Computational Fluid Dynamics. Lecture 6

Transcription

1 omputtionl Fluid Dynmics Lecture 6 Spce differencing errors. ψ ψ + = 0 Seek trveling wve solutions. e ( t) ik k is wve number nd is frequency. =k is dispersion reltion. where is phse speed. =, true solution is non dipersive for constnt. k note tht the group velocity,, the speed of energy propgtion is defined s g = nd order sptil differences. g = g, in this cse φ j φj+ φj + = 0 seek discrete solutions. e ( t) ikj () t ik ik e e ie + e k = φ ( ) ( ) ik ik e e = i = sin ( k) = sin k = k sin k = k j ikj t ikj t Note tht this is the sin c ( ) sin sin function,, which hs the nice property s 0.

2 Which is function of k, dispensive unlike solutions to the true dvection eqution. For good resolution k nd Tylor series sin 6 ( k) 6 Phse speed error is time lgging nd second order. Phse error lrger for lrger π π k =, λ, k = λ π sin = = 0 π k, worst cse is: The phse speed of the wve is zero. Noise doesn t propgte t ll. Since is rel, there is no chnge in wve mplitude with time, nd no mplitude error. The group velocity of wves = cos kis pproimte correct for smll k. π But for poorly resolved wves, lrger k, 0< k < = cosπ = nd the energy propgtion for poorly resolved wves is = The energy propgtes bckwrds! Fourth Order Accurcy φ 4 φ φ φ φ = sin k sin k 6 j j+ j j+ j. No mplitude error. - rel. Wves re dispersive.

3 . Phse speed of wve is 0. ( k) = k 0 hs smller phse speed errors tht * for smll k. 4 4 = cos k cos k for λ = 4 =, even lrger error. Evidently schemes hve indequte performnce for poorly resolved flow fetures.

4 All finite difference schemes re unble to propgte the wve, higher order schemes hve better properties for better resolved wve lengths. One-sided differences hve similr (worse) problems. onsider φ ( φj φj ) j + = 0 ik e i e + e ik f = ( e ) = i f = sin k + i( cos k ) ( ) ( ) ikj t ikj t rel prt is sme s for = 0 but now there is n imginry prt lso, so tht there is n mplitude ( ) cos k t error (growth or decy) e for > 0 the solution dmps (stble). For < 0, flow unstble; domin of dependence is not cptured. Lrgest wve numbers get dmped the most. (This is good) A third order upstrem difference. φ j + 6 gives f ( φj+ + φj 6φj + φj ) 4 i = sin k sin k cos k 6 ( ) 4

5 rel prt is identicl to 4 th order scheme, but the imginry prt shows tht it is mplifying for < 0 nd dmping for > 0. The dmping is much smller for rd order thn st order. Dissiption, Dispersion nd the Finite Difference Eqution onsider first order sptil difference uj u u u u 6 j = + + σ nd order sptil difference u u u u 6 j+ j 4 = + + σ rd order ( ) ( ) u + u 6u + u u u u 6 0 ( ) 4 4 j+ j j j 6 = + + σ 4 th 4 order 4 u u u u u u j+ j j+ j 6 = + σ ( ) If ny of these formuls re used in dvection eqution, we relly get modified eqution. u u u u + = + b + m+ m+ m+ m+ + m+ m m ( ) ( ) σ ( ) when pproimted with scheme tht is of order ( ) m. So s 0 we get good representtion of desired PDE, but it pproches the modified eqution more rpidly thn the PDE. Qulittive response of the leding error terms Suppose. m u u = m If the leding error term is even (m) s occurs for upwinding, nd then dditionl diffusion or dissiption is the leding order behvior of the differencing scheme. Shorter wvelengths decy more rpidly thn longer modes.

6 This is desirble nd referred to s numericl dissiption, in the pproimte solution of the dvection eqution. Solutions re of the form (, ) u t m ik k t = e e If the odd-order derivtive terms (m + ) re the leding order trunction error, s in the cse of centrl differences, it hs effect like m+ u u = m+ which hs solution (, ) u t = e ( t) ik ( ) where = m k m+ ( ( )) for m> 0 these re dispersive wves. f k So this would yield numericl dispersion (undesirble). 4 u entered differences (most common) hve, i.e. ( ) type errors s leding order. entered sptil differences do not produce numericl dissiption becuse there re no even derivtives in trunction errors. There is numericl dissiption in one-sided, upwind, differences, nd lso numericl dispersion. Note tht one-sided schemes hve the sme dispersion errors s the higher order centered scheme, but they dd dissiption lso. The net behvior is different; dispersion is diffused in upwind schemes. Net Topic: Eploiting trunction errors to our dvntge. 6

7 Emple for HW # M= = 0 b c = 0 u + = n 0 b c n n n µ u + + u + + µ u + = u b c n n n 4 n n µ u + u + µ u = u µ u + u + µ u = u u n+ n+ n+ n+ n 4 4 = 0 [ ] n+ n A u = u = 0 b = c = 0 = µ b = c = µ = µ b = c = µ = µ b = c = µ = b = c 0 = 0 N = ( bcu f N) ll tridg,,,,, 7