz = ( B A )( C A ) ( C A )( B A ) [ ] The flux components at the th face can be evaluated ether usng the values of the of the surroundng cell centers (.e. and n Fg. 3) or by usng the values of the conserved varables. In the present work the convectve fluxes at the cell face- are evaluated by usng the flow varables on the cell face tself. E ( W) = E( W ) F( W) = F( W ) G ( W) = G( W ). (7) Where: W s the value of the conserved varable at the th face. The smplest approach to calculate the dependent varables on the cell face s to average the values of the two cell- centers (see Fg. 3): W +W W =. (8) Equatons (7) and (8) lead to a second-order central dfference formulaton on an orthogonal unform Cartesan grd whch s satsfactory for the mean-flow equatons. However for non-orthogonal shapes they requre especal consderatons whch are detaled n the followng secton. 4. Central Dfference Scheme If the fnte volume scheme of Equaton (7) s used to evaluate the convectve fluxes as n Equaton (8) the resultng scheme becomes dentcally equal to the standard three-pont central dfference scheme n three dmensons. Thus the present fnte-volume scheme s second order accurate at least for orthogonal meshes. However fnte-volume schemes depart from the standard fnte-dfference schemes when appled to curvlnear reference systems. Ths s due to the fact that the fnte-volume schemes do not nvolve any coordnate transformaton. The fnte-volume scheme descrbed n the prevous secton s essentally a central scheme as ponted out here. Therefore the scheme s nondsspatve and when appled to convecton problems such as the Euler equatons spurous oscllatons are not damped. Ths can affect the qualty of the numercal soluton as well as the convergence rate. To overcome ths problem artfcal dsspaton terms may be added to the fnte-volume equatons as descrbed n the followng secton. 5 Numercal Dsspaton Second-order central dscretzaton schemes are essentally non-dsspatve. Hence when appled to the convectve terms of the mean-flow equatons [.e. Equaton (5)] they can generate undamped oscllatons that affect the qualty of the soluton and can result n a deteroraton of the convergence. Oscllatons are generally produced by the odd-even pont decouplng whch s present n central schemes. Subsequently the spatal errors change the sgn at consecutve ponts producng hgh-frequency ISSN: Issue Volume 5 January 00

6 Seyed Saed Bahranan oscllatons that wll not be damped as the steadystate soluton s approached. Another type of oscllaton can be trggered by dscontnutes n the soluton (.e. shock waves) whch can be rather ntense and propagate rapdly throughout the flow doman. In order to remove these types of nstabltes one can employ an upwnd scheme whch s only frst-order accurate. Second-order spatal accuracy can be acheved by ntroducng more upwnd ponts n the dfference formulaton. Ths however leads to the generaton of oscllatons around dscontnutes smlar to those encountered wth central schemes. Therefore extensve research has been carred out to acheve the goal of oscllaton-free and hgher-order upwnd schemes capable of predctng shock waves as well as contact dscontnutes accurately. A varety of methods have been developed that are based on Total Varaton Dmnshng (TVD) propertes. These methods are now well defned for one-dmensonal flows. However the complexty of the method ncreases as multdmensonal flows are consdered. In general mult-dmensonal problems are approxmated by a seres of one-dmensonal approaches. Ths approach s sutable for use wth structured grds but leads to some dffcultes when unstructured grds are consdered. An alternatve more straghtforward approach to elmnaton of the spurous oscllatons s to retan the central dscretzaton for the convectve fluxes and add just enough numercal dsspaton where needed. Ths method bascally ntroduces numercal dsspaton terms nto the Euler soluton of nvscd flows to control numercal nstabltes and to enable the clean capture of shock waves. A blendng of second and fourth dfferences of the flow varables W has been ntroduced and added to the system of mean flow equatons n such a way that the conservaton form of the equatons s preserved: dw dt = ( F D ). (9) Where F s the dscrete flux vector and D s the dsspaton functon. The dea s to add thrd order dsspaton terms throughout the doman to provde a base level of dsspaton suffcent to prevent non-lnear oscllatons apart from near shock waves where t may cause nstabltes. In order to capture shock waves addtonal frst-order dsspaton terms are added locally by a sensor desgned to detect dscontnutes. The dsspaton functon D can now be wrtten for unstructured grds (cf. Jameson and Mavrpls [8]) as: faces = d + d = D. (0) wth: d = αε ( W W ) d () = αε ( W W ). () where s the undvded Laplacan operator on a three-dmensonal tetrahedral grd: faces = = W ( W W ). (3) and α s an approprate scalng factor related to the maxmum egenvalues of the convectve Jacoban matrces on the th face. α = u Sx + v Sy + w Sz + c Sx + Sy + Sz. (4) where u v and w are components of the velocty vector at the face- c s the local speed of sound c = γ ρ and ( Sx Sy Sz) are components of the area outward normal assocated wth face- whch were expressed by relatons (6) for a trangular face. The adaptve coeffcents ε and ε are defned as follows: ε = k υ ε (5) = max 0( k ε ). (6) Where k and k are emprcally-chosen constants whch usually take values n the ranges (0.5 < k < ) and (/56 < k < /3) and υ s a shock sensor. The constructon of the shock sensor s a crucal matter snce t has an effectve control on the level of dsspaton ntroduced n the flow doman. The orgnal structured grd based sensor was devsed n ISSN: Issue Volume 5 January 00

7 Seyed Saed Bahranan the form of a non-dmensonal second dfference of pressure whch takes nto account the contrbuton of dsspaton along each drecton ndvdually. However the unstructured equvalent of ths formulaton proposed by Jameson et al [5] swtches on or off n all drectons at the same tme: υ = faces = faces = ( ) ( + ). (7) As nvestgated by Stolcs and Johnston [7] ths sotropc behavor not only ncreases the level of artfcal dsspaton n the regons where the soluton s smooth but results n an nsuffcent level of dsspaton n the proxmty of dscontnutes. In order to remove ths problem the sngle shock sensor for each cell s replaced by a non-sotropc sensor for each face: ( ) υ =. ( + ) (8) The above sensor has worked well wth several Euler and Naver-Stokes problems as reported by Stolcs and Johnston [7] when employed wth the edges of two-dmensonal unstructured trangular grd cells. In smooth regons of the flow where the pressure gradents are small ν s also small ( ν ) therefore ε and d are neglgble whle ε s of order one formng the man contrbuton to D. In the neghborhood of a shock wave ν s of order one where the fourth dfference term s swtched off through the relaton between ε and ε whch s gven by Equaton (6). 6 Tme Dscretzaton Once the spatal dscretzaton s performed as descrbed n the prevous secton the governng mean-flow equatons (.e. the Euler equatons) become a system of ordnary dfferental equatons n tme: W = R. (9) d dt Where R s the resdual or devaton from the steady-state soluton for each cell k and can be wrtten as: R = ( F D ) (30) and the rght-hand sde terms are as descrbed n equaton (9). The methods avalable for solvng a system of ordnary dfferental equatons can be classfed nto two dfferent categores: mplct and explct methods. Explct methods evaluate the resdual R usng only values obtaned from the prevous teraton step whlst mplct methods use also values at current teraton step. Several mplct methods are avalable and are revewed by Hrsch []. These methods are actually sutable for steadystate calculatons because they do not have restrctons on the maxmum allowable tme step and thus requre less teratons to reach the fnal soluton. On the other hand explct methods can be easly mplemented on unstructured grds requre less memory and can be effcently adapted for use wth parallel computers. It should be noted however that explct methods requre more teratons n order to reach the steady-state soluton snce the sze of ther maxmum allowable tme-step s lmted due to stablty constrants. Therefore technques such as local tme steppng resdual smoothng and multgrd can be employed to accelerate convergence (Jameson [5]). In the present work an explct tmemarchng scheme s used whose detals are gven n the followng secton. 6. Explct Tme marchng Scheme Tme ntegraton of equaton (9) s performed usng an explct mult-stage scheme. The standard form of an M-stage Runge-utta scheme can be wrtten as: (0) = n W W W = W + α tr ( n + ) M W = W ( m) (0) ( m ) m. (3) Where n s the current tme level (n+) s the new tme level (m) s the ntermedate stage and M s the total number of stages. A 4-stage scheme s adopted here because of ts stablty propertes wth coeffcents of: ISSN: Issue Volume 5 January 00

8 Seyed Saed Bahranan α = α = α = α 4 = (3) However n order to mnmze the computatonal tme the expensve calculatons of the dsspaton term D s only performed once at the frst stage (0). The values at all remanng stages are frozen; thus the resdual n Equaton (30) at each ntermedate stage (m) s gven by: R = ( F D ). (33) ( m) ( m) (0) The maxmum CFL number for the above scheme s about f appled to the Euler equatons cf. Jameson [5]). 7 Intal and Boundary Condtons The next and fnal task before start of computatons s to defne the approprate ntal and boundary condtons of the flow doman. These condtons are gven n the followng sectons. For the soluton of most problems n compressble flow the contnuty momentum and energy equatons are suffcent. However for an nvscd adabatc flow t can be shown that the entropy of a movng flud element s constant. ds dt = 0 (35) s = constant. If the flow s steady the entropy (s ) s constant along a streamlne n an adabatc nvscd flow. Moreover f the flow orgnates n a constant entropy reservor such as the free stream far ahead of a movng body each streamlne has the same value of entropy and hence Equaton (35) holds throughout the complete flow feld. For sentropc flows Equaton (35) s frequently convenent and may be used to substtute for ether the energy or momentum equatons. Therefore the varaton of energy on the surface of sold geometry can be expressed by the varaton of entropy. 7. Intal Condtons As far as the numercal ntegraton scheme s concerned the most mportant consequence s that the steady-state soluton becomes ndependent of the ntal condtons employed. The ntal condtons at tme (t=0) appled n the present method consst of settng all the quanttes equal to ther free stream values. Ths corresponds to a sudden nserton of the sold body (.e. cylnder wng etc.) n an undsturbed flow wth free stream condtons everywhere. The non-dmensonal free stream values are expressed n terms of the free stream densty ρ ressure Mach number M ncdence angle α and sde slp or yaw angle β. Wth the yaw angle β set to zero and havng a non-dmensonal value of for the free stream densty and pressure the ntal condtons become: = = ρ = ρ = γ ( ρ ) s = s. (36) The ntal value for the entropy s calculated by applyng the ntal condtons as shown below. γ ( ρ ) s = =. (37) It should be noted that Equaton (35) s vald for both steady and unsteady flows and has some advantages when shock waves have to be nvestgated. 7. Sold-wall Boundary Condtons The nvscd wall boundary condton mposes flow tangency at the surface boundary (wall surface). The component of the velocty normal to the wall boundary s set to zero. u = M γ cosα v = 0 (34) uω = vω = wω = 0. (38) w = M γ snα. Where the subscrpt denotes ntal condtons. A further condton for the pressure s obtaned by usng a smple zero-th order extrapolaton and the pressure at the wall s set equal to the pressure at the center of the near wall cell: ISSN: Issue Volume 5 January 00

9 Seyed Saed Bahranan y n ω = 0. (39) (c = γ ρ ). For supersonc outflow and nflow all of the varables are extrapolated from the nteror or set to ther free stream values respectvely. Where the subscrpt ω ndcates condtons at the wall and y s the surface normal dstance. n 7.3 Outer Boundary Condtons On the outer boundary t s not possble to defne the boundary condtons n a unque manner. Snce the computatons are nvscd the characterstc based boundary condtons developed for nvscd flows can be appled. The condtons employed are based on one-dmensonal Remann-nvarant theory for the flow normal to the boundary (cf. Jameson and Baker [6]). The Remann nvarants are gven as: c R + = un + γ c R = un. (40) γ Where u n s the mean-velocty component normal to the boundary surface (the outer-boundary trangular face n ths case) and c s the speed of sound. At each pont on the outer boundary the values are ether extrapolated from the nner ponts (ext) f the flow s outgong or set equal to the free stream values ( ) f the flow s ncomng. Therefore R + and R can be the re-wrtten as follows: cext R + ext = un + ext γ c R = un. γ For subsonc nflow and outflow the normal mean-velocty component and the speed of sound are gven by the Remann nvarants as: ( + un = Rext + R ) γ + c = ( Rext R ). (4) 4 If the flow s outgong two addtonal condtons are requred concernng entropy and tangental velocty whch are extrapolated from the nteror. These quanttes are set to ther free stream values f the flow s ncomng. The pressure s determned usng the defnton of the speed of sound 8 Computatonal Results The mathematcal formulaton of the Euler equatons and the fnte-volume soluton technques along wth the ntal and boundary condtons descrbed n the prevous sectons have been mplemented n computer codes. These codes are wrtten n the FORTRAN language and work n parallel wth the two- and three-dmensonal unstructured grd generators. As dscussed n the prevous Secton the Euler flow solvers employ fnte-volume spatal dscretzaton and an explct mult-stage Runge-utta tme dscretzaton. The unsteady Euler equatons are marched n tme to reach a steady state soluton. The objectve of ths paper s to present the flow solutons wth emphass on the two numercal dsspaton coeffcents k and k as a result of employng the Euler flow solvers wth unstructured grds. The flow solvers gve the soluton results over the entre flow doman. Here the surface solutons whch gve the requred flow characterstcs are presented for each computatonal case. The frst and second case nvolved the flow over a -dmensonal crcular cylnder at free stream Mach numbers of 0.3 and 0.45 whle the thrd and forth computatonal cases descrbe 3-dmensonal flow over a crcular cylnder attached to a plane of symmetry at free stream Mach numbers of 0.3 and 0.8. These cases were selected to demonstrate the effect of varyng the values of artfcal dsspaton terms. The computatonal unstructured grd shown n Fg. (b) was employed for the frst case wth freestream Mach number of 0.3. The resultng flow solutons wth varous dsspaton coeffcents k and k are gven n Fg. 4. The convergence hstory for three dfferent solutons up to 5000 teratons s shown n Fg. 4(a). The pressure coeffcents for these solutons are plotted over the cylnder secton and are shown n Fg. 4(b) where these Cp values are compared wth an nvscd analytcal soluton. The optmal values for the dsspaton coeffcents k and k are found to be 0.5 and 0.05 respectvely. The pressure and Mach number contours for the soluton wth these optmal values are plotted over the cylnder surface and are shown n Fg.s 4(c) and 4(d). ISSN: Issue Volume 5 January 00

10 Seyed Saed Bahranan Log (Average Densty Resdual) =0.00 k 4 =0.03 =0.00 k 4 =0.00 =0.5 k 4 = Number of Iteratons (a) Convergence hstory for flow solutons wth varous dsspaton coeffcents k and k for the set of dsspaton coeffcents. Fgs. 5(c) and 5(d) show the pressure and Mach number contours for the soluton run wth the k and k values of 0.5 and Log (Average Densty Resdual) =0.5 k 4 =0.00 =0.45 k 4 =0.00 =0.05 k 4 = =0.5 k 4 = θ Number of Iteratons (a) Convergence hstory for flow solutons wth varous dsspaton coeffcents k and k C p - -3 =0.00 k 4 =0.03 =0.00 k 4 =0.00 =0.5 k 4 =0.05 Analytcal Soluton (b) ressure coeffcent plots for varous dsspaton coeffcents k and k ress Mach C p θ =0.5 k 4 =0.00 =0.45 k 4 =0.00 =0.05 k 4 = =0.5 k 4 = (b) ressure coeffcent plots for varous dsspaton coeffcents k and k (c) ressure contours k =0.5 k = (d) Mach No contours k =0.5 k =0.05 Fg. 4 Frst case Two-dmensonal flow soluton results around a crcular cylnder M = 0.3 The same computatonal grd shown n Fg. (b) was employed for the second case wth a freestream Mach number of Ths was done to reach transonc condtons. The resultng flow solutons wth varous dsspaton coeffcents k and k are gven n Fg. 5. Hstory of the soluton convergence s shown n Fg. 5(a) where four dfferent solutons are presented. The pressure coeffcents for these four solutons are plotted over the cylnder secton and are shown n Fg. 5(b). It s clear that two strong shocks are present over the cylnder surface for all the runs n ths test case. The shock waves are best captured wth 0.5 and ress (c) ressure contours k =0.5 k = MACH (d) Mach No contours k =0.5 k = Fg. 5 Second case Two-dmensonal flow soluton results around a crcular cylnder M = 0.45 To show consstency wth hgher order spatal dscretzaton sample solutons are presented n three-dmensons. The computatonal grd for the crcular cylnder of dameter and length 4 attached to a plane of symmetry shown n Fg. was employed for the three dmensonal cases. Euler flow solutons were obtaned at free-stream Mach number of 0.3 and 0.8. ISSN: Issue Volume 5 January 00

11 WSEAS TRANSACTIONS on FLUID MECHANICS Seyed Saed Bahranan wth k = 0.5. It should be noted that the drecton of the free-stream flow n all of the computatonal cases s from negatve x-axs towards the postve x-axs. In other words the front stagnaton pont occurs at locaton θ = 0 on the cylnder. At a free-stream Mach number of 0.3 solutons were obtaned for dfferent values of the st order dsspaton coeffcent k and are shown n Fg. 6. It s shown that by reducng the st order adjustable dsspaton coeffcent k coeffcent of pressure values over the entre span of the cylnder move closer to ther analytcal nvscd soluton. ressure dstrbuton on a crcular cylnder D = L = 4 M = 0.3 C θ 90 Cp Cp (a) ressure coeffcents -3 CFD Smulaton Exact Soluton Expermental Data * (a) k M = 0.5 ressure dstrbuton on a crcular cylnder D = L = 4 M = 0.3 M θ C - (b) Mach number - -3 CFD Smulaton Exact Soluton Expermental Data * S (b) k = 0.5 Fg. 6 ressure dstrbuton on a crcular cylnder wth varous st order dsspaton k Cylnder: D = L = 4 α = 0º M = 0.3 * Expermental data Schlchtng [9] at Re= S (c) Entropy Fg. 7 ressure Mach number and entropy contours on the upper and lower surfaces of the crcular cylnder M = 0.3 and k = 0.5 Fg. 7 shows the pressure Mach number and entropy contours on the upper and lower surfaces of the crcular cylnder for the soluton at M = 0.3 ISSN: Issue Volume 5 January 00

12 WSEAS TRANSACTIONS on FLUID MECHANICS Seyed Saed Bahranan change at the leadng edge. Snce the flow started wth a free stream Mach number of 0.8 a strong shock s present on the cylnder shoulder and s shown n Fg. 9 (c). The entropy contoures shown n Fg. 9(d) show a rapd change at the cylnder showlder and reman constant elswere. Agan ths s due to the presence of shock on the cylnder shoulder and nvscd behavour n other regons. To show the effect of adjustable dsspaton coeffcents for a more realstc transonc flow solutons wth a free stream Mach number of 0.8 are obtaned wth the st order dsspaton k values of 0.5 and 0.5 as the forth and last case. Coeffcents of pressure ponts along the span of the cylnder are compared wth an analytcal nvscd soluton n Fg. 8. Agan t can be seen that by reducng the amount of the st order dsspaton k coeffcent of pressure ponts along the span of move closer to the analytcal nvscd curve. ressure dstrbuton on a crcular cylnder D = L = 4 M = 0.8 Cp θ C C* - - (a) ressure coeffcents -3 CFD Smulaton Exact Soluton Crtcal Value M (a) k = 0.5 ressure dstrbuton on a crcular cylnder D = L = 4 M = θ C C* - (b) Mach number - -3 CFD Smulaton Exact Soluton Crtcal Value (b) k = 0.5 Fg. 8 ressure dstrbuton on a crcular cylnder wth varous st order dsspaton k Cylnder: D = L = 4 α = 0º M = 0.8 * Expermental data Schlchtng [9] at Re= (c) Entropy The pressure Mach number and entropy contours on the upper and lower surfaces of the crcular cylnder for ths case are dsplayed n Fg. 9. The pressure contours n ths case show a strong ISSN: S Fg. 9 ressure coeffcent Mach number and entropy contours on the upper and lower surfaces of the crcular cylnder M = 0.8 and k = 0.5 Issue Volume 5 January 00

13 Seyed Saed Bahranan 9 Concluson The dscretzed Euler flow equatons are essentally non-dsspatve. Hence convectve terms of the flow equatons can generate spurous oscllatons that can affect the qualty of the soluton and can result n a deteroraton of the convergence. A small amount of numercal dsspaton s added to damp the oscllatons enablng the flow soluton to reach ts steady state condtons. Thus by reducng the amount of numercal dsspaton (st order dsspaton n ths case) that s needed to control the spurous oscllatons nstabltes wll occur. The objectve of ths paper s to llustrate the effect that the amount of added dsspaton has on the qualty of the flow soluton. For ths objectve Euler flow solutons of varyng frst- and thrd-order dsspaton coeffcents have been nvestgated. The frst example consders a subsonc flow soluton wth a free-stream Mach number of 0.3 over a crcular cylnder. Intally frst-order dsspaton coeffcents were turned off and an arbtrary small value was selected for the thrd-order dsspaton coeffcent. Ths caused nstabltes n the flow soluton and s clearly shown n Fg. 4(b). Although at ths Mach number no shock s present dsturbances are caused by the strong change n the pressure at the leadng edge stagnaton pont. The small amount of the thrd-order dsspaton coeffcent k was suffcent enough to enabled the flow soluton to converge. Ths thrd-order dsspaton coeffcent was reduced emprcally to obtan an optmal value for k n the absence of frst order coeffcent. Snce the thrd order dsspaton coeffcent s present throughout the entre flow doman t s essental to keep ths coeffcent at a mnmal value to avod over dampng the regons of hgh pressure gradents. Fnally to properly control the oscllatons n these regons frst-order dsspaton coeffcents were turned on and the solutons are shown to match ther analytcal values. To demonstrate the above procedure for a more realstc condton expected of nvscd flows characterzed by shock waves smlar analyss were performed on the cylnder wth a free stream Mack number of In ths case values of 0.5 and were found to be the optmal values for the set of dsspaton coeffcents k and k. Ths can be seen n Fg. 5(b) where pressure coeffcent plots of dfferent test runs are compared. The thrd case presented here nvolved the soluton of Euler equatons on a three dmensonal cylnder secton attached to a plane of symmetry. Ths geometry whch was shown n Fg. was selected for analogy wth the two dmensonal cases. Intal condtons for ths case ncluded the free stream Mach number of 0.3 at zero attack angle and varyng values of the dsspaton coeffcents. The forth and last computatonal case presented here ams to demonstrate the extent of these studes to transonc flow condtons. For ths reason and analogy wth the two-dmensonal cases presented earler the sample tetrahedral grd generated for a crcular cylnder was used agan. Several Euler flow solutons were obtaned by employng ths unstructured tetrahedral grd. Two sample runs at transonc Mach number of 0.8 were presented. These solutons were obtaned by varyng the st order dsspaton coeffcent from 0.5 to 0.5. It s shown that the reducton of the st order dsspaton coeffcent can gve results closer to nvscd condtons. The transonc sample flow soluton wth values 0.5 and for the dsspaton coeffcents k and k revealed the shock capturng capablty of the method descrbed here. Ths was llustrated by plottng the pressure coeffcent Mach number and entropy contours on the upper and lower surfaces of the cylnder (see Fg. 9). There are numerous sources that can affect a numercal soluton. These may nclude the local tme steppng used wth the tme marchng scheme or the Courant numbers (CFL) used for the mplct resdual smoothng. To consder only the effect of dsspaton terms on the numercal flow solutons Courant numbers were kept the same for local tme steppng and resdual smoothng was not employed. It was shown that proper adjustment of dsspaton coeffcents k and k s requred for every sold boundary shape consdered. Furthermore any change n the flow condtons such as the free stream Mach number requre new dsspaton constants. It s therefore concluded that by adjustng the values of dsspaton coeffcents k and k accurate Euler flow solutons can be obtaned. References: [] MacCormak R. W. and Baldwn B. S. A Numercal Method for Solvng the Naver- Stokes Equatons wth Applcaton to Shock- Boundary Layer Interacton AIAA aper [] Hrsch C. Numercal Computaton of Internal and External Flows Vol. : Computatonal Methods for Invscd and Vscous Flows John Wley and Sons New ork USA 99. ISSN: Issue Volume 5 January 00

14 Seyed Saed Bahranan [3] Steger J. L. Implct Fnte Dfference Smulaton of Flow about Two-Dmensonal Geometres AIAA Journal [4] Jameson A. Schmdt W. and Turkel E. Numercal Smulaton of the Euler Equatons by Fnte Volume Methods Usng Runge-utta Tme Steppng Schemes AIAA 5th Computatonal Flud Dynamcs Conference [5] Jameson A. Transonc Aerofol Calculatons Usng The Euler Equatons In. L. Roe (edtor) Numercal Methods n Aeronautcal Flud Dynamcs Academc ress New ork USA 98. [6] Jameson A. Baker T. J. and Weatherll N.. Calculaton of Invscd Transonc Flow Over a Complete Arcraft AIAA aper [7] Stolss L. and Johnston L. J. Soluton of the Euler Equatons on unstructured grds for Two- Dmensonal Compressble Flow Aeronautcal Journal Volume 94 pages [8] Jameson A. and Mavrpls D. Fnte Volume Soluton of the Two-Dmensonal Euler Equatons on a Regular Trangular Mesh AIAA Journal Vol. 4 No. 4 Aprl 986. [9] Schlchtng H. Boundary Layer Theory 7th edton McGraw Hll Book Co. N N 987. [0] Bahranan Seyed Saed Daneh Dezful Alreza An Unstructured Grd Generaton Approach for Invscd Flow Solutons WSEAS Transactons on Flud Mechancs Issue Vol. 4 pp [] Bahranan S. S. Constructon of Hexahedral Block Topology and ts Decomposton to Generate Intal Tetrahedral Grds for Aerodynamc Applcatons Journal of Aerospace Scence and Technology (JAST) Vol. 5 No. pp E F G W vectors F flux vector H total enthalpy L length or span k M k adjustable dsspaton coeffcents Mach number n unt outward normal vector statc pressure R vector of resduals R Remann nvarant s entropy T temperature u v w Cartesan velocty components α angle of attack Ω control volume cell volume volume ε st - order pressure swtches ε 3rd - order pressure swtches ν pressure sensor for cell γ rato of specfc heats ρ densty t tme step W vector of ndependent varables w... w 5 components of the unknown vector x y spatal ncrement n x and y drectons Nomenclature A cell area c speed of sound chord c p specfc heats at constant pressure c v C D E specfc heats at constant volume pressure coeffcent artfcal dsspaton vector dameter total energy per unt mass or volume ISSN: Issue Volume 5 January 00