Computational Methods for the Euler Equations
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- Gerard Daniels
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1 Computatioal Methos for the Euler Equatios Before iscussig the Euler Equatios a computatioal methos for them, let s look at what we e leare so far: Metho Assumptios/low type -D pael -D, Icompressible, Irrotatioal Iisci Vortex lattice Potetial metho Pral-Glauert -D, Icompressible, Irrotatioal Iisci, Small isturbace -D, Subsoic compressible, Irrotatioal, Iisci, Small isturbace Euler CD -D, Compressible (o M limit, Rotatioal, Shocks, Iisci The oly major effect missig after this week will be iscous-relate effects. -D Euler Equatios i Itegral orm Cosier a arbitrary area (i.e. a fixe cotrol olume through which flows a compressible iisci flow: y x c S δ C outwar poitig ormal (uit legth S elemetal (ifferetial surface legth S S = yi xj y S x Note: Path arou surface is take so that iterior of cotrol olume is o left.
2 Computatioal Methos for the Euler Equatios Coseratio of Mass rate of chage rate of mass of mass ic out of C Mass ic = rate of chage = C of massic ρa C ρa flow = where ρ estiy of flui Now, the rate of mass flowig out of C : Mass flow out of C = ρu S u = elocity ector C δ A ρu S = C ρ δ C Coseratio of x-mometum Recall that: total rate of chage mometum = forces or x-mometum this gies: rate of chage of rate of x momflow = orces i x-irectio x mometum i C out of C ρua ρuu S = orces i x-irectio C δ C Now, lookig closer at x-forces, for a iisci compressible flow we oly hae pressure (igorig graity. Normal to surface Recall pressure acts ormal to the surface ps S Ito surface orce si x = p i S δc Gies x-irectio 6.
3 Computatioal Methos for the Euler Equatios ρua ρuu S = p C δc δc i S Coseratio of y-mometum This follows exactly the same as the x-mometum: ρa ρu S = p C δc δc Coseratio of Eergy Recallig your thermoyamics: total rate of chage work oeo heat ae = of eergy i C flui i C to C or the Euler equatios, we igore the possibility of heat aitio. total rate of chage rate of chage of rate of eergy = of eergy i C eergy i C flow out of C The total eergy of the flui is: ρ E = ρe ρ( u js Total eergy Iteral eergy Kietic eergy Note: e = c T where c specific heat at costat olume Static temperature So, total rate of chage = ρea ρeu S of eergy i C C δc The work oe o the flui is through pressure forces a is equal to the pressure forces multiplie by (i.e. actig i the elocity irectio: work = p us ( ( δc Pressure force 6.
4 Computatioal Methos for the Euler Equatios ρea ρeu S = p C δc δc us Summary of -D Euler Equatios ρa ρu S = C δc ρua ρuu S = p i S C δc δc ρa ρu S = p js C δc δc ρea ρeu S = C δc δc p S These are ofte writte ery compactly as: UA ( i Gj s = c sc U ρ ρu ρ ρe ρu ρu ρu ρuh p ρ ρu G ρ ρh p Coseratie state ector lux ector for x-irectio lux ector for y-irectio H total ethalpy E p ρ Ieal gas: p = ρ RT = ( γ ρe ρ( u A iite Volume Scheme for the -D Euler Eqs. Here s the basic iea: ( Diie up (i.e. iscretize the omai ito simple geometric shapes (triagles a quas 6. 4
5 Computatioal Methos for the Euler Equatios Lookig at this small regio: Cell is surroue by cells,, &. i.e. cell has eighbors: cell,, &. Nearest eighbors 6. 5
6 Computatioal Methos for the Euler Equatios ( Decie how to place the ukows i the gri. (a Cell-cetere: cell-aerage alues of the coseratie state ector are store for each cell. (b Noe-base: poit alues of the coseratie state ector are store at each oe. The ebate still rages about which of these optios is best. We will look at cellcetere schemes because these are easiest (although ot ecessarily the best. Also, they are ery wiely use i the aerospace iustry. ( Approximate the -D itegral Euler equatio o the gri to etermie the chose ukows. 6. 6
7 Computatioal Methos for the Euler Equatios 6. 7
8 Computatioal Methos for the Euler Equatios 6. 8
9 Computatioal Methos for the Euler Equatios 6. 9
10 Computatioal Methos for the Euler Equatios Let s look i etail at step (: f b e c a Cells:,,, Noes: a, b, c,, e, f Cell-aerage ukows: ρ ( ρu U = U ( ρ ( ρe ρ = ( ρu ( ρ ( ρe U =.... U =... Specifically, we efie U as: U A UA where C C A cell area of cell Now, we apply coseratio eqs: UA ( i Gj S = C δc The time-eriatie term ca be simplifie a little: U UA = A C The surface flux itegral ca also be simplifie a little: b ( i Gj s = ( i Gj S δc a c ( i Gj S b a ( i Gj S c 6.
11 Computatioal Methos for the Euler Equatios Combiig these expressios: A U b c ( i Gj S ( i Gj S a b a ( i Gj S = c No approximatios so far! Now, we make some approximatios. Let s look at the surface itegral from a b : b b ( i Gj s a c a ab The ormal ca easily be calculate sice the face is a straight lie betwee oes a & b. Recall, the ukows are store at all ceters. So, what woul be a logical approximatio for : b ( i Gj S ab =??? a Optio #= Optio #= Note: Optio # optio # i geeral. There is ery little ifferece i practice betwee these optios. Let s stick with: b Iab ( i Gj abs = ( ( ab ab a i G G j S c Ibc ( i Gj bcs = ( i ( G G j bc Sbc b a Ica ( i Gj cas = ( i ( G G j ca sca c Where ( U G G( U ( U G G( U ( U G G( U ( U G G( U 6.
12 Computatioal Methos for the Euler Equatios ially, we hae to approximate forwar Euler: A U U somehow. The simplest approach is A Iab Ibc Ica = A I ab etc. are efie as: Iab i U U A Iab Ibc Ica = t U U t a t t, iteratio Where ( o U U ( ( G G ( etc. ( o j ab S ab or steay solutio, basic proceure is to make a guess of U at t = a the iterate util the solutio o loger chages. This is calle time marchig. Questio What assumptios hae we mae i eelopig our -D Euler Equatio iite Volume Metho? 6.
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