The Advection-Diffusion equation!

Transcription

1 ttp:// Te Advectio-iffusio equatio! Grétar Tryggvaso! Sprig 3! Navier-Stokes equatios! Summary! u t + u u x + v u y = P ρ x + µ u + u ρ y Hyperbolic part! u x + v y = Elliptic equatio! Parabolic part! Te Navier-Stokes equatios cotai tree equatio types tat ave teir ow caracteristic beavior! epedig o te goverig parameters, oe beavior ca be domiat! Te differet equatio types require differet solutio teciques! For iviscid compressible flows, oly te yperbolic part survives! Advectio/diffusio equatio! t + U x = f Forward i time/cetered i space (FTCS)! f + + U f + = f + f + Δt Stability limits! UΔt Δt & Δt = U & Δt = Δt For ig ad low! R = UL FTCS! L-W! C-N! t + U x = f O( Δt, ) O( Δt,) O( Δt, ) O( Δt, ) UΔt Δt & UΔt + Δt UΔt Δt Ucoditioally stable! Steady state solutio to te advectio/diffusio equatio! U x = f U f = f = L R L =! R L =! R L =! R L =! Exact solutio! ( ) ( ) f = exp R L x /L exp R L R L = UL

2 Te Cell Reyolds umber! Numerical solutio of:! U + U x = f Cetered differece approximatio! = + + U = + + Cetered differece approximatio! U + = + + Rearrage:! U + Rearrage:! Were:! ( ) = + + ( R ) ( R + ) = R = U Solutio! Substitute:! ivide by! ( R ) ( R + ) = = q ( R )q + + 4q ( R + )q = q ( R )q + 4q ( R + ) = ( R )q + 4q ( R + ) = Solvig for q gives two solutios:! q = ad! Te geeral solutio is:! or! = C q + C q q = + R R + R = C + C R Apply te boudary coditios! + R f = C + C R + R f N = C + C R N = C + C = = Te fial solutio is:! + R R = + R R N +

3 U f or! Solutio! = + + ( R + ) ( R +) + = Try solutios! givig! = q q ( R + )q + ( R +) = ( ) ( ) N = + R + R R = U Exact solutio! ( ) ( ) f = exp R L x /L exp R L Cetered differeces! + R = R + R R = ( + R) ( + R) N N + R L = UL R = U Exact! Cetered! Exact! Cetered! Exact! Cetered! We cetered differecig is used for te advectio/diffusio equatio, oscillatios may appear we te Cell Reyolds umber is iger ta. For upwidig, o oscillatios appear. I most cases te oscillatios are small ad te cell Reyolds umber is frequetly allowed to be iger ta wit relatively mior effects o te result.! R = U <

4 example! +U t x + V y = f + f y f = =.! t=.88! Re cell =3.8! Flow! f = f = Computatios usig cetered differeces o a 3 by 3 grid! =.! t=.88! Re cell =6.46! =.! t=.88! Re cell =.93! =.! t=.!. Fie grid! Re cell =3.8!. Coarser grid! Re cell =6.676! Stability i! terms of Fluxes!.. 3 3

5 Stability i terms of fluxes! Stability i terms of fluxes! Cosider te followig iitial coditios:! + F + / = U = F / = U = U = urig oe time step, U t of f flows ito cell, icreasig te average value of f by U t/.! Δt =. U =. Cosider te followig iitial coditios:! F / = U = U f + = f Δt (F + / f i f i+ F + / = U = F / ) =.( ) =. Stability i terms of fluxes! Stability i terms of fluxes! F / = U = U F + / = Uf =.U F + / = U =.7U F / = U = U F + / = U =.U + + f + = f Δt (F + / + = + Δt (F + 3/ + F / ) =.(. ) =.7 F +/ ) =.(.) =. Takig a tird step will result i a eve larger positive value, ad so o util te compute ecouters a NaN (Not a Number).! Stability i terms of fluxes! If U t/ >, te average value of f i cell will be larger ta i cell -. I te ext step, f will flow out of cell i bot directios, creatig a larger egative value of f. Takig a tird step will result i a eve larger positive value, ad so o util te compute ecouters a NaN (Not a Number).! MOVIE FROM MATLAB! Stability i terms of fluxes! % oe-dimesioal advectio by first order upwid.! =8; step=; dt=.; legt=.;! =legt/(-);y=zeros(,);f=zeros(,);f()=.;! for m=:step,m! old off, plot(f); axis([,, -.,.]);! pause(.);! y=f;! for i=:-,! f(i)=y(i)-(dt/)*(y(i)-y(i-)); %upwid! ed;! ed;!

6 Stability i terms of fluxes! By cosiderig te fluxes, it is easy to see wy te cetered differece approximatio is always ustable.! Cosider te followig iitial coditios:! F / = U ( f + f ) =. F +/ = U ( f + f + ) =. + Advectio by Higer Order Metods! f + = f Δt (F +/ F / ) =..(. ) =. So cell will overflow immediately!! For te advectio terms, te metods described for yperbolic equatios, icludig ENO, ca all be applied, yieldig stable ad robust metods tat ca be forgivig for low resolutio.! QUICK, were a tird order upstream differecig is used is also popular.! s=! f! s=! f! s=/! s=3! f 3! At s = /! f / = ( /8) [ 3 f f f ] s=4! f 4! Use to solve:! t + x = f x f { i +/ f i / } i = {[ 64 3 f i f i f i ] [ 3 f i + 6 f i f i ] } Cetered! QUICK! t + x = f t + x = f Cetered! QUICK! Re_cell=! Re_cell=!

7 Secod order ENO sceme for te liear advectio equatio! f t + u f x = f * = f Δt u + = Δt f ( +/ / ) f +/ f * * * ( ( / ) + u ( +/ / )) u +/ = ( ) = ami a,b a, a < b b, b a + ami Δf + (, Δ ), if u + u + ami Δf + ( +, Δ + ), if u + u + ( ) > ( ) < Secod order ENO! t + x = f Cetered! QUICK! Re_cell=! ENO! Δ + = + Δ = Secod order ENO sceme for te liear advectio equatio! f t + u f x = ENO! Higer order! i space! Higer order fiite differece approximatios! Te simplest approac is to use more poits:! f(x-) f(x-) f(x) f(x+) f(x+)! Cetered x f Skewed x = O( 4 ) = O( 4 ) = O( 3 )

8 Compact Scemes! Compact scemes! Te stadard way to obtai iger order approximatios to derivatives is to iclude more poits. Tis ca lead to very wide stecils ad ear boudaries tis requires a large umber of gost poits outside te boudary. Tis ca be overcome by compact scemes, were we derive expressios relatig te derivatives at eigborig poits to eac oter ad te fuctio values.! Compact Scemes! By a Taylor series expasio te followig fort order relatios betwee te values of f ad te derivatives of f ca be derived! f i+ = f i + f i x Δx + f i Δx + 3 f i x 3 Δx f i x 4 Δx O(Δx ) f i = f i f i x Δx + f i Δx 3 f i x 3 Δx f i x 4 Δx O(Δx ) Addig! f i+ + f i = f i + f i Δx iv Δx 4 + f i + O(Δx6 ) Takig te secod derivative:! f i+ + f i = f i + f iv i Δx vi Δx 4 + f i + O(Δx6 ) Elimiatig te fourt derivative! ( ) f i+ + f i + f i = ( f Δx i+ f i + f i ) + O Δx 4 ()! ()! (3)! (4)! Compact Scemes! By a Taylor series expasio te followig fort order relatios betwee te values of f ad te derivatives of f ca be derived! f i+ = f i + f i x Δx + f i Δx + 3 f i x 3 Δx f i x 4 Δx O(Δx ) f i = f i f i x Δx + f i Δx 3 f i Δx 3 x f i Δx 4 x O(Δx ) Addig ad takig te first derivative:! f i+ + f i = f i + f i Δx iv Δx 4 + f i + O(Δx6 ) Subtractig from! Δx 3 f i+ f i = f i Δx + f i 3 + O(Δx ) Elimiatig te tird derivative! ( ) f i+ + 4 f i + f i = 3 ( Δx f f i+ i ) + O Δx 4 ()! ()! (3)! (4)! Compact Scemes! To solve te oliear advectio-diffusio equatio! i t = f i i x + f i we first solve fid te first ad secod derivatives usig te expressios derived above:! ( ) i+ x + 4 i x + i x = 3 ( Δx f f i+ i ) + O Δx 4 f i+ + f i x + f i ( ) + O( Δx ) 4 = f Δx i+ f i + f i Ad use te values to compute te RHS. Te time itegratio is te doe usig a ig order time itegratio metod.! A large umber of advaced umerical metods ave bee developed for yperbolic, parabolic ad elliptic equatios. Tese metods ca be applied directly to te Navier-Stokes equatios, altoug te structure of te equatios geerally requires us to pay close attetio i wic order te solutio proceeds.!