Derivation of the basic equations of fluid flows. No. Conservation of mass of a solute (applies to non-sinking particles at low concentration).

Transcription

1 Deriation of the basic eqations of flid flos. No article in the flid at this stage (net eek). Conseration of mass of the flid. Conseration of mass of a solte (alies to non-sinking articles at lo concentration). Conseration of momentm. Alication of these basic eqations to a trblent flid.

2 A fe concets before e get to the meat ensor (Stress), Vectors (e.g. osition, elocit) and scalars (e.g., S, CO ). We need to define a coordinate sstem, and an (infinitesimal) element of olme. We assme a continos flid, and that all the fields of interest are differentiable.

3 he Lagrangian frameork is the frameork in hich the las of classical mechanics are often stated. he coordinates of a oint (t) describe the trajector from ( t ). Densit,, can eole along the trajector. B the chain rle, along a arcel trajector: d d dt dt t d( ( t), t) dt D Dt tt t const const t t (, t) Conersion from Lagrangian to Elerian

4 Eamle: Let s assme that e are in a rier that feeds on glacial melt. he ater arms at a constant rate that is a fnction of distance from the sorce. If e drift don rier (A la Hckleberr Fin ), the temeratre increases ith time (D/Dt>). At one oint along the rier, hoeer, e ma see no change in temeratre ith time (/t), as the ater arriing there is alas at the same temeratre. he heat fl is adectie, (/>). In short, the conectie deriatie is: D Dt t t

5 Mass conseration (Elerian, differential aroach): Acconting for the change in mass inside a fied, constant- sie olme: (-Δ/,Δ/,Δ/) (Δ/,Δ/,Δ/) (-Δ/,-Δ/,Δ/) A ΔΔ AΔΔ AΔΔ AΔΔ Δ Δ AΔΔ (Δ/,Δ/,-Δ/) MassV A ΔΔ ( V ) ( ) ( ) A A A ( ) t 1 t Δ t (-Δ/,-Δ/,-Δ/) (Δ/,-Δ/,-Δ/) / / / / Δ Δ Δ Δ Δ / Δ / 1 ( ) ( ) 1 ( ) Δ Δ Δ / Δ / Δ / Δ / Δ / t ( ) ( ) ( ) ( ) Δ /

6 Mass conseration (Elerian, integral aroach): Acconting for the change in mass inside a fied, constant-olme olme (V ): d dt V dv t V t dv V V nds dv Where e sed the diergence theorem: It states that the olme total of all sinks and sorces, the olme integral of the diergence, is eqal to the net flo across the olmes bondar (WIKI).

7 Reiteration (no sinks/sorces): Mass conseration (Lagrangian, integral): D Dt dv V Mass conseration (Elerian, integral): d dt V dv nds V

8 Mass conseration: Note that: Can be ritten as: tt r ( ) 1D r DtD he nd term is the flid diergence (rate of otflo of olme er nit olme). his can be nonero onl for comressible flids. It is the rate of loss of densit de to comression/eansion. r For both ater and air e can assme that t in terms of their dnamics (e need comressibilit to ass sond ).

9 Mass balance for consered scalar: Adding moleclar diffsion: C dv t V r ( C K C ) S r nds Where V is the olme of the control olme and S its srface, and sing Fick s la. B the hel of the diergence theorem: V C C t r dv ( C K C) Since the olme is arbitrar, this can be tre if and onl if: C C t r ( C) ( K C)

10 Momentm balance (Naier-Stokes): Neton s nd la of motion states that the time rate of change of momentm of a article is eqal to the force acting on it. his la is Lagrangian, the time rate of change is ith resect to a reference sstem folloing the article. d dt dv gdv V ( t ) V ( t ) V ( t ) ds Where g is the bod force er nit mass (e.g. grait) and is the srface force er nit srface area bonding V. If the olme is small enogh, the integrands can be taken ot of the integral: d dt d dt d d dv dv dt dt V ( t) V ( t) d d( δ V ) δv δv δv dt dt ( ) ( δ V ) ( ) d d dt

11 Momentm balance (Naier-Stokes): he bod force is similarl il l treated: t gdv V ( t) gδv Defining a stress tensor (eanded on the net slide): Τ n And aling the diergence theorem: ds V ( t) V ( t) Τ dv ΤδV D Dt g Τ

12 Srface forcing: For an iniscid flid, the srface force eerted b the srronding flid is normal to the srface, i.e. n, and is called the ressre force. In general, iscos stress force, S, is also resent. For iscos flids: n S. B definition Τ n, and e no hae I, here S n and I is the identit tensor. For Netonian flids, Σ And the resltant Naier-Stokes eqations for incomressible flids are: D Dt g

13 Rotational smmetr: 1 ( ) 3 1

14 otal stress tensor, Netonian flid: j i δ ij j i i j ij δ i, j{1,, 3}{,, } Stokes, 1845: 1 Σ linear fnction of elocit gradients,j {,, } {,, } 1. Σ ij linear fnction of elocit gradients.. Σ ij shold anish if there is no deformation of flid elements. 3. Relationshi beteen stress and shear shold be isotroic. ( ) i ˆ ( )

15 Naier-Stokes eqations:, g Dt D 1 t 1 t 1 g t Coriolis is added hen moing the frameork to an accelerating frameork. Hae to add bondar & initial conditions.

16 Naier-Stokes eqations (Bossinessq aroimation): Searate balance of flid at rest from moing flid. ( ) (,,, ) t (,,, t) First order balance (hdrostatic): 1 g nd order balance: D g, Dt

17 Eamle: stead flo nder grait don an inclined lane. g α From: Acheson, ν g sinα BCs: 1 : cos α g h :, a

18 Eamle: stead flo nder grait don an inclined lane. g α Soltion: a g g ν ( h ) cosα ( h ) sinα Q: hat ν shold e se?

19 Renolds decomosition of the N-S eqations Assme a trblent flo. At an gien oint in sace e searate the mean flo (mean can be in time, sace, or ensemble) and deiation from the mean sch that:,,, ensemble) and deiation from the mean sch that:,, etc Sbstitting into the continit eqation (linear): ;

20 1 Sbstitting into -momentm Naier-Stokes eqation: t 1 ( ) ( ) ( ) he eoltion of the mean is forced b correlations of flctating roerties. he correlation terms time densit are the Renolds stresses. Sbstitting into a scalar conseration eqation: Sbstitting into a scalar conseration eqation C C C K C C C t C ( ) ( ) ( ) C C C t

21 Note that the Renolds stress tensor is smmetric (as is the iscos stress tensor): the iscos stress tensor): he closre roblem: to deelo eqations for the eoltion of the Renolds stresses theseles, higher order correlation are needed (e.g. ) and so on. For this reason theories hae been deised to describe ij in terms of th fl the mean flo. For more, see: htt://.cfdonline.com/wiki/introdction to trblence/renolds aeraged eqations onl ne.com/w k /Introdct on_to_trblence/renolds_aeraged_eqat ons

22 One soltion to the closre roblem is to link the Renolds stress to mean-flo Qantities. For eamle: K K edd U edd his te of formlation is aealing becase it: a. Proide for don-gradient fl. b. Is reminiscent of moleclar l diffsion and iscosit. c. Proide closre to the eqations of the the mean fields. his te of formlation is roblematic becase: a. K edd is a roert of the flo and not the flid. b. K edd is likel to ar ith orientation, nlike moleclar rocesses. Ho is K edd related to the trblence?

23 Assme a gradient in a mean roert (momentm, heat, solte, etc. Remember: no mean gradient no fl). Assme a flctating elocit field: l is the distance a arcel traels before it loses its identit. he rate of ard ertical trblent transfer of <Ψ> is don the mean gradient: ψ l ψ l ψ K edd ψ

24 Ho is K edd related to the trblence? U Kedd γ ennekes and Lmle (197) aroach this roblem from dimensional i analsis based on assming a single length scale-l and a single elocit scale ω< > 1/. c ω ; c ~ O(1) he eddies inoled in momentm transfer hae orticities, ω/l; this orticit is maintained b the mean shear (l is the length scales of the eddies, e.g. the decorrelation scale). U U ω / l c ; c ~ O(1)

25 It follos that: Ke dd ~ l ω ~ l U In analog ith momentm fl, for heat e hae: c c γ It is most commonl assmed, and erified that γ K edd.

26 Edd-diffsion: ersectie from a de atch (figres from lectre notes of Bill Yong, UCSD) De atch << dominant scale of eddies. Dashed circle denotes initial osition of atch.

27 De atch dominant scale of eddies De atch >> dominant scale of eddies

28 Cheat sheet: 1 G di f l ( ) k j i φ φ φ φ ˆ ˆ ˆ 1. Gradient of a scalar (a ector): k j i φ. Diergence of a ector (a scalar): 3 Diergence of a ensor : j i ˆ ˆ 3. Diergence of a ensor, 1 (face)(direction) : k j ˆ k

29 Cheat sheet (cotined): 4 L l i f ( ) 4. Lalacian of a ector (a ector): ( ) î ( ) ˆ ˆ k j k j