1 Equatio of Motio i Streamlie Coordiate Ai A. Soi, MIT 2.25 Advaced Fluid Mechaic Euler equatio expree the relatiohip betwee the velocity ad the preure field i ivicid flow. Writte i term of treamlie coordiate, thi equatio give iformatio about ot oly about the preure-velocity relatiohip alog a treamlie (Beroulli equatio), but alo about how thee quatitie are related a oe move i the directio travere to the treamlie. The travere relatiohip i ofte overlooked i textbook, but i every bit a importat for udertadig may importat flow pheomea, a good example beig how lift i geerated o wig. A treamlie i a lie draw at a give itat i time o that it taget i at every poit i the directio of the local fluid velocity (Fig. 1). Streamlie idicate local flow directio, ot peed, which uually varie alog a treamlie. I teady flow the treamlie patter remai fixed with time; i uteady flow the treamlie patter may chage from itat to itat. Fig. 1: Streamlie coordiate I what follow, we implify the expoitio by coiderig oly teady, ivicid flow with a coervative body force (of which gravity i a example). A coervative force per uit ma G i oe that may be expreed a the gradiet of a time-ivariat calar fuctio, G = U(r ), (1)
2 ad the teady-tate Euler equatio reduce to 1 V V = p U(r). (2) ρ A uiform gravitatioal force per uit ma g poitig i the egative z directio i repreeted by the potetial U = gz. (3) A treamlie coordiate ytem i ot choe arbitrarily, but follow from the velocity field (which, we ote, i ot kow à priori). Aociated uiquely with ay poit r ad time t i a flow field are (Fig. 2): the treamlie that pae through the poit (treamlie caot cro), the treamlie local radiu of curvature ad ceter of curvature, ad the followig triad of orthogoal uit vector: i : i the flow directio i : i the ormal directio, away from the local ceter of curvature i l : i the bi-ormal directio, ( i l = i i ). The uit vector defie icremetal ditace d meaured alog the treamlie i the flow directio, d meaured i the ormal directio, away from the ceter of curvature, ad dl meaured i the bi-ormal directio. The radiu of curvature i defied a poitive if i poit away from the ceter of curvature, ad egative if i poit toward it. The uit vector, the radiu of curvature, ad the ceter of curvature all chage from poit to poit ad i uteady flow from time to time, depedig o the velocity field. To traform Euler equatio ito treamlie coordiate, we ote that i thoe coordiate 1, ad = i + i + i (4) l V = i V (5) where V i the magitude of the velocity vector V. From (4) ad (5), V = V (6) 1 f f f The gradiet of a calar fuctio f (,,l ) i defied by f dr df (,, l ) = d + d + dl. Equatio (4) follow l from thi defiitio ad the expreio dr = i d + i d + i l dl for a icremetal diplacemet i treamlie coordiate.
3 ad thu V 2 i (V )V = V (Vi ) = i + V 2 2. (7) The uit vector i the lat term of (7) chage orietatio a oe move alog the treamlie. The chage di i i from to +d i obtaied with the cotructio how i Fig. 2 a d di = i dθ = i (8) Fig. 2: Icremetal chage i the treamwie uit vector from to +d. from which we ee that i = i (9) Uig (9) i (7), we obtai the covective acceleratio a ( V ) V = i V 2 2 V 2 i (10) The firt term o the right i the covective acceleratio i the directio of the velocity, ad the ecod i the cetripetal acceleratio, toward the ceter of curvature. The preure gradiet i treamlie coordiate i p p p p = i + i + i l (11) l
4 Uig (10) ad (11) i (2), we obtai the equatio of motio i treamlie coordiate for teady, ivicid flow a -directio: V 2 = 1 p 2 ρ U (12) -directio: V 2 = 1 p ρ U l-directio: 0 = 1 p ρ l U l (13) (14) I a uiform gravitatioal field U=gz ad thee equatio read -directio: 1 ( 2 ρv 2 ) = 1 p ρ g z (15) -directio: V 2 = 1 p ρ g z l-directio: 0 = 1 p ρ l g z l (16) (17) For cotat-deity flow i a uiform gravitatioal field, the equatio implify further to ρv 2 -directio: p + ρgz + = 0 (18) 2 -directio: ( p + ρgz) = ρv 2 (19) l-directio: ( p + ρgz) = 0 (20) l The -directio equatio (18) tate Beroulli theorem: the total preure the um p + ρgz + ρv 2 2 of the tatic, gravitatioal, ad dyamic preure remai ivariat alog a treamlie. The -directio equatio (19) tate that whe there i flow ad the treamlie curve, the um p + ρgz (which i cotat i whe the fluid i tatic) icreae i the - directio, that i, a oe move away from the local ceter of curvature.
5 The l-directio equatio (20) tate that p + ρgz remai cotat for mall tep i the biormal directio, that i, the preure ditributio i quai-hydrotatic ditributio i the l-directio. EXAMPLE Coider the imple cae of 2D, ivicid air flow over a mooth hill (Fig. 3). Far uptream of the hill the icidet velocity i uiform at V. The hill deflect the air aroud it, ad a uiform flow i agai etablihed far dowtream. Far uptream, above, ad dowtream of the hill, the preure i cotat at p ad the treamlie are traight (the hill doe ot perturb the flow at ifiity ). We hall aume that gravitatioal effect are egligible (the medium i air ad the hill elevatio i modet) ad the free tream Mach umber i mall, o that ad the deity ca be take a cotat. Baed o the available equatio, what ca we ay about the preure ad velocity ditributio over the hill where i the velocity higher tha V, for example, ad where lower? Fig. 3: Sketch of treamlie i a 2D flow over a hill. To awer thi quetio accurately we eed to kow the hape of the treamlie throughout the flow field or, at leat, i the regio that i perturbed by the hill. We do t have thi iformatio, o we proceed by drawig a rough etimate of the treamlie patter, a how i Fig. 3. The differece betwee the preure at ifiity ad at the top of the hill, poit (3), ca be etimated by itegratig equatio (19) alog the vertical path from (3) to ( ). Sice thi path follow the local -directio, >0 everywhere alog it. Neglectig the gravitatioal term, (19) give from which we ee that p ρv 2 = (21)
6 ρv 2 d p p 3 = > 0 (22) 3 Thu p 3 < p, ad accordig to Beroulli equatio (18), it follow that V 3 > V. Uig imilar argumet, we coclude that p 1 = p adv 1 = V, ad p 2 > p ad V 2 < V, etc. I priciple, if () ad V() ca be etablihed or etimated, the itegral i (21) ca be evaluated. For example if we fid that the flow perturbatio caued by the hill i egligible at elevatio greater tha ome multiple of the height of the hilltop, we might write for the path from (3) to ( ) H hill e (23) where hill i the treamlie radiu of curvature i the viciity of the hilltop, i meaured from the top of the hill upward, ad H = βh i ome multiple β of the actual height h of the hilltop, the coefficiet β beig a empirical umber. From Beroulli equatio (18) we alo have that ρv 2 ρv 2 p + = p +, (24) 2 2 Subtitutig for ad V ito (21) from (23) ad (24), repectively, we itegrate (23) ad obtai 2 2 H p p 3 = ρv e hill 1 (25) 2 For a low hill uch that 2H<< hill, the expoetial term ca be expaded ad (25) implified to ρv 2 H p p 3 (26) hill The velocity at poit (3) ow follow from (24) ad (25) a or, i the ame low-hill approximatio a (26), H V hill 3 = V e (27) V 3 V 1 + H hill (28)
MIT OpeCoureWare http://ocw.mit.edu 2.25 Advaced Fluid Mechaic Fall 2013 For iformatio about citig thee material or our Term of Ue, viit: http://ocw.mit.edu/term.