Astrophysical Fluid Dynamics: I. Hydrodynamics

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1 Witer School o Computatioal Astrophysics, Shaghai, 2018/01/29 Astrophysical Fluid Dyamics: I. Hydrodyamics Xueig Bai ( 白雪宁 ) source: J. Stoe Istitute for Advaced Study (IASTU) & Tsighua Ceter for Astrophysics (THCA)

2 Geeral refereces Ladau & Lifshitz, Fluid Mechaics (Vol. 6 of Course of Theoretical Physics), 1987, 2 d Eglish Editio Shu, F. H., The Physics of Astrophysics, Vol. 2: Gas dyamics, 1992, Uiversity Sciece Books Ogilvie, G. I., Lecture otes o Astrophysical Fluid Dyamics, 2016, Joural of Plasma Physics, vol. 82, Cambridge Uiv. Press Spruit, H. C., Essetial Magetohydrodyamics for Astrophysics, 2013, arxiv: Kulsrud, R., Plasma Physics for Astrophysics, 2005, Priceto Uiversity Press Note: here we oly cosider Newtoia (o-relativistic) fluid dyamics 2

3 Outlie Formulatio Fluid basics Coservatio laws Viscosity Liear waves Shocks ad discotiuities Example: Bodi accretio 3

4 Outlie Formulatio Fluid basics Coservatio laws Viscosity Liear waves Shocks ad discotiuities Example: Bodi accretio 4

What is a fluid? Fluid is a idealized cocept i which the matter is described as a cotiuous medium with macroscopic properties varyig with positio.

5 What is a fluid? Fluid is a idealized cocept i which the matter is described as a cotiuous medium with macroscopic properties varyig with positio. A local physical quatity ca be thought to be associated with a fluid elemet that is: l l sufficietly large to cotai umerous particles to defie macroscopic properties sufficietly small compared with the legth scales of iterest (ca be cosidered as ifiitely small) a fluid elemet It ca be applicable to 3 of the 4 states of matter: liquid, gas, ad sometimes, plasmas. 5

6 Validity of the fluid approximatio Geerally requires the molecular mea free path to be much smaller tha scales of iterest. At microscopic level, this also meas that collisios are sufficietly frequet so that particle distributio fuctios approach Maxwellia. Size scale Air ~10 6 cm ~10-6 cm particle mea free path Solar iterior ~10 10 cm ~10-8 cm (Coulomb) Solar wid ~AU~10 13 cm ~10 13 cm (Coulomb) Galactic ceter ~10 13 cm ~10 18 cm (Coulomb) Galaxy clusters ~Mpc~10 24 cm ~10 22 cm (Coulomb) Whe fluid approximatio fails, kietic treatmet is geerally required. 6

7 Astrophysical fluid/gas dyamics Differs from laboratory ad/or egieerig fluid dyamics i the relative importace of certai effects. l l l l l l l l Compressibility Gravity Magetic fields Radiatio forces Relativistic effects Viscosity Surface tesio Presece of solid boudaries Geerally importat Sometimes importat sometimes ivoked to mimic turbulece Geerally uimportat 7

8 How to describe a fluid? Macroscopic physical quatities are momets of the distributio fuctio: Z Desity: (x) =m d 3 vf(x, v) (mass) Velocity: v(x) = Pressure: P(x) =m R d 3 vvf(x, v) R d3 vf(x, v) Z d 3 v(v v)(v v)f(x, v) (mometum) (eergy) For isotropic DF, pressure is a scalar: P(x) =P (x)i Note: with frequet collisios, f becomes Maxwellia, which ca be exactly specified by the above 3 quatities. Occasioally, oe cosiders multi-species, requirig oe copy of the above quatities for each species => multi-fluid. 8

9 How to describe a fluid? Euleria view: Follow physical quatities at fixed locatios. Temporal chages of a quatity Q are described by a partial Q(r,t+ t) = Q(r,t) t Lagragia view: Follow physical quatities as oe travels alog the flow. Temporal chages of a quatity Q are described by a Lagragia or covective time derivative: DQ Dt Q(r + r,t+ t) = Q(r,t) + v r Q where r = v t 9

10 Equatio of cotiuity Pick a volume V bouded by a closed surface S. The total mass cotaied i the volume is: M = Z V dv This mass chages due to masses flowig through the surface S: dm dt = dv = I S v ds = Z V r ( v)dv This must hold for ay volume elemets, which gives the + r ( v) =0 mass flux 10

11 Two remarks I geeral, coservatio law for a quatity A is expressed (desity of A)+r (flux of A) Source ad sik terms, if preset, eter oto the right had side. The cotiuity equatio ca be + r ( v) =0 The compressio term: r v < or > 0: r v 0: D Dt = compressio or dilatio flow is icompressible r v 11

12 Equatio of motio Pick a fluid elemet withi volume V bouded by a closed surface S, with total mass M. Newto s 2d law of motio over this volume gives: The force ivolves pressure: F P = I S PdS = M Dv Dt = F Z V rp dv g PdS For a ifiitesimal volume, we arrive at Euler s equatio: Dt + v rv = + v rv = rp Additioal forces/acceleratio (e.g., gravity) ca be added to + v rv = rp + g = rp r 12

13 Thermal eergy equatio With the cotiuity equatio + Euler equatio, the system is ot closed uless pressure is kow. For the special case of a barotropic fluid (which is ofte adopted), the equatio of state (EoS) is give by P=P(ρ), which closes the system. More geerally, the EoS also depeds o temperature P=P(ρ,T), a separate equatio is eeded to determie temperature evolutio. With thermodyamic relatios, there are multiple equivalet forms for this equatio. 13

14 Thermal eergy equatio For a ideal fluid without eergy dissipatio or exchage, ay fluid elemet behaves adiabatically, which coserves etropy: Ds +(v r)s =0 s: specific etropy (etropy per uit mass) Etropy is related to iteral eergy ε via the first law of thermodyamics: d = Tds P dv = P 2 d D (=0) Dt = D Dt = P r v +(v r) = P r v Source terms icorporatig additioal irreversible processes, e.g. thermal coductio, viscous dissipatio ca be added to the right had side. 14

15 Thermal eergy equatio For ideal gas, we have: = c V T P = kt P =( where 1) = c P /c V =(c V + k)/c V ratio of specific heat From the thermal eergy equatio We obtai: 1 1 DP Dt P D Dt = P r v D = Dt P r v It further + v rp + P r v =0 which is a commoly used form of the thermal eergy equatio. 15

16 Outlie Formulatio Fluid basics Coservatio laws Viscosity Liear waves Shocks ad discotiuities Example: Bodi accretio 16

17 Hydrostatics For a fluid at rest i a gravitatioal field, the mechaical equilibrium of the fluid (hydrodyamic equilibrium) is described by: rp = g l l Widely applicable to astrophysical systems, e.g., stellar iterior, atmospheres. Commoly serve as iitial coditios for stability aalysis, ad for settig up umerical simulatios. Accurate solutios requires kowledge about temperature T (thermodyamics) with a equatio of state P=P (ρ,t). I simple cases oe may cosider a isothermal equatio of state P=ρc s2. For a oe-dimesioal problem alog the z directio, we have d dz = g(z) which is directly itegrable oce g(z) is kow. isothermal soud speed (costat) 17

18 Hydrostatics: simple exercises Assumig isothermal EoS, solve for the followig 1D hydrostatic problems: 1. Plaetary atmosphere with g(z)=-g 0 =costat. c 2 s d dz = g 0 = 0 e z/h 0 Q: how thick is the Earth s atmosphere? where H 0 =c s2 /g 0. Scale height 2. Vertical structure of a accretio disk: g(z)=-ω K z. c 2 s d dz = Disk aspect ratio: Kz H R = 0 e z2 /2H 2 0 c s K R = c s v K where H 0 =c s /Ω K. Scale height Q: how thick is the solar ebula? 18

19 Vorticity Defie vorticity:! r v Physically, it describes the rotatio/circulatio of the fluid elemet. Cosider: v! v (r v) = 1 2 rv2 (v r)v Combied with Euler s equatio: We arrive + r 2 + v rv = + rp rp r v! =0 Takig a curl, we obtai the = r (v w)+ 1 r rp 2 19

Vorticity The vorticity equatio reads: @w @t = r (v w)+ 1 r rp 2 barocliic term (vorticity geeratio) If the gas is barotropic, meaig P=P(ρ), the the barotropic term vaishes, ad hece we simply have:

20 Vorticity The vorticity = r (v w)+ 1 r rp 2 barocliic term (vorticity geeratio) If the gas is barotropic, meaig P=P(ρ), the the barotropic term vaishes, ad hece we = r (v w)+ This is related to Kelvi s circulatio theorem: The velocity circulatio: I C v dl alog a closed cotour is coserved as the cotour moves with the fluid. C dl This is exactly aalogous to the cocept of magetic flux freezig, which will be proved i the ext lecture o MHD. 20

21 Beroulli s theorem For a ideal fluid (o dissipatio): + r 2 v2 + Assumig a =0 dh = Tds+ VdP = dp/ ethalpy + rp v! =0 Restrict this equatio alog the flow streamlie i steady state: 1 v r 2 v2 + + h =0 alog the flow =0 => The Beroulli costat B 1 2 v2 + + h is coserved alog streamlies. 21

22 Beroulli s theorem: simple applicatios Beroulli costat: B 1 2 v2 + + Z dp water flow from the faucet De Laval ozzel Loop drive 22

23 Compressibility Cosider a flow i which fluid variables (e.g. P, ρ, v) vary over some characteristic scale L ad time T (with L~vT). We ca compare the magitude of terms i the Euler equatio: Note that P c 2 s where c s is the soud speed, we fid: v2 + c 2 s The flow ca be cosidered to be largely icompressible if v<<c s ad gravitatioal potetial does ot vary greatly alog streamlies. For icompressible flow, the cotiuity equatio is simplified to: r v =0 23

24 Icompressible flow Fluid equatios are substatially simplified for a compressible flow. Cotiuity equatio becomes: r v =0 For homogeeous backgroud medium, it implies ρ=cost. (ot true otherwise) Euler s + v rv = r(p/ ) r Kelvi s circulatio theorem applies, ad the system ca simply be closed with the vorticity equatio aloe. Beroulli s theorem is simplified to B = 1 2 v2 + P + Methods for solvig icompressible fluid equatios are geerally very differet from those for solvig compressible equatios. We will focus o the latter. 24

25 Outlie Formulatio Fluid basics Coservatio laws Viscosity Liear waves Shocks ad discotiuities Example: Bodi accretio 25

26 Coservatio laws: I geeral, coservatio law for a quatity A is expressed (desity of A)+r (flux of A) Source ad sik terms, if preset, eter oto the right had side. We already have derived the equatio of + r ( v) =0 Give the fudametal importace of mometum ad eergy coservatio, they should be applicable to fluids as well. 26

27 Mometum coservatio Startig from the cotiuity equatio ad the Euler equatio with o + r ( v) It is straightforward to + v rv = ( @t v = v rv rp r ( = r ( vv + P I) mometum flux desity tesor idetity tesor With exteral forces, source terms ca be added o the + r ( vv + P I) = g 27

28 The eergy equatio Eergy desity has cotributios from iteral, kietic ad gravitatioal eergies. D Dt 1 2 v2 D Dt = v r = v Dv Dt = v r v rp (kietic) (gravitatioal) Assumig static gravitatioal potetial (ca be relaxed) D Dt = P r v (iteral) D 1 Dt 2 v2 + + = r (P v) Summig over the above: With further maipulatio usig apple 1 2 v r apple 1 v 2 v P v =0 eergy desity eergy flux desity 28

29 Outlie Formulatio Fluid basics Coservatio laws Viscosity Liear waves Shocks ad discotiuities Example: Bodi accretio 29

gives a mometum flux: xy v th dvx dy mfp This mometum flux attempts to reduce backgroud shear.

30 Origi of viscosity Radom motio of molecules i a fluid leads to mometum exchage at the scale of molecular mea free path. Cosider a backgroud shear flow: y x λ mfp Exchage of x mometum due to thermal motio i y ad molecular collisios gives a mometum flux: xy v th dvx dy mfp This mometum flux attempts to reduce backgroud shear. I geeral, viscosity should be proportioal to rate of strai, ad acts to reduce velocity gradiets ad drive the system towards uiform motio. 30

Viscous stress Recall the equatio of mometum coservatio i a ideal fluid: Viscosity should cotribute to the mometum flux tesor, which becomes: viscous stress tesor As we have see, π should be

31 Viscous stress Recall the equatio of mometum coservatio i a ideal fluid: Viscosity should cotribute to the mometum flux tesor, which becomes: viscous stress tesor As we have see, π should be proportioal to velocity gradiets. I additio, it should vaish i uiform rotatio. The most geeral tesor satisfyig these is dyamic viscosity shear viscosity (traceless) secod viscosity bulk viscosity 31

32 Navier-Stokes equatio Recall the viscous stress: Whe viscosity does ot vary appreciably i the fluid, This is the Navier-Stokes equatio, which becomes cosiderably simpler for a icompressible fluid: where is the kiematic viscosity. From our aalysis earlier: (more detailed aalysis would give a additioal factor 1/3) 32

33 Whe is viscosity importat Order of magitudes from the Navier Stokes equatio: Note: ~V 2 /L ~V 2 /L ~c ~νv/l 2 s2 /L Defie the Reyolds umber: Re VL viscosity is domiat whe Re~1 or less. For flow speeds ot far from soud speed, V~v th, hece Re is large whe L>>λ mfp. Eve with large Re for the bulk flow, viscosity ca still play importat roles at small scales (e.g., at boudaries, discotiuities, turbulet dissipatio).

34 Eergy dissipatio i viscous fluid Viscosity correspods to iteral frictio ad leads to irreversible eergy dissipatio. Recall eergy coservatio i ideal gas: where Oe ca follow the procedure ad update the eergy coservatio equatio: where I the process we ca fid stress tesor rate of viscous dissipatio 34

35 Outlie Formulatio Fluid basics Coservatio laws Viscosity Liear waves Shocks ad discotiuities Example: Bodi accretio 35

36 Liear waves Cosider perturbatios o top of a homogeeous fluid at rest (ρ=ρ 0, v=v 0 =0), ad assumig isothermal EoS (P=ρc s2 ) for simplicity. Recall the geeral fluid = v r @t + v rv = c2 sr / Note, for adiabatic gas, dp = γ(p/ρ)dρ, ad we have the same equatio except that c s2 = γ(p/ρ). To first-order, perturbatio equatios read = 0r = c 2 s r 0 36

37 Liear waves Cosider 1D problem, ad also restrict velocity to that dimesio, perturbatio equatios becomes: This is a wave equatio, with geeral solutio: where R, L are arbitrary fuctios. Disturbaces propagate to right ad left at velocity c s. Q: for a icompressible fluid, are there soud waves? 37

38 Eigemode aalysis Ay disturbaces ca be decomposed ito a series of Fourier modes. Pick up oe mode with waveumber k, ad look for solutios of the form Recall the perturbatio equatios (agai restrictig to 1D): We arrive a system of liear algebraic equatios. No-trivial solutios exist oly whe its determiat vaishes: Dispersio relatio 38

39 Eigemode aalysis Eigevector of a give mode must satisfy the liear equatios: + Eigevector gives the relative amplitude ad phase of perturbatios amog differet physical quatities. Recoverig the origial form of perturbatio by takig the real part: This describes the propagatio of a sigle mode of soud waves, with Phase velocity: V p =! k = c s Group velocity: V = c s V g =V p : soud wave is o-dispersive 39

Wave aalysis: geeral approach Startig from the full equatios to obtai liearized fluid equatios (ow also iclude the eergy equatio). We still cosider 1D (i.e.,

40 Wave aalysis: geeral approach Startig from the full equatios to obtai liearized fluid equatios (ow also iclude the eergy equatio). We still cosider 1D (i.e., wave propagatio i a give directio), but iclude all vector compoets (so that the aalysis + r ( + v rv + v rp + P r v =0 40

41 Wave aalysis: geeral approach Defie a vector of the primitive fluid variables: W (,v x,v y,v z,p) T The liearized equatios ca be cast ito the form (we omit subscript =0 where P Coduct eigemode aalysis assumig variables are of the form e i(kx!t) We obtai AW 1 =! k W 1 Clearly, the eigevalues of matrix A give the speeds of all possible wave modes. 41

42 Wave aalysis: geeral approach The matrix A has 5 eigevalues: =(v x c s,v x,v x,v x,v x + c s ) where c s = p P/ (adiabatic soud speed) They correspod to 5 waves: what are they? P Lookig at the correspodig eigevectors of each mode: RecallW (,v x,v y,v z,p) T c s c s l Two for soud waves (forward / backward propagatig) l l Two vortical modes (perturbatios i v y, v z advected alog x) Oe etropy wave (with desity but o pressure perturbatios) c s soud vortical etropy c s soud 42

43 Outlie Formulatio Fluid basics Coservatio laws Viscosity Liear waves Shocks ad discotiuities Example: Bodi accretio 43

44 No-liear steepeig of soud waves Recall that i a ideal gas, soud speed is give by c s = P I liear aalysis, this is cosidered to be costat (homogeeous backgroud). To the ext order, however, c s is slightly larger i higher-desity regios. => the crest of a wave propagates faster tha the leadig/trailig edge s = s P Slightly o-liear soud waves will steepe to form a discotiuity: shock 44

45 Shocks ad discotiuities Fluid equatios are PDEs: discotiuous solutios do ot hold i a classical sese. Physically, more fudametal are the itegral form of the coservatio laws, which do admit Z (Desity of A) dv + Z (Flux of A) ds =0 l l Shock is oe type of discotiuity, which usually forms whe the flow speed becomes supersoic. There are other types of discotiuities as we shall see. Microscopic dissipatio (e.g., viscosity) is importat across the discotiuities. 45

46 Surface of discotiuity We work i the co-movig frame with the surface of discotiuity. The fluxes of mass, mometum, eergy across this surface must be cotiuous. Mass coservatio: 1 2 ρ 1,v 1, P 1 ρ 2,v 2, P 2 x Mometum coservatio: Eergy coservatio: where we have used 46

47 Tagetial discotiuity If there is o mass flux through the surface: Mass coservatio: 1 2 ρ 1,v 1, P 1 ρ 2,v 2, P 2 Mometum coservatio: x P 1 =P 2, while v y ad v z ca be discotiuous by ay amout. For discotiuous tagetial velocities, the iterface is subject to the Kelvi-Helmholtz istability: The special case with cotiuous tagetial velocities:, all other quatities are cotiuous. This is called a cotact discotiuity. 47

48 Shocks jump coditios If the mass flux across the surface is ozero, we obtai the shock jump coditios: upstream (1) dowstream (2) Mass coservatio: Mometum coservatio: ρ 1,v 1, P 1 ρ 2,v 2, P 2 shock frot x Eergy coservatio: This is a set of 3 algebraic equatios with 3 ukows, which are called Rakie-Hugoiot coditios. 48

4pc Note: if we are i the shock frame, the upstream flows ito the shock

49 Superova remat shock dowstream (shocked) upstream (ushocked) Tycho shock frot shock velocity: ~ km/s Chadra 4 ~ 2.4pc Note: if we are i the shock frame, the upstream flows ito the shock at the observed shock velocity. Shock (soic) Mach umber is defied as (based o upstream flow properties)

50 Example: a strog shock For a strog shock, the upstream kietic eergy >> thermal eergy: upstream (1) dowstream (2) ρ 1,v 1, P 1 ρ 2,v 2, P 2 shock frot x It is straightforward to obtai: For mooatomic gas, γ=5/3, leadig to: compressio ratio substatial kietic eergy from upstream coverted to thermal eergy dowstream 50

51 Dissipatio at shock frot upstream (1) dowstream (2) For a ideal fluid, the shock is ifiitesimally thi ad the dissipatio process is ot captured i fluid equatios. ρ 1,v 1, P 1 ρ 2,v 2, P 2 shock frot x I reality, molecular viscosity starts to domiate oce the shock thickess reaches about molecular mea free path: Re~1 at the scale of shock thickess O the other had, most astrophysical shocks are collisioless, where particle mea free path >> characteristic scale of the shock. Need additioal dissipatio resultig from kietic effects. 51

52 Outlie Formulatio Fluid basics Coservatio laws Viscosity Liear waves Shocks ad discotiuities Example: Bodi accretio 52

53 Applicatio: Bodi Accretio Bodi (1952) Cosider a object of mass M adiabatically accretig spherical symmetrically from a uiform medium of gas (desity=ρ 0, temperature=t 0 ) i a steady state. Questio: what is the accretio rate? Mass coservatio: Adiabaticity: Ds Dt = D Dt l P =0 where the costat K ca be determied by T 0. Euler s equatio: (we recovered Beroulli theorem) 53

54 Applicatio: Bodi Accretio Bodi (1952) From the Euler equatio: ad usig:, We obtai: Note: c s is a fuctio of ρ, which is related to v through mass coservatio. This is a ODE ad ca be itegrated from ifiity except whe v=c s. This is called the soic poit. If it is achieved at radius r=r s, the the RHS must also vaish at this radius, hece r s = GM 2c s (r s ) 2 54

55 Possible types of solutios M v c s Outer boudary coditio: v 0 at ifiity => Types 1, 3 are viable. There are a family of solutios as oe varies Ṁ util some maximum value correspodig to Type 1 solutio. Ier boudary coditio: For a sufficietly compact object, expect supersoic accretio flow. => Choose the type 1 solutio with uique accretio rate. 55

56 Property of the solutio Bodi accretio rate: This may also be obtaied from order-of-magitude cosideratios: Characteristic radius where ambiet gas is strogly affected by the cetral object: r b ' GM c s (1) 2 (Bodi radius) Order-of-magitude accretio rate: Ṁ 4 r 2 b (1)c s (1) =4 G 2 M 2 (1) c 3 s(1) 56

57 Additioal remarks l l l l Exactly the same set of equatios apply to describe spherical wids (simply reversig v), which correspods to the Parker wid solutio (Parker, 1958). Boudary coditios play a importat role i determiig the type of solutios. Whe solvig steady-state problems, oe always ecouters critical poits oce flow speed reaches some characteristic wave speed. If we solve the time-depedet fluid equatios, (e.g., ruig hydro simulatios), oe geerally circumvets this mathematical difficulty ad the system ca relax to the desired steady-state solutio. 57

58 Summary Fluid: describes cotiuum medium o macroscopic scales. There are multiple forms of fluid equatios, with the coservatio form beig the most fudametal. Viscosity attempts to smear out velocity gradiets, ad is a importat source of dissipatio. A ideal fluid supports soud waves (compressible) ad etropy waves (related to cotact discotiuity). No-liear ature of fluid equatios leads to shocks ad discotiuities, with Rakie-Hugoiot jump coditios. Bodi accretio: spherical accretio flow that trasitios from subsoic to supersoic. 58

Fluid Physics 8.292J/12.330J % (1)

Fluid Physics 8.292J/12.330J % (1) Fluid Physics 89J/133J Problem Set 5 Solutios 1 Cosider the flow of a Euler fluid i the x directio give by for y > d U = U y 1 d for y d U + y 1 d for y < This flow does ot vary i x or i z Determie the