Spectral Simulation of Turbulence. and Tracking of Small Particles

Transcription

1 Specra Siuaio of Turbuece ad Trackig of Sa Parices

2 Hoogeeous Turbuece Saisica ie average properies RMS veociy fucuaios dissipaio rae are idepede of posiio. Hoogeeous urbuece ca be odeed wih radoy sirred urbuece i a cubic periodic box. The urbuece i a periodic box is hoogeeous bu o isoropic. Diagoas ad edges are differe.

3 avier-sokes Equaio wih Rado Sirrig Force u u u p ρ f υ u

4 Periodic Cubic Box u x L y L z pl u x y z The veociy is a periodic fucio of x y ad z wih period L.

5 Tie Discreizaio Tie Spiig Each ie sep ivoves hree sub-seps. Firs sub-sep: he o-iear er is copued expiciy. Secod sub-sep: he pressure er is copued ipiciy. Third sub-sep: he viscous er is copued ipiciy.

6 Roaioa For of avier- Sokes Equaio u u ω Π f ν u ω u Π p u ρ

7 Firs Fracioa Sep ~ u u u ω f p p p p p is he ie sep

8 Secod Fracioa Sep ^ ~ u u Π Π ~ u

9 Third Fracioa Sep u p ^ u υ u p

10 Fourier Represeaio of he veociy ad pressure fieds We represe he veociy ad pressure fieds by hree-diesioa Fourier series. Sice Fourier series are periodic he veociy ad pressure fieds are periodic. Cacuaios wih he veociy ad pressure are i physica space. Cacuaios wih he Fourier coefficies are i specra space

11 Three-Diesioa Fourier Series q L z y x i L z y x i e P z y x e z y x Π π π U u

12 Fas Fourier Trasfor FFT Fas Fourier Trasfor Cooey & Tookey 966 If a Fourier series ivovig ers is copued direcy he uber of operaios is proporioa o Wih he FFT agorih he uber of operaios is proporioa o og

13 Procedure for Cacuaig he Veociy ad Pressure The firs fracioa sep is perfored wih he veociy fied o a 3D grid wih 3 pois physica space. The Fourier coefficies of he pressure are copued i specra space. The Fourier coefficies of he veociy are copued i specra space.

14 FFT s eeded for Each Tie Sep A he ed of a ie sep we have he Fourier coefficies of he veociy. Therefore we eed a 3D FFT o copue he veociy ad voriciy for he firs fracioa sep of he ex ie sep. Afer copuig he firs ierediae veociy fied he ~ fied do a iverse 3D FFT o obai he Fourier coefficies.

15 Pressure Sep g Π Π ~ ~ ^ u u u I specra space: 4 ~ ^ G P p L P L i U U π π

16 Viscous Sep ^ p p u u u υ I specra space. ^ 4 p p L U U U π υ

17 Chae Fow-Ihoogeeous Turbuece Pressure drive fow bewee wo fa ifiie parae paes. Le us assue ha he fow is i he x- direcio ad ha he was are ocaed a zh ad z-h. Assue ha he fow is periodic i x ad y ad use Fourier series i hose direcios bu use a Chebyshev series i z.

18 Chebyshev Poyoias The covergece of Chebyshev series is idepede of he boudary codiios uike Fourier series because Chebyshev poyoias are souios of a siguar Sur-Liouvie probe. We ca si use FFT ehods for Chebyshev series.

19 Chebyshev Poyoias T z h cos θ θ cos z h

20 Specra Represeaio for Chae Fow q L y x i L y x i h z T e P z y x h z T e z y x 0 0 Π π π U u

21 Trackig of Sa Parices Low coceraios of sa parices have ie effec o he uderyig fow: oeway coupig To rack a parice we eed o kow he forces acig o i. We sove 6 ODE s i ie for each parice s coordiaes ad veociy copoes.

22 The parices ay be see o he ef ad he agiude of he fuid voriciy is show o he righ. The parices are cerifuged ou of regios of high voriciy io regios of ow voriciy.

23 Equaio of Moio for Sa Spherica Parices dv d τ τ 9 ρ v ρ f p u gk a υ C c

24 Cuigha Correcio Facor The Cuigha facor C c depeds o he oecuar ea-free pah ad he diaeer of he parice Uder ora codiios he Cuigha facor is cose o uiy for aerosos ha are arger ha icro.

25 Cacuaio of Aeroso Trajecories To copue he rajecory of a aeroso parice we eed o sove 6 siuaeous ODE s for he coordiaes ad veociy copoes of he parice. Sice he drag force ivoves he fuid veociy a he ocaio of he parice i is ecessary o ierpoae he fuid veociy o he coses grid pois.

26 Herie Ierpoaio i D i i i- h G h G H H h x x G f G f H f H f x f i i i i i ' ' 3

27 Accuracy The grid spacig is deoed by h o he previous side. The error ivoved i he ierpoaio of he fucio f is Oh 4. oe ha he ierpoaio reduces o he correc vaues for he fucio ad is firs derivaives a he eds of he grid ierva.

28 Two or Three-Diesioa Herie Ierpoaio I D oe eeds he vaues of he fucio he firs derivaives of he fucio ad he secod ixed derivaive of he fucio a he four eighborig pois 6 ubers. I 3D he hird ixed derivaive is used. A oa of 64 ubers are eeded.

29 Advaages of Herie Ierpoaio Herie ierpoaio has he advaage ha i avoids discoiuiies as a parice crosses he arificia boudaries bewee grid ces. I wo-diesios oe eeds o kow he veociy copoes heir firs derivaives ad heir secod ixed derivaive a he 4 coses grid pois.