Analysis of a Numerical Scheme An Example

Transcription

1 Computatioal Fluid Dyamics Lecture 3 Jauary 5, 7 Aalysis of a Numerical Scheme A Example Grétar Tryggvaso Numerical Aalysis Example Use the leap-frog method (cetered differeces to itegrate the diffusio equatio t = D f D > i time. Use the stadard cetered differece approximatio for the secod order spatial derivative. (a Write dow the fiite differece equatio. (b Write dow the modified equatio (c Fid the accuracy of the scheme (d Use the vo Neuma's method to derive a equatio for the amplificatio factor g. Hit: assume that the amplificatio is the same for each step: g = ε + ε = ε ε f + f Δt Numerical Aalysis Example (a Write dow the fiite differece equatio t = D f Approximate both terms by cetered differeces = D f + f + f h Numerical Aalysis Example (b Write dow the modified equatio ( f + = f + t Δt + f Δt t + 3 f Δt 3 t ( f = f t Δt + f Δt t 3 f Δt 3 t ( 3 f + = f + h + f h + 3 f h ( 4 f = f h + f h 3 f h Substitute ito f + f = D f + f + f Δt h Substitute ( ( Numerical Aalysis Example ( f + ( 4 = D 3 Δt h yieldig t + 3 f Δt t 3 6 = D f 4 + D f h 4 + Rearrage t D f 4 = D f h 4 3 f Δt t (c Fid the accuracy of the scheme O( Δt,h Modified Equatio

2 Numerical Aalysis Example (d Use the vo Neuma's method to derive a equatio for the amplificatio factor g. Substitute ε = ε e ito ikx ε + ε Δt givig ε + e ikx ε e ikx Δt = D ε + ε + ε h = D ε h (eikh e ikx e ikx + e ikh e ikx ε + e ikx ε e ikx Δt Numerical Aalysis Example Cacel the commo factor Rearrage ε + ε ε + ε Usig that Gives: Δt ε ε = D ε h (eikh e ikx e ikx + e ikh e ikx = D ε h (eikh + e ikh = ΔtD h (e ikh + e ikh g = ε + ε = ε ε g g = ΔtD kh 4si h e ikh + e ikh = coskh si θ = cosθ Numerical Aalysis Example g g = ΔtD kh 4 si h Puttig B = 8 ΔtD kh h si gives g = Bg Solutio g = B ± B + Which is always > g + Bg = x + bg + c = x = b ± Ucoditioally ustable b c Aalysis of a Numerical Scheme Aother Example The followig fiite differece approximatio is give f + f = U Δt h f + f ( + U h f + + ( + f (a Write dow the modified equatio f + f = U Δt h f + f ( + U h f + + ( + f (a Write dow the modified equatio (b What equatio is beig approximated? (c Determie the accuracy of the scheme (d Use the vo Neuma's method to derive a equatio for the stability coditios + f f h f + f h + f + f + Δt

3 ( f + = f + t Δt + f Δt t + 3 f Δt 3 t 3 6 +! ( f + = f + h + f h + 3 f h ! ( 3 f = f h + f h 3 f h ! ( 4 f + + = f ( 5 f + = f + + h + + f h h + + f h f h f h ! 6 +! Substitute ( f = U Δt h ( ( (3 + U h ( (4 (5 yieldig t + f Δt t +! = U + 3 f h 3 6 +! f h 3 6 +! + Usig ( agai we ca write = + f t Δt + 3 f Δt t + 4 f Δt 3 t 3 6 +! 3 + f = 3 f + 4 f 3 3 t Δt + 5 f Δt 3 t f Δt 3 t ! + Substitute agai: t + f Δt t +! = U + 3 f h 3 6 +! + + f t Δt + 3 f Δt t +!+ 3 f +! h 3 6 +! Collectig the terms t + f Δt t +! = U + 3 f h f t Δt + 3 f Δt t + 3 f h 3 6 +! = U t U f t = U = f t t + Rearrage t + U = U 3 f h 3 6 U 3 f Δt t 4 +! (b What equatio is beig approximated (c Fid the accuracy of the scheme O( Δt,h t + U = (d Use the vo Neuma's method to derive a equatio for the amplificatio factor g. Substitute ε = ε e ito ikx ε + ε = U Δt h ε + givig ( ε + U h ε ( ε ( + ε + e ikh e ikx e ikh e ikx ( ( ε + e ikx ε e ikx = U Δt 4h ε e ikh e ikx e ikh e ikx ε + e ikx ε e ikx = U Δt 4h ε e ikh e ikx e ikh e ikx Cacel the commo factor Rearrage writig g = ε + ε Gives: ( + ε + e ikh e ikx e ikh e ikx ( ( ( ( + ε + ( e ikh e ikh ε + ε = U Δt 4h ε e ikh e ikh ε + ε = ΔtU 4h ( e ikh e ikh + ε + ( e ikh e ikh ε g = ( + g ΔtU 4h ( isikh e ikh e ikh = isikh

4 g = ( + g ΔtU 4h ( isikh g( A( isikh =+ A( isikh + A isikh g = ( A( isikh g = 4A si kh + iasikh + 4A si kh g = B + ib + B g = ( B + B ( + B B + B 4 <+ B + B 4 z = x + iy z = ( x + iy ( x iy = x + y Ucoditioally Stable Notes: You will do a few other schemes as homework problems. Geerally we assume that results for the liear equatios hold for the oliear oe as well. The algebraic equatio for the amplificatio factor ca ofte be fairly complicated. It is, however, routiely solved. Obectives A Fiite Differece Code for the Navier-Stokes Equatios i Vorticity/Streamfuctio Form Develop a uderstadig of the steps ivolved i solvig the Navier-Stokes equatios usig a umerical method Write a simple code to solve the drive cavity problem usig the Navier-Stokes equatios i vorticity form Short discussio about why lookig at the vorticity is sometimes helpful Outlie The Drive Cavity Problem The Drive Cavity Problem The Navier-Stokes Equatios i Vorticity/ Streamfuctio form Boudary Coditios The Grid Fiite Differece Approximatio of the Vorticity/ Streamfuctio equatios Fiite Differece Approximatio of the Boudary Coditios Iterative Solutio of the Elliptic Equatio The Code Results Covergece Uder Grid Refiemet Movig wall Statioary walls

5 The vorticity/streamfuctio equatios: The vorticity/streamfuctio equatios: y u t + u u + v u y = p + v t + u v + v v y = p y + ω + u ω t + v ω y = Re ω = v u y u Re + u y v Re + v y ω + ω y Solve the icompressibility coditios u + v y = by itroducig the stream fuctio Substitutig: u = ψ y ; v = ψ ψ y ψ y = Substitutig The vorticity/streamfuctio equatios: u = ψ y ; v = ψ ito the defiitio of the vorticity yields ω = v u y ψ + ψ = ω y ω t = ψ y The vorticity/streamfuctio equatios: The Navier-Stokes equatios i vorticity-stream fuctio form are: Advectio/diffusio equatio Elliptic equatio ω + ψ Recall the advectiodiffusio equatio ω y + Re ψ + ψ = ω y ω + ω y t + u + v y = D f + f y Boudary Coditios for the Streamfuctio Boudary Coditios for the Streamfuctio At the right ad the left boudary: u = ψ y = ψ = Costat At the top ad the bottom boudary: v = ψ = ψ = Costat

6 Boudary Coditios for the Streamfuctio Boudary Coditios for the Vorticity Sice the boudaries meet, the costat must be the same o all boudaries ψ = Costat The ormal velocity is zero sice the streamfuctio is a costat o the wall, but the zero tagetial velocity must be eforced: At the right ad left boudary: v = ψ = At the bottom boudary: u = ψ y = At the top boudary: u = U wall ψ y = U wall Boudary Coditios for the Vorticity The wall vorticity must be foud from the streamfuctio. The stream fuctio is costat o the walls. Summary of Boudary Coditios ψ y = U wall ; = ψ y At the right ad the left boudary: ψ + ψ = ω y = ψ Similarly, at the top ad the bottom boudary: ψ + ψ = ω y = ψ y ψ = = ψ ψ y = ; ψ = Costat = ψ y ψ = = ψ Discretizig the Domai To compute a approximate solutio umerically, we start by layig dow a discrete grid: =NY Fiite Differece Approximatios The we replace the equatios at each grid poit by a fiite differece approximatio ψ i, ad ω i, stored at each grid poit = = i= i= i=nx Grid boudaries coicide with domai boudaries ω t i, = ψ ω y i, ψ i, + ψ ω y + ψ y i, i, + ω + ω Re y = ω i, i,

7 Fiite Differece Approximatios Fiite differece approximatios Fiite Differece Approximatios Use the otatio developed earlier: (x = f (x = (t t = f (x + h f (x h h 3 f (x 3 h + f (x + h f (h + f (x h 4 f (x h h 4 xx + f (t + Δt f (t Δt f (t Δt t + (x, y + - i - i i+ f i, + = f (x, y + h f i, = f (x,y f i +, = f (x + h, y Laplacia Fiite Differece Approximatios f + f y = f i +, f i, + f i, + f i, + f i, + f i, = h h f i +, + f i, + f i, + h + f i, 4 f i, ω t The advectio equatio is: ω + i, ω i, = Δt ψ ψ i, + i, h + Re ω i+, Fiite Differece Approximatios = ψ ω y + ψ ω + ω i, ω i+, ω i, h + ω i, + h + ω i, y + ω Re + ω y + ψ i+, h 4ω i, ψ i, ω i, + ω i, h ω i, Fiite Differece Approximatios The vorticity at the ew time is give by: + = ω i, + Δt ψ ψ i, + i, ω i+, h h + ψ ψ i+, i, ω i, + h h + Re ω i+, + ω i, + ω i, + h ω i, ω i, + ω i, 4ω i, ψ i+, Fiite Differece Approximatios The elliptic equatio is: ψ + ψ i, + ψ y = ω + ψ i, + h + ψ i, 4ψ i, = ω i,

8 Fiite Differece Approximatios These equatios allow us to obtai the solutio at iterior poits =y = = i= i= i=x ψ i, = o the boudary Need vorticity o the boudary! =3 = = Discrete Boudary Coditio Cosider the bottom wall (=: Need to fid = ω i, = U wall i - i i+ give: ψ = Costat ψ y = U wall ; = ψ y =3 = = i - i i+ Discrete Boudary Coditio U wall Expad the streamfuctio give: ψ = Costat ψ y = U wall; = ψ y ψ i, = =ψ i, = + ψ i, = h + ψ i, = h y y + O(h3 Discrete Boudary Coditio ψ i, = =ψ i, = + ψ i, = h + ψ i, = h y y + O(h3 Usig: = ψ i, = ; U y wall = ψ i, = y this becomes: h ψ i, = =ψ i, = + U wall h + O(h 3 Solvig for the wall vorticity: ( = ψ i, = ψ i, = h +U wall h +O(h The elliptic equatio: ψ i +, Rewrite as Solvig the elliptic equatio +ψ i, ψ + i, =.5 (ψ i +, +ψ i, +ψ i, + +ψ i, 4ψ i, h = ω i, +ψ i, + +ψ i, + h ω i, Time Step Limitatios o the time step νδt h 4 ( u + v Δt ν Solve by SOR ψ i, α + α = β.5 (ψ i +, α + ( βψ i, α +ψ + α i, +ψ i, + α +ψ + i, + h ω i,

9 Solutio Strategy Solutio Strategy Iitial vorticity give Solve for the stream fuctio Fid vorticity o boudary Fid RHS of vorticity equatio Update vorticity i iterior t=t+δt Iitial vorticity give Solve for the stream fuctio Fid vorticity o boudary Fid RHS of vorticity equatio Update vorticity i iterior t=t+δt For l=:maxiteratios for i=:x-; for =:y- s(i,=sor for the stream fuctio ed; ed ed v(i,= for i=:x-; for =:y- rhs(i,=advectio+diffusio ed; ed v(i,=v(i,+dt*rhs(i, The Code Results: % Drive Cavity by Vorticity-Stream Fuctio Method Nx=7; Ny=7; MaxStep=; Visc=.; dt=.5; time=.; h=./(nx-; MaxIt=; Beta=.5; MaxErr=.; % parameters for SOR sf=zeros(nx,ny; vt=zeros(nx,ny; vto=zeros(nx,ny; x=zeros(nx,ny; y=zeros(nx,ny; for i=:nx,for =:Ny,x(i,=h*(i-;y(i,=h*(-;ed,ed; for istep=:maxstep, for iter=:maxit, % Time loop vto=sf; for i=:nx-; for =:Ny- % solve for the stream fuctio by SOR iteratio sf(i,=.5*beta*(sf(i+,+sf(i-,+sf(i,++sf(i,-+h*h*vt(i,+(.-beta*sf(i,; ed; ed; Err=.; for i=:nx; for =:Ny, Err=Err+abs(vto(i,-sf(i,; ed; ed; % check error if Err <= MaxErr, break, ed % stop if coverged ed; vt(:nx-,=-.*sf(:nx-,/(h*h; % vorticity o bdrys vt(:nx-,ny=-.*sf(:nx-,ny-/(h*h-./h; % top wall vt(,:ny-=-.*sf(,:ny-/(h*h; % right wall vt(nx,:ny-=-.*sf(nx-,:ny-/(h*h; % left wall vto=vt; for i=:nx-; for =:Ny- vt(i,=vt(i,+dt*(-.5*((sf(i,+-sf(i,-*... (vto(i+,-vto(i-,-(sf(i+,-sf(i-,*(vto(i,+-vto(i,-/(h*h+... Visc*(vto(i+,+vto(i-,+vto(i,++vto(i,--4.*vto(i,/(h^ ; ed; ed; time=time+dt subplot(, cotour(x,y,vt,4, axis('square'; subplot(, cotour(x,y,sf, axis('square'; pause(. ed; 7 by 7 Dt=. D=. Results: Results:..8 Streamfuctio 5 Vorticity 7 by 7 Dt=. D= u i, = ψ y ψ ψ i, + i, h v i, = ψ ψ i+, ψ i, h

10 Results: 9 by 9 grid 7 by 7 grid Streamfuctio at t =. Vorticity at t = The total volume flow to the right i the drive cavity versus time, computed as the differece betwee the value of the stream fuctio o walls ad its miimum value, for three resolutios. The velocity i the drive cavity, foud by differetiatig the stream fuctio, alog with the horizotal velocity o a vertical axis through the ceter of the domai ad the vertical velocity o a horizotal axis through the ceter of the domai, for three resolutios. Fiite Differece Approximatios Periodic Domai i the x-directio Periodic Boudaries Set poit x+ = poit Update i=;x Set poit = Poit x =y ψ = Q Boudary of domai Same Same = = i= i= i=x ψ = Oe colum of ghost poits i=x+ Simulatios of the rollup of a periodic vortex sheet x = 65; y = 65; MaxStep = ; Visc =.; dt =.; sf(i,=sf(i,y=; % Iitial vorticity distributio vt(:34,(y-/-=./h; vt(33:x-,(y-/+=./h; Why is vorticity importat?

11 Vorticity Helmholtz decompositio: Ay vector field ca be writte as a sum of Take the curl u = φ + Ψ Take divergece u = φ = φ = ( = ω u = Ψ By a Gauge trasform this ca be writte as Ψ = ω ω t Vorticity For icompressible flow with costat desity ad viscosity, takig the curl of the mometum equatio yields: or: + u ω = ( ω u + ν ω Dω = ( ω u + ν ω Dt Helmholtz s theorem: Iviscid Irrotatioal flow remais irrotatioal Vorticity Flow over a body I two-dimesios: Ψ = (,,ψ ω = (,,ω Irrotatioal outer flow ω = v u y or: Dω Dt =ν ω ψ = ω Rotatioal wake Zero viscosity: Dω = Dt The vorticity of a fluid particle does ot chage! Boudary layer: the flow is viscous ad rotatioal Boudary Layers Advectio ad diffusio Boudary layers u t + u u + v u y = P ρ + t + U = D f µ u + u ρ y

12 Boudary Layers Cosider the steady state balace of advectio ad diffusio f = x = Govered by: U = D f Solve this equatio aalytically Itegrate: dx = D d f U dx f D U U dx = C f = x = L d f D dx U dx = Rearrage or Itegrate f C = D U Boudary Layers dx ( = U D dx f C ( f C dx = U D = U ( f C D dx l( f C = U D x + C f = exp( Ux / D exp( C + C At At Boudary coditios Boudary Layers f = exp( Ux / D exp( C + C ( + C ( x = : f = = exp C C = exp C x = L : f = = exp( UL/ D exp( C + C = exp( UL/ D exp( C exp( C ( [ exp( U / D ] = exp C exp( C = exp( UL/ D Boudary Layers f = exp( Ux / D exp( C + C C = exp( C exp( C = ( ( exp Ux / D f = exp UL/D exp Rx / L f = ( exp( R exp UL/D ( R = UL D Boudary Layers Boudary Layers r=;for i=:,x(i=(i-/99;ed; for i=:,f(i=(exp(r*x(i-/(exp(r-;ed; plot(x,f R= R=5 R= R= Scalig: dx = D d f U dx Estimate the thickess of the boudary Layer R= L/R= R=5 L/R=. R= L/R=. R= L/R= δ = D δ DL U δ UL = L R

13 D f= Boudary Layers Solutio of: U= D=.5 U f = D f + f y f= δ D U = Obectives Develop a uderstadig of the steps ivolved i solvig the Navier-Stokes equatios usig a umerical method Write a simple code to solve the drive cavity problem usig the Navier-Stokes equatios i vorticity form Short discussio about why lookig at the vorticity is sometimes helpful Proect I Time evolutio of a oe-dimesioal equatio Write a program to compute the usteady behavior of the followig equatio t + f = ν f i a periodic domai of legth with ν =.. This is usually called the oliear advectio-diffusio equatio or the Burgers equatio. Take the iitial coditios to be f (x,t = = si(πx +. ad approximate the equatio usig a forward i time approximatio for the time derivative, ad cetered approximatios for the first ad secod derivative. Follow the evolutio up to time. usig at least three differet grid resolutios, the fiest of which should use at least grid poits. You ca start from the program used i class for the liear advectio- diffusio equatio.