Joural of Computatioal Physics 149, 418 422 (1999) Article ID jcph.1998.6131, available olie at http://www.idealibrary.com o NOTE Defiig Wave Amplitude i Characteristic Boudary Coditios Key Words: Euler compressible equatios; characteristic boudary coditios; oreflectig coditios; iitial coditios. Characteristic treatmet of boudary coditios for the Euler equatios relies o determiig the stregth of the waves eterig the computatioal domai as a fuctio of the stregth of the outgoig waves ad the physical boudary coditios. The purpose of this ote is to demostrate how critical the defiitio chose for the wave amplitudes ca be. The 2D Euler equatios may be expressed i quasi-liear form as V + A V x + B V = 0. (1) y Here V = (ρ, u,v,p) T is the vector of primitive variables ad each of the matrices A ad B has its ow complete set of real eigevalues ad right ad left eigevectors. The matrix E defied as A x + B y ca be itroduced, where is chose as the outward ormal to the boudary uder cosideratio. By diagoalizig E the eigevalue matrix Λ = L E L 1 = diag ( λ 1,λ2,λ3,λ4 ) =diag(u, u, u + c, u c), (2) is obtaied, where u = u ad c is the speed of soud. The matrices L (L 1 ) with left (right) eigevectors as rows (colums) relate variatios i the characteristic variables W to variatios i the primitive vector V through the relatios δw = L δv, δv = L 1 δw. (3) I 2D, the four characteristic variables satisfy a set of covectio equatios with the speed of propagatio give by Eq. (2), with source terms related to pressure ad velocity variatios i the s-directio, where s forms a orthoormal basis (, s) with. These equatios are 0021-9991/99 $30.00 Copyright c 1999 by Academic Press All rights of reproductio i ay form reserved. 418
WAVE AMPLITUDES 419 obtaied by premultiplyig Eq. (1) by L. The fourth equatio reads + (u c) W 4 + cs W 2 = 0. (4) Applyig a explicit Euler time discretizatio to Eq. (1), the update of primitive variables ca be writte as [ V = V +1 V = tr= t A V ] x +B V. (5) y For a give boudary with ormal, the full residual R of Eq. (5) ca be split ito a ormal compoet R (ivolvig oly ormal derivatives) ad a tagetial compoet R s (ivolvig oly derivatives alog s). Let us defie V as the boudary value at time level, ad δv P the predicted boudary update from the iterior scheme, prior to applicatio of the boudary coditio. Defie also δvw P as the compoet of δvp to which the boudary coditio will be applied, ad δv U = δv P δvw P as the part of the boudary update which is ot affected by the characteristic boudary coditio. Typically, a characteristic based boudary treatmet is applied as follows: 1. Choose the part of the residual (δvw P ) to which the boudary coditios are to be applied. If it is the complete residual, δv U = 0. 2. From Eq. (3), decompose δvw P, ito characteristic variatios δwi,p igoig ad outgoig waves, with correspodig primitive variatios δv i,p w 3. Modify the amplitude of the icomig wave(s) δw i,p ad δw out due to ad δv out w. accordig to the physical. Retai the requiremets at the boudary. This produces the corrected amplitudes, δw i,c outgoig waves δw out or δv out w as they are. 4. Combie the waves δw i,c primitive variables. This gives δv C w ad δw out, ad usig L 1, Eq. (3), trasform back to. The boudary poit is the updated as V +1 = V + δv U + δv C w = V + δv U + δv i,c + δv out. The decompositio of the Euler equatios ito a set of waves travelig ormally to the boudary provides a theoretical basis to derive proper boudary coditio treatmets, followig steps 2 3 4 above. However, such theory gives o idicatio of the best defiitio for the part of the update to which the boudary coditios are to be applied. This is the pricipal reaso why so may differet formulatios are discussed i the literature [1 8]. Let us defie a approach, we shall call the full residual approach, as a boudary treatmet such that i step 1 above δvw P = tr. Followig steps 2 4, i the case of a 2D subsoic outlet, this leads to the oreflectig boudary coditio = 0 = u + 1 P ρc which is equivalet to that proposed i [1 4]. Similarly, we shall defie the ormal approach as the boudary treatmet such that for step 1, δv P w = tr. This leads to the oreflectig coditio (u c) [ = 0 = (u c) u + 1 ρc ] P (6) (7)
420 F. NICOUD which is equivalet to the forms i [5, 6], although preseted i a completely differet formalism. Hirsch [7] argues that the oreflectig coditio has to be applied to the advectio terms of the bicharacteristic equatios (u c) + u s = 0. (8) Followig Giles [8], the aalysis of the liearized Euler equatios based o a Fourier decompositio of the solutio at the boudary gives W 2 = u u s ; (9) Eqs. (6), (7), (8), ad (9) are all oreflectig boudary coditios based o characteristic aalysis. I 1D, they all reduce to oe of the forms (u c) ( /) = 0or / = 0 which are equivalet sice the last characteristic equatio is simply / + (u c) ( /) i this case. However, these boudary treatmets are ot equivalet i 2D. It traspires that uder certai circumstaces they ca eve produce completely differet results. For example, cosider the computatio domai defied spatially by 0 < x < 1 ad 0 < y < 1. The iitial coditio is uiform for the desity ad the static pressure, ad zero for the velocity i the y-directio. For the streamwise velocity, we impose u(x, y) = U 0 (1.5 + tah(10(y 0.5))) for x = 0 ad u(y) = 0 elsewhere. U 0 is chose such that the flow is subsoic everywhere. The full residual approach is used at the ilet to impose the velocity compoets ad the temperature while a oreflectig coditio is tested at the outlet. The ormal oreflectig characteristic coditio is used for both y = 0 ad y = 1 to allow acoustic disturbaces i the y-directio to leave the domai. The velocity profile is expected to propagate dowstream durig the computatio. The steady solutio is obviously u(x, y) = u(0, y) for all x. Typical velocity profiles obtaied after covergece with both the full residual ad ormal formulatios at the outlet boudary are show i Fig. 1. Clearly the full residual outlet coditio prevets the give velocity profile from propagatig alog the x-directio. Istead, the u-velocity teds to be uiform ear the exit. O the other had, the use of the ormal approach leads to the correct velocity profiles. Both the Hirsh ad the Giles formulatios allow the hyperbolic taget profile to propagate as FIG. 1. Velocity profiles at differet abscissa for the full residual (left) ad the ormal (right) o-reflectig outlet boudary coditio.
WAVE AMPLITUDES 421 FIG. 2. Time evolutios of the streamwise (left) ad the ormal (right) velocity compoet at a poit at the outlet boudary for the Giles (dashed lie) ad ormal (solid lie) formulatios. expected, but the relaxatio time to the steady state is loger with the Giles coditio (see Fig. 2). These dramatic differeces may be explaied by formulatig all the boudary coditios i the same framework. Ay (oreflectig) boudary coditio ca be writte either i terms of time derivatives (temporal form) or i terms of ormal derivatives (spatial form). These two forms are liked through the compatibility relatio, Eq. (4), ad imposig a boudary coditio o the time derivative ca be traslated ito a coditio o the ormal derivative ad vice versa. A overview of the various coditios cosidered, writte i their two equivalet forms (temporal ad spatial), is give i Table I. This table provides a formal compariso of these boudary coditios. Of course, the results of a computatio depeds oly o the choice of the boudary (the rows i the table) ad ot o the form uder which it is writte (the colums i the table). The full residual formulatio imposes W 4 / = 0 ad thus forces the temporal evolutios of the streamwise velocity ad the pressure to remai early proportioal. At the iitial time, both quatities are uiform at the exit, so that their profiles keep the same shape durig the computatio if 1/ρc does ot deped o y. This feature of the solutio at the boudary is well predicted by the computatio (ot show) but is ot compatible with the preset physical cofiguratio. Actually the full residual approach is likely to give icorrect results as soo as the iitial coditios at the boudary are icosistet with the actual flow structure. The temporal forms of the other three boudary treatmets (see Table I) show that the temporal evolutio TABLE I Correspodece betwee the Temporal ad the Spatial Form for Some Noreflectig Boudary Coditios; 2D Case Name Temporal form Spatial form Thompso [4] Poisot [6] Hirsh [7] Giles [8] = 0, Eq. (6) W = u 4 s = c W2 = u W 2 u s, Eq. (9) = 1 u u c s W4 +c W2 =0, Eq. (7) = us (u c) = W 2 +c W2,Eq. (8)
422 F. NICOUD of the velocity ad the pressure are o loger proportioal. Three differet terms appear i the right-had side of the temporal forms, amely A = u s ( /), B = c( W 2 /) ad C = u ( W 2 /). At least oe term is eeded to esure that the give boudary coditio ca reach the correct steady state. For Hirsh s coditio which leads to results that are equivalet to the ormal approach, the term deoted above as A is ot critical (this term is the differece betwee two boudary coditios which give the same results; see Table I). Thus the term B is resposible for the success of the computatios with those coditios. Oe observes also that the term C i the Giles treatmet is othig but a factor of u /c smaller tha B i the preset subsoic test case. Accordigly, with the Giles treatmet the relaxatio time to the steady state has bee foud to be loer tha for the other coditios (see Fig. 2). More details of the preset study are available i [9], icludig the 3D versio of Table I ad implemetatio details of the boudary treatmets for a flow solver based o hybrid meshes. Some prelimiary rus were performed by Dr. G. Heradez. All the computatios were doe with the Fortra library AVBP/COUPL developed at CERFACS. REFERENCES 1. S. Chakravarthy, Euler equatios implicit scheme ad boudary coditio, AIAA J. 21, 699 (1983). 2. M. Hayder ad E. Turkel, Noreflectig boudary coditios for jet flow computatios, AIAA J. 33(12), 2264 (1995). 3. M. Hayder ad E. Turkel, High Order Accurate Solutios of Viscous Problems, AIAA Paper 93-3074, 1993. 4. K. W. Thompso, Time depedet boudary coditios for hyperbolic systems. II. J. Comput. Phys. 89, 439 (1990). 5. K. W. Thompso, Time depedet boudary coditios for hyperbolic systems, J. Comput. Phys. 68, 1 (1987). 6. T. J. Poisot ad S. K. Lele, Boudary coditios for direct simulatios of compressible viscous flows, J. Comput. Phys. 101, 104 (1991). 7. C. Hirsh, Numerical Computatio of Iteral ad Exteral Flow (Wiley, New York, 1990), Vol. 2. 8. M. Giles, No-reflectig boudary coditios for euler equatio calculatio, AIAA J. 28(12), 2050 (1990). 9. F. Nicoud, O the Amplitude of the Waves i the Characteristic Boudary Coditios, Techical Report TR/CFD/98/21, CERFACS, 1998. Received Jauary 20, 1998; revised August 10, 1998 F. Nicoud 1 CERFACS Cetre Europée de Recherche et de Formatio Avacée e Calcul Scietifique 42, Aveue Gaspard Coriolis 31057 Toulouse cedex, Frace E-mail: icoud@cerfacs.fr 1 Preset address: Ceter for Turbulece Research, Staford Uiversity, Staford, CA 94305. E-mail: icoud@ ctr.staford.edu.