Basic Equations of Fluid Dynamics

Transcription

1 Basc Equatons of Flud Dynamcs Sergey Pankratov A lecture to the Practcal Course Scentfc Computng and Vsualzaton (June 17, 2004) Sergey Pankratov, TU München 1

2 Focus areas Euler and Lagrange descrptons of flud flows The contnuty equaton (CE) The Naver-Stokes (N.-S.) equaton 2

3 Part1: Knematcs of flud flows 3

4 Mathematcal descrpton of a flow The materal flud s consdered to be: - a contnuous substance havng a postve volume (V > 0) - dstrbuted over a doman Ω of the d = 3 Eucldean space A flud flow: s represented by a one-parameter famly of mappngs of a 3d doman Ω, flled wth flud, nto tself, Ω t Ω t+τ Propertes of the mappngs: - smooth: twce contnuously dfferentable n all varables, 2 ; usually mappngs are assumed thrce contnuously dfferentable everywhere, wth the excepton of some sngular ponts, curves or surfaces where a specal analyss s requred; - bjectve (one-to-one) Geometrcally: a dffeomorphsm (nvertble, dfferentable), Ω t Ω t+τ 4

5 The flud flow transform The real parameter t desgnatng the mappng of flud domans s dentfed wth the tme, - < t < + Analytcal and numercal descrpton of the flud moton: 1) To ntroduce a fxed rectangular coordnate system {x } 2) A flud partcle an nfntesmal element of a flud dstrbuton 3) Each coordnate trple {x } denotes a poston of a flud partcle Wthout loss of generalty, t = 0 an arbtrary ntal nstant Observaton: the partcle that was ntally n the poston {ξ j } has moved to poston {x } The flud flow transformaton: x = x (ξ j,t) (*) If ξ j s fxed whle t vares, (*) specfes the path of a flud partcle (a pathlne or a partcle lne): x = x (t) = x (ξ j,t) j ξ =const, x (0) = ξ 5

6 Flud flow a mappng A flud partcle movng wth the flud 3 x M () t { x, x, x } Mt ( + τ ) M () t = gtm gt : { ξ, ξ, ξ } M Transformatons g t make up a one-parameter group of dffeomorphsms, g t+τ =g t +g τ 2 x 1 x 6

7 Flud flow s nvertble If t s fxed, the transformaton of the flud doman, Ω 0 Ω t : j j x ( ξ ) = x ( ξ, t), x (0) = ξ s exactly the flud flow t= const x (0) = ξ x ( ξ j, t) Functons x (ξ j,t) are sngle-valued (1-to-1) and are assumed to be thrce dfferentable n all varables. Then ( x, x, x ) x J = = det j ( ξ, ξ, ξ ) ξ Hence, there exsts a set of nverse functons: ξ j = ξ j ( x, t) (**) defnng the ntal poston of a flud partcle, whch occupes any poston {x } at tme t. Such nverse transform has the same propertes as the drect one 7

8 Two equvalent descrptons An mportant consequence of nvertblty: A flud flow can be equvalently descrbed by a set of ntal postons of partcles as a functon of ther postons at later tmes Thus, there are two equvalent descrptons of the flud moton: - The Euleran descrpton n terms of space-tme varables (x,t) Throughout all tme, a gven set of coordnates {x } remans attached to a fxed poston. The spatal coordnates {x } are called the Euler coordnates; they serve as an dentfcaton for a fxed pont - The Lagrangan descrpton chroncles the hstory of each flud partcle unquely dentfed by varables {ξ } Throughout all tme, a gven set of coordnates {ξ } remans attached to an ndvdual partcle. They are called the Lagrangan coordnates 8

9 Physcal nterpretaton of the Euler and Lagrange descrptons Dfferent physcal nterpretatons In the Euler s pcture, an observer s always located at a gven poston {x } at a tme t observes flud partcles passng by In the Lagrange pcture, an observer s movng wth the flud partcle, whch was ntally at the poston {ξ } vews changes n the flow co-movng wth the observer s partcle Euler Lagrange The Euler descrpton mportant for CFD; The Lagrange descrpton mportant for ecologcal modelng (passve tracers, contamnaton spread, etc.) 9

10 What s a flud partcle? Flud dynamcs the macroscopc dscplne studyng the contnuous meda The meanng of the word contnuous: any small flud element a flud partcle s stll bg enough to contan N >>1 molecules Dmensons of a flud partcle: a << δx << L a a typcal ntermolecular dstance δ x a flud partcle dameter L a characterstcal system dmenson L s an external parameter, e.g. the ppe dameter, the wng length,... a ~ N -1/3, where N s the total number of molecules n the system In gases, one more parameter s mportant: the free path length 10

11 Feld descrpton of flud flows The flud moton s completely determned ether by transform (*), Slde 5 the Euler pcture, or by ts nverse (**), Slde 7 the Lagrange pcture. The crucal questons: How to: - descrbe the state of moton at a gven poston n the course of tme? - characterze the flud moton n terms of some mathematcal objects defned at {x,t}? The smplest object for ths role s the nstant flud velocty u u-functons can be defned n both descrptons, u (x j,t) and u (ξ j,t) A hstorcal remark: though the descrptons of flud flows n terms of spatal and materal coordnates are attrbuted, respectvely, to Euler and Lagrange, both of them are, n fact, known to be due to Euler (see J.Serrn n Handbuch der Physk, Band 8/1, Sprnger1959) 11

12 Objects for flud felds Consder a functon f characterzng a flud n the Euler or Lagrange frameworks. Any such quantty (e.g. the temperature) may be vewed as a functon of spatal varables, f = f(x,t) and also of materal varables, f = f (ξ,t).e. f = f( x ( ξ j,),) t t or f = f( ξ j ( x,),) t t Geometrcal meanng: - f = f(x,t) s the value of f experenced by the flud partcles nstantaneously located at {x } - f = f (ξ,t) s the value of f felt at tme t by the flud partcle ntally (at t = 0) located at {ξ } Queston: why do we need u and other objects, f we have (*) and (**)? Answer: n most stuatons the actual moton of flud partcles s not gven explctly.e. nether x =x (ξ j,t) nor ξ j =ξ j (x,t) s known 12

13 Change of varables The varaton of quantty f when the flud flows two dfferent tme dervatves: 1) f f( x, t) = t t x = const smply a partal dervatve - the rate of change of f wth regard to a fxed poston {x } 2) df f ( ξ, t) materal or convectve dervatve the = dt t rate of change of f wth regard to a ξ = const movng flud partcle For f = x, by defnton: j j dx x ( ξ, t) u ( ξ, t) = = dt t ξ = const - the velocty of a flud partcle 13

14 The velocty feld Usually t s the flud velocty u (x j,t) that s a measurable quantty. The velocty dentfes a flud partcle (n terms of ts ntal poston {ξ }) An observer s sttng n a poston {x } to measure the speed of movng partcles (ntally were at {ξ }) Polze The flud velocty s a vector feld (transformaton propertes!). Throughout all tme, a set of velocty components {u (x j,t)} remans attached to a fxed poston the feld varables In numercs or n experment: a flud velocty u (1d, 2d, 3d, 4d) for the tme nterval t to determne x x (ξ j,t), x u t 14

15 The velocty feld (contnued) Analytcally: dx j = u ( x, t), x (0) = ξ (***) dt - a dynamcal system (the Cauchy problem) A unque soluton exsts on U T for u 1 (Cauchy-Lpschtz- Peano), the soluton depends contnuously on ntal values {ξ } It means that the flud moton s characterzed completely,.e. the descrpton n terms of flud velocty s equvalent to the Euler or Lagrange pctures Thus: the state of a flud n moton s fully represented by the velocty feld {u (x j,t)} of a flud partcle at {x j,t} Fundamental mportance of the feld descrpton: provdng the possblty to formulate the flud dynamcs n terms of PDEs. 15

16 Other objects How can one calculate the materal dervatve of any quantty f(x,t)? j j df f ( ξ ) f ( ξ ( x,),) t t f ( x ( ξ,),) t t = = = = dt t t t ξ = const ξ = const ξ = const j f f x ( ξ, t) = + or df j t ξ = const x t f j f = + u ξ = const j dt t x 1. Ths formula nterconnects the materal and spatal dervatves through the velocty feld 2. Ths formula expresses the change rate of any local parameter f(x,t) wth respect to a movng flud partcle located at {x,t} Example: the materal dervatve flud acceleraton a (x,t) du u j u j j Notaton: a = = + u dt t x j Du : = u + u u = u, + u u, t t j t j 16

17 A remark on acceleraton In classcal mechancs constructed by Newton, the acceleraton plays the central role n dstncton to the Arstotelan pcture In flud dynamcs, although t s just a branch of mechancs, acceleraton s much less mportant The reason for t: n most physcal and engneerng settngs acceleraton of a flud partcle a s unquely determned by ts coordnate x and velocty u Therefore, acceleraton s usually not consdered a state parameter So, we mght formulate the followng feld descrpton framework: The flud moton can be characterzed n terms of objects local parameters of the meda - defned at {x,t}. The smplest object s the flud velocty u 17

18 Streamlnes and trajectores A streamlne - a curve defned for a gven t > 0, for whch the tangent s collnear to the velocty vector u (x j,t): dx dx dx = = u ( x, t) u ( x, t) u ( x, t) Streamlnes depend on tme t (parameter), for whch they are wrtten Trajectory of a flud partcle {ξ j,t} (path lne, partcle lne) a locus of the partcle postons for all moments t > 0 For a statonary flow, u (x j,t) = u (x j ), streamlnes do not depend on t and concde wth partcle lnes (trajectores) The nverse s not true: the flow may be non-statonary, however streamlnes can concde wth path lnes, for example: Qt () x u =, R: = ( x x ) 3 4π R 1/2 18

19 Compressble and ncompressble fluds Incompressble: when the flow does not result n the change of the flud volume for all {x,t} dj 0 dt = Otherwse - compressble: A remnder: the flud moton s a dffeomorphsm,.e. contnuously dfferentable and nvertble Ths s suffcent for 0 < J < dj dt J = dv/dv 0 the flud dlataton j j How to calculate dj/dt? dj ξ d x ξ u = J = J = j k j j ξ u x dt x dt ξ x ξ J = J k j u = Jdvu x x ξ 0 J s the Jacoban of the transform (*), dv = JdV 0 dv = dx 1 dx 2 dx 3 - the partcle volume at {x } dv 0 = dξ 1 dξ 2 dξ 3 - the ntal partcle volume 19

20 Two fundamental statements What s the rate of change: (I) of an nfntesmal volume of a flud partcle? (II) Of any local quantty f(x,t) ntegrated over an arbtrary closed flud doman (volume) Ω(t) movng wth the flud? Answer to (I): the Euler expanson theorem d ln J = dvu dt Answer to (II): the Reynolds transport theorem A volume ntegral: df() t Ft () = f( x,) tdv, =? dt Ω() t The dea: to transform F(t) nto an ntegral over the ntal volume Ω(0), whch s fxed F d d dj = f( x ( ξ,),) t t JdV f( x ( ξ,),) t t J f( x ( ξ,),) t t dv dt dt = + dt dt j j j 0 Ω(0) Ω(0) 0 20

21 The Reynolds transport theorem Usng the Euler expanson theorem, we get k df() t df dj df u = J f dv0 f dv k dt + = dt dt + dt x Ω(0) Ω( t) Another form of the Reynolds theorem: usng the nterconnecton between the materal (Lagrange) and feld (Euler) dervatves (see Slde 16), we obtan d dt Ω () t Ω () t ( t ) f( x, t) dv = f + ( fx ) dv There exsts one more form useful for numercal methods (e.g. for so called mmetc schemes): d k f( x, t) dv Here 2-surface element t fdv fu dsk dt Ω = () t Ω + () t Ω () t s a vector normal to Ω(t) enclosng Ω(t) 21

22 Physcal nterpretaton of the Reynolds theorem The total rate of change of any quantty f (ntegrated over the movng volume) = the rate of change of f tself + the net flow of f over the surface enclosng the volume A movng (control or materal) volume Ω(t) s a volume of flud that moves wth the flow and permanently conssts of the same materal partcles Specfc case: f = const what s the flud volume rate of change? dv ( Ω, t) k V( Ω, t): = dv; = u dv = u ds k k dt Ω() t Ω() t Ω() t k 22

23 Geometry of flud flows A flud partcle movng along ts pathlne changes ts volume, shape, and orentaton (e.g. rotates) these are geometrc transforms of a flud partcle (wth respect to ts ntal poston) From the geometrcal vewpont, such changes can be descrbed as the tme evoluton of the flow governed by the geodesc equaton on the group of volume-preservng dffeomorphsms. So, flud moves along a geodesc curve A dynamcal system (***), Slde 15, correspondng to flud moton, can be vsualzed as a vector feld n the phase space, on whch the soluton s an ntegral curve Geometry enables us to descrbe global propertes of the famly of soluton curves fllng up the entre phase space The geometrc approach (orgnally suggested by V.I. Arnold) s a natural framework to descrbe the flud moton 23

24 Stran rate and vortcty The smplest geometrcal object descrbng spatal varatons of a flud partcle s the tensor τ j (x k, t) := u / x j constructed from the flud velocty u = δ j u j In general, τ j has no symmetry, however t can be decomposed nto a symmetrc and antsymmetrc parts: τ j = θj + ζj, θj = θ j, ζj = ζ j Here the stran rate θ and vortcty ζ are defned as 1 u uj 1 u u j θj : = +, ζj : j = j 2 x x 2 x x Frequently the vortcty vector s defned as 1 jk jk ζ = ε ζ s the unt antsymmetrc pseudotensor of rank 3 jk, ε 2 These defntons are used when treatng the Naver-Stokes equaton 24

25 The consttuton of flud knematcs 25 The law of flud moton x = x (ξ j,t) The flud velocty u (ξ j,t) = tx (ξ j,t) IVP for flud moton dx /dt = u (x j,t), x (0)= ξ The Euler expanson J -1 dj/dt = dvu The Reynolds transport theorem d/dt Ω(t) fdv = Ω(t) (df/dt+fdvu)dv The contnuty equaton ρ/ t + dv ρu = 0 Transton to physcs ρ(x,t)

26 Control questons on flud knematcs 1. Can one fnd the law of moton of a contnuous medum, f partcle lnes are known? 2. Can one fnd the velocty feld, f streamlnes are gven? 3. Can the partcles move wth acceleraton, f a) veloctes of all partcles are equal? b) the velocty does not change wth tme n each pont x Ω? 4. The densty of each ndvdual partcle of an ncompressble medum s constant. Can the densty vary wth tme n some pont x Ω? 26

27 Answers to control questons on flud knematcs 1. Can one fnd the law of moton of a contnuous medum, f partcle lnes are known? No, snce although a curve (trajectory), along whch a partcle moves, s known for each partcle, but the velocty of the moton may be dfferent (s not defned) 2. Can one fnd the velocty feld, f streamlnes are gven? No, n each pont the straght lne, along whch the velocty s drected s known, but the velocty value may be dfferent 3. Can the partcles move wth acceleraton, f: a) veloctes of all partcles are equal? Yes b) the velocty does not change wth tme n each pont x Ω? Yes 4. The densty of each ndvdual partcle of an ncompressble medum s constant. Can the densty vary wth tme n some pont x Ω? Yes 27

28 Flud knematcs summng t up All partcles are ndvdualzed (classcal mechancs!) Lagrange coordnates {ξ } are partcle dentfers Moton of contnuous meda and accompanyng processes are descrbed by physcal felds (velocty, pressure, temperature, etc.). In case these felds are consdered as functons of ξ, such descrpton s called Lagrangan or materal Events n the Lagrangan pcture occur wth ndvdual partcles The law of moton of a contnuous medum: x = x (ξ j,t) Velocty of partcles: u (ξ, t) = t x (ξ, t) the velocty feld,.e. the velocty of partcles located n {x } at tme t Acceleraton of partcles: a (ξ, t) = t u (ξ, t) Felds consdered as functons of {x, t} the Euler descrpton 28

29 Knematcs summng up (cont d) Tme dervatve: df f f ( = + u = ) t + u f dt t x Transton from the Lagrangan to the Euleran descrpton: ( ) ( ) ( ) Transton from the Euleran to the Lagrangean descrpton - to solve the Cauchy problem (***), Slde 15: The soluton (f t could be found): ( ) ( ) x = x ξ j, t ξ = ξ x j, t f ξ x j, t, t = ϕ x j, t dx dt (, ), = 0 Then for any f(x, t) whose Euler s descrpton s known: x j = u x t x = ξ ( j ξ, t) ( ( ξ j, ), ) = φ( ξ j, ) f x t t t t 29

30 Part2: Dynamcs of flud flows 30

31 Transton to flud dynamcs No physcs so far - no nformaton about the nature or structure of a flud was requred How to fnd the equatons governng the flud behavor? 1) Mathematcal axoms and theorems (mostly knematcs) 2) Physcal hypotheses (structure, nteracton law, etc.) 3) Expermental evdence 4) Mathematcal and computatonal models Transton to dynamcs: takng nerta nto account (recall the Newton s law!) Flud specfcty: lqud mass s contaned wthn the doman Ω(t): Mt () = ρdv> 0 for any doman Ω(t). Ω() t Hence ( ρ x, t) > 0, { x, t} Ω( t) 31

32 The contnuty equaton (CE) Extremely mportant! The underlyng physcal prncple conservaton of mass (strctly speakng, not a physcal conservaton law not assocated wth any symmetry, a phenomenologcal prncple) Leads to a very profound queston what s a mass? dm () t d (n fact along the flud partcle paths) = dv 0 dt dt ρ = Ω() t From here the contnuty equaton: dρ u + ρ = 0 dt x - a necessary and suffcent condton for the mass of any flud doman to be conserved Proof s based on the Reynolds transport theorem (see next slde) 32

33 The contnuty equaton - proof Suffcency: to substtute the contnuty equaton nto the Reynolds theorem dm () t d ρ u = ρ dv dt + dt x Ω() t dm () t Necessty: dt for any arbtrary Ω(t), hence the ntegrand must be zero. A more customary form: The contnuty equaton, despte ts smplcty, brngs about many nontrval results, e.g.: - How to fnd the rate of change of a quantty f(x,t) averaged wth the medum densty dstrbuton? = 0 ρ+ ( ρ ) = 0 t u Qt (): = Ω() t ρ fdv 33

34 Some consequences of the contnuty equaton The rate of change of the weghted functon f: We have d Qt () = ρ( x,) t f( x,) tdv=? dt Ω() t k d d( ρ f) u ρ fdv = + ρ f dv k = dt dt x Ω() t Ω() t df d ρ = ρ + f + ρ fdvu dv = dt dt Ω() t df d ρ df = ρ + f + ρdvu dv dv dt dt = ρ dt Ω() t Ω() t The result a very smple formula: dq Due to CE, the dfferentaton operator = ρ dt dt Ω() t gnores the densty dstrbuton. df dv 34

35 Some more dynamcs Note: although the medum densty s not a knematc object, the contnuty equaton may be derved from knematcs Gven a flud densty dstrbuton, one can determne a velocty feld u (x j,t) for the flud moton, whch would conserve the mass n each doman Ω(t) In such formulaton, the contnuty equaton s the frstorder PDE wth respect to u (x j,t) not suffcent to fnd 3 unknown functons Flud dynamcs cannot be descrbed by the contnuty equaton alone one needs more physcs 35

36 Forces n flud meda Physcal forces, both nternal and external, are actng on flud partcles to set them n moton The Cauchy model: two types of forces nternal contact forces, act on a flud partcle over ts surface (e.g. stress, pressure) external body forces, act over a medum (e.g. gravty, electromagnetc feld) External forces descrbed per unt mass by a vector F (x j,t,u k ) dependng on poston, tme, and state of flud moton (e.g. magnetc feld, relatvsm, etc.) Cauchy: nternal forces are more nterestng - for any closed surface Ω(t), there exsts a stress vector dstrbuton s (x j,t,n k ) dependng on the normal n k 36

37 The Cauchy stress The dea: the stress acton exercsed by the stress vector s (x j,t,n k ) s equvalent to the acton of nternal forces P P The Cauchy stress s = lm σ 0 σ vector s σ n The model: s s a lnear functon of the normal vector n (x j,t): j j So, the stress vector for a surface element wth the normal n can be calculated as s n = sn The coeffcents T j - the flud stress tensor In general, no symmetry s = Tn 37

38 The Cauchy equaton of moton An adaptaton of the Newton s law to flud flows: acceleraton of a flud partcle du ρ dt T x = ρf + external force densty j j nternal force densty A system of 1st-order PDE wth respect to flud velocty components Includes the densty dstrbuton satsfyng the contnuty equaton External forces F (x k,t) and the flud stress tensor T j (x k,t) are consdered to be known, together wth ntal and boundary condtons. In ths case the soluton can be obtaned 38

39 The general system of equatons n flud dynamcs If the external forces and the stress tensor components are known, ntal and boundary condtons are gven, then the flud velocty u (x j,t) can be obtaned from the followng system: 1 tu + u ju = F + jt ρ j tρ + j( ρu ) = 0 j j - four frst-order PDEs over four unknown functons Once the soluton s obtaned, the law of moton x = x (ξ j,t) can be found from u by solvng the system of frst-order ODEs (***): dx j = u ( x, t), x (0) = ξ dt 39

40 The total acceleraton Thus, n prncple, the problem of flud moton can be solved (theoretcally, n practce ths s rather dffcult) A reduced form of flud moton equaton analogous to the D Alembert prncple n classcal mechancs: j du - the total acceleraton, a combned A ( x, t): = F dt response to nertal and external forces j Physcal nterpretaton: total acceleraton s ρ A = jt determned by nternal forces (actng on a flud partcle) One needs physcal assumptons regardng the nature of nternal forces (actng across flud surface elements) Such assumptons would specfy the T j tensor components 40

41 Specfc forms of moton equatons No a pror propertes of T j (x k,t): no algebrac symmetry (ndces), k k l no geometrc symmetry ( x = R x ), no physcal symmetry (t -t) An example of a physcal assumpton the Boltzmann postulate: j j T = T l an mmedate consequence: conservaton of angular momentum Holds not for all fluds: e.g. for polar fluds the asymmetry of T j may be essental Now, we restrct ourselves to fluds wthout heat transfer: the temperature T = const, the heat flux q (x,t) = 0 Tree man flud equatons: 1. The Euler equaton 2. The Stokes equaton; 3. The Naver-Stokes equaton 41

42 The Euler equaton The perfect flud, T j = -pδ j : j 1 j tu + u ju = F δ jp ρ p(x j,t) s the flud pressure depends on the thermodynamc state A control queston: can the pressure be < 0? Answer: yes expanson of a flud partcle (n fact nstablty) What s a perfect flud? ( Nobody s perfect! ) No dsspaton, no nternal frcton, no nternal heat transfer Heat conductvty and vscosty are neglected adabatc moton: ds s s u dt t x = + = Combnng wth the contnuty equaton: ( ρs) + ( ρsu ) = 0 t x 0 the entropy flux 42

43 Another form of the Euler equaton Introducng h enthalpy, dh = Tds + Vdp: Usng the formula: we can wrte or (applyng curl): t x x j u u h + u = ( u ) u= ( u ) [ u curlu] 2 u u [ u curlu] = ( h + ) t 2 t curlu= curl[ u curlu] (for ds = 0, (1/ρ) ρ/ x = h/ x ) - only the velocty s present Boundary condton: u n = 0 (fxed boundary) In general, 5 varables: u, ρ, p; ρ = ρ(p,t) the state equaton 43

44 The Stokes equaton The model: The vscostes: T = ( p+ α) δ + βθ + γδ θ θ j j j k jl kl α(x,t), β(x,t), γ(x,t) scalar functons, n general depend on the thermodynamcal state of the flud ( ) ρ u + u u = ρf δ ρ+ j j t j j ( j ) ( k jl βθ δ γθ θ ) + + j kl j - a very complex expresson. The Stokes equaton s used to model non-elastc fluds 44

45 The Naver-Stokes equaton The model: T = ( λθ p) δ + 2µθ j j j ( j ) j j ( ) 2 ( j tu + u ju = F j + j + j ) ρ ρ δ ρ δ λθ µθ Here λ(x,t), µ(x,t) are scalar functons - the frst and the second vscostes. For λ = const and µ = const: u u 1 p λ + µ u u t x ρ x ρ x x x x 2 k 2 j j j kl + u = F δ + δ + µδ j j j k k l For ρ = const (ncompressble), dv u = 0. Then x u x - the man equaton of u p µ + ( u ) u= F + ν u ν= the classcal flud t ρ ρ dynamcs k k 45

46 Another form of the Naver-Stokes equaton Smlarly to the Euler equaton an alternatve form Incompressble flud dv u = 0 the velocty feld s rotatonal or transversal, u = u. Applyng curl (or rot) to the N.-S. equaton: u curlu= curl[ u curlu] + ν curlu+ curlf t u curlu - helcty or curlu+ ( u ) curlu ( curlu ) u= ν curlu+ curlf t Ths s a closed equaton wth respect to u no pressure nvolved On the contrary, equaton for the pressure: 2 ( j uu ) p = ρ j or, transformng, x x j u u p ρ = =, := j x x ρτ τ τ u j j j j 46

47 Dervaton of the pressure equaton Applyng dv to the Naver-Stokes equaton: u j p F + u u u j = + ν j + t x x x ρ x x x j x Assumng dv F =0 (no charges). The second term n LHS: u u u u u u u u u x x x x x x x x x x x x x 2 2 u u 0 ν j = ν = j x x x j x xj x j j 2 j j j j j u u = = + u = + u = j j j j j j The second term n the RHS: Thus: j u u p ρ = =, := j x x ρτ τ τ snce ( ) u j j j j uu u u u u x x x x x x x 2 j j j j = u u j + j j = j 2 or, n an equvalent form, 2 ( j uu ) p = ρ x x j 47

48 A closed system of equatons of classcal flud dynamcs A closed system of equatons conssts of: - The Naver-Stokes (N.-S.) equaton - The contnuty equaton - The thermodynamc state equaton ρ = ρ(p,t) 5 equatons for unknown functons: u (x j,t), p(x j,t), ρ(x j,t) In prncple, the system s closed, provded the external forces F (x j,t) and the vscosty coeffcents λ, µ are gven Intal and boundary condtons must be mposed to select an approprate soluton Although consderable dffcultes do exst n solvng the N.-S. equaton, t has been successfully appled to descrbed varous regmes of flud moton (ncludng turbulence to some extent) 48

49 Elementary model of the vscosty Many practcally mportant problems cannot be solved wthout vscosty beng taken nto account, e.g. - moton of bodes n fluds - flud flow through channels, tubes, etc. x - flows past an obstacle x du F ( y + dy) = µ dxdz y F dy y+ dy dy x du F ( y) = µ dxdz dy force appled to a unt volume: x F ( y+ dy) F ( y) u f = = µ 2 dxdydz y x x 2 x x y A 2d shear flow x To N.-S.equaton 49

50 Newtonan fluds When a sold s slghtly deformed so that t s straned, a restorng force opposng the stran s observed; for small stran t s proportonal to the stran (the Hooke s law) Fluds also resst strans, but for a flud t s not only the stran magntude that s mportant, but the stran rate as well Fluds respondng to a sheer stress ndependent of ts rate are known as Newtonan fluds. More generally, n a Newtonan flud the sheer stress s proportonal to the velocty gradent (see Slde 49),.e. components of the stress tensor are lnear functons of τ j = u / x j For Newtonan fluds, the vscosty s constant (does not depend on how fast the upper plate s sldng past the bottom) For non-newtonan fluds, vscosty depends on relatve velocty. Such fluds (studed n rheology) may have no defned vscosty 50

51 The physcal meanng of vscosty Vscosty s a macroscopc manfestaton of ntermolecular nteractons. Vscosty always leads to energy dsspaton All phenomena nvolvng vscosty are rreversble We can calculate the energy dsspaton rate n an ncompressble vscous flud no work s spent on compresson (or rarefacton) of flud partcles. The rate of change of knetc energy nsde the control volume Ω: Snce k 1, ρ k k k k k ρ x xk x x x x ρ x x 2 u u u u p u p u = u u u + = u + ρ we get 2 u k u p u E ( Ω ) = ρu dv = u ρ u + µ dv k k t x x x x Ω Ω k 2 k u p u u E ( Ω ) = dσ ρ uk + µ u µ dv k k 2 ρ x x Ω Ω 2 51

52 The physcal meanng of vscosty (cont d) The surface ntegral n E ( Ω) descrbes the energy change due to the flud nflow/outflow (the term ρu k u 2 /2) and wth the work produced by surface stresses The dsspated power s gven by the volume ntegral, whch s always postve 2 u k The vscous dsspaton rate per unt volume s Q = µ k = µτ τ x.e. proportonal to the dynamc vscosty The vscous coeffcent µ 0 (ths s the consequence of the second law of thermodynamcs) When can one neglect vscosty? The vscous force µ u ~ µu/l 2 must be << p ~ p/l ~ ρu 2 /L or µ << ρul (here L s the system s characterstc sze, see Slde 10) Ths crteron can be wrtten as 1/Re <<1, Re:= ρul/µ = ul/ν 52 k

53 The Reynolds number In 1883 Reynolds made the systematc study of of the water flow through a crcular tube and found that the flow pattern changes from a lamnar to a turbulent state at a certan value of ul/ν Later, ths quantty was called after hm the Reynolds number, Re Usng the Reynolds number, one can wrte the N.-S. equaton n dmensonless form: Du j 1 j = δ jp + Re ju + F Dt (D:= t + u j j, see Slde 16 n fact a covarant dervatve) Physcally, Re s a measure of the rato between nertal and vscous forces n the flow; for Re>>1 nerta prevals, for Re<<1 frcton (vscosty) prevals Both forces are balanced by the pressure gradent Smple steady flows at small values of Re vs. complcated nonstatonary flows at large values of Re 53

54 Flow past an obstacle The transton from a lamnar to a turbulent mode can be observed n the flow past an obstacle, the smplest obstacle beng a sphere Re 1 Re~10 Re~ Re~ Re >

55 Descrpton of flow patterns Re 1: The flow s statonary and passes regularly around the sphere (wthout separaton) 1 Re 10: The flow s stll statonary but separates at the back sde of the sphere. A rng vortex s formed, whch ncreases wth Re 10 Re 10 2 : The rng vortex deforms nto a helcal vortex, whch rotates about the flow axs. The flow becomes non-statonary, but perodc (pulsatons) 10 2 Re 10 5 : The helcal vortex breaks up and s replaced by a turbulent wake. The flow s totally aperodc n the wake Re > 10 5 : Not only the wake, but the entre flow (ncludng the boundary layer) becomes turbulent. The flow s completely random everywhere. The wake s thnner and the drag reduced 55

56 Mathematcal status of the N.-S. equaton No exstence theorem s known as yet for a smooth and physcally reasonable soluton to a 3d Naver-Stokes equaton Two major sources of mathematcal dffcultes: 1) a nonlnear (quaslnear) system of equaton due to the convecton (nertal) term u j j u Ths nonlnearty s of knematc (geometrcal) orgn, arsng from mathematcal dervaton (the chan rule), not from physcal consderatons. A unque case n physcs 2) vscosty terms ntroduce the second-order operator nto the frst-order equaton leads to sngular perturbatons The queston: would the solutons to the N.-S. equatons n general converge to the solutons of the Euler equatons n a doman Ω wth boundary Ω, as the vscosty tends to zero? 56

57 Physcal status of the N.-S. equaton A phenomenologcal macroscopc equaton to descrbe the dynamcs of fluds vewed as contnuous meda The flud flow at the mcroscopc level s treated n nonequlbrum statstcal mechancs and physcal knetcs Physcs of fluds can be reconstructed from the frst prncples only for very specal cases (e.g. dlute gases) The N.-S. equaton, the Euler equaton, etc. can be obtaned from the Boltzmann equaton (the moments of Boltzmann equaton) The Boltzmann equaton s also a specal case of a more general dsspatve knetc equaton, the latter beng n ts turn a specfc case of a more general equaton (the Louvlle equaton), etc. So, the Naver-Stokes equaton s accepted only as a reasonable mathematcal model for flud flows 57

58 Phenomenology of flud moton From the physcal vewpont, the flud n moton s a macroscopc non-equlbrum system When physcsts are studyng such systems, they apply two dstnct approaches: phenomenologcal and mcroscopc In the phenomenologcal method, the problem s reduced to establshng the relatonshp between macroscopc parameters, wthout referrng to atomc or molecular consderatons The concept of contnuous medum s an dealzaton. When the atomc structure s taken nto account, fluctuatons become essental. One can consder ther nfluence only wthn the framework of knetc theory. The meanng of stress tensor, vscosty, heat conductvty, etc., as well as the applcablty of the N.-S. equaton can be also elucdated only wthn the framework of knetc theory 58

59 Examples of usng the dstrbuton functon The knetc equaton s wth respect to dstrbuton (one-partcle) functon, PDF: f(r, v, t) = f(x,v,t) Defnton of flud dynamcs quanttes n terms of PDF: ( ) ( ) 3 ρ x, t = f x, v, t d v The Boltzmann knetc For an ncompressble flud: equaton serves as the ( ) 3, base for a so-called lattce f x v u, t d v= const Boltzmann numercal where approach ( ) 3 u = v f x, v, t d v the average flow velocty (the man quantty n the flud dynamcs) The flow temperature: m ( ) ( ),, 2 3 T = v u f x v u t d v 2 We assumed t to be constant, thus dsregardng the heat transfer 59

60 What else? Physcal flud dynamcs: Heat transfer Thermodynamcs of contnuous meda Surface phenomena and free boundares Dmensonal analyss, scalng, and self-smlarty Boundary layers Comressble fluds and gas dynamcs Turbulence Computatonal flud dynamcs (CFD) and numercal modelng 60