Module 6. Lecture 2: Navier-Stokes and Saint Venant equations

Transcription

1 Modle 6 Lectre : Naer-Stokes and Sant Venant eqatons

2 Modle 6 Naer-Stokes Eqatons Clade-Los Naer Sr George Gabrel Stokes St.Venant eqatons are dered from Naer-Stokes Eqatons for shallo ater flo condtons. The Naer-Stokes Eqatons are a general model hch can be sed to model ater flos n man applcatons. general flood ae for -D staton can be descrbed b the Sant-Venant eqatons.

3 Naer-Stokes Eqatons Contd It conssts of 4 nonlnear PDE of med hperbolc-parabolc tpe descrbng the fld hdrodnamcs n 3D. Epresson of Fma for a fld n a dfferental olme The acceleraton ector contans local and conecte acceleraton terms here :,, :,, j :,, Modle 6 ( ) ( ) ( ) t a t a t a j j t a

4 The force ector s broken nto a srface force and a bod force per nt olme. The bod force ector s de onl to grat hle the pressre forces and the scos shear stresses make p the srface forces(.e. per nt mass). Modle 6 (6.) (6.0) (6.9) p g f p g f p g f ρ ρ ρ Naer-Stokes Eqatons Contd

5 The stresses are related to fld element dsplacements b nokng the Stokes scost la for an ncompressble fld. ( ) ( ) ( ) (6.3) 6.,, µ µ µ µ µ µ Modle 6 Naer-Stokes Eqatons Contd

6 Sbstttng eqs nto eqs , e get, Modle 6 notaton Ensten p g f p g f p g f p g f j j (6.8) (6.7) (6.6) ν ρ ν ρ ν ρ ν ρ Naer-Stokes Eqatons Contd

7 The eqaton of contnt for an ncompressble fld The three N-S momentm eqatons can be rtten n compact form as Modle 6 (6.9) j j j j g p t ν ρ (6.0) 0 0 Naer-Stokes Eqatons Contd

8 Sant Venant Eqatons The Sant Venant Eqatons ere formlated n the 9th centr b to mathematcans, de Sant Venant and Bosnnesqe. Joseph Valentn Bossnesq The solton of the St. Venant eqatons s knon as dnamc rotng, hch s generall the standard to hch other methods are measred or compared. Contnt eqaton: Momentm eqaton: Q t Q Q g 0 t g( S o S f ) Jean Clade Sant-Venant Q-Dscharge throgh the channel -rea of cross-secton of flo - Depth of flo S 0 -Channel bottom slope S f - Frcton slope 0

9 Modle 6 ssmptons of St. Venant Eqatons Flo s one-dmensonal Hdrostatc pressre preals and ertcal acceleratons are neglgble Streamlne cratre s small. Bottom slope of the channel s small. Mannng s and Che s eqaton are sed to descrbe resstance effects The fld s ncompressble Channel bondares are consdered fed and therefore not ssceptble to eroson or deposton. D gradall ared nstead flo n an open channel s gen b St. Venant eqatons: Contnt Eqaton ( based on Conseraton of Mass) Momentm Eqaton ( based on Conseraton of Momentm)

10 Modle 6 -D Open channel flo In the dagrams gen, Q nflo to the control olme q lateral nflo Q Q Q d (ρd) t Rate of change of flo th dstance Otflo from the C.V. Change n mass Eleaton Ve Plan Ve

11 Modle 6 St. Venant eqatons Contnt eqaton: Q t 0 Q-Dscharge throgh the channel -rea of cross-secton of flo Conseraton of Mass In an control olme consstng of the fld (ater) nder consderaton, the net change of mass n the control olme de to nflo and otflo s eqal to the net rate of change of mass n the control olme

12 Contnt Eqaton-Deraton Q V olme ater dscharge [L 3 /T] ρq Mass ater dscharge ρv [M/T] /t(mass n control olme) Net mass nflo rate (assmng q0) ( ρ) ( V ) ρv ρv ρ t ( ρ) ( V ). e ρ 0 t V ρ 0; Here V t Q, dsch arg e throgh the cross secton t Q 0 Modle 6

13 In -D open channel flo contnt eqaton becomes, 0 ) ( t V Non-conseraton form (eloct s dependent arable) 0 q t Q Conseraton form Modle 6 0 t V V

14 Modle 6 Eample Problem Calclate the nlet eloct Vn from the dagram shon. 0 d dt CV d ( ρ dt tank ρd dh dt tank h) V 0.* 0.0 ρv n CS ρv n n V V n n ot ρv ot ot (0.005) ot g *(0.005) V n 4.47 m / s

15 Momentm In mechancs, as per Neton s nd La: Net force tme rate of change of momentm Change n momentm n the s drecton F ( m ) s s Sm of forces n the s drecton mass Veloct n the s drecton

16 Modle 6 Momentm Eqaton The change n momentm of a bod of ater n a flong channel s eqal to the resltant of all the eternal forces actng on that bod. d F Vρd dt c.. c. s. VρV. d Sm of forces on the C.V. Momentm stored thn the C.V Momentm flo across the C. S.

17 Conseraton of Momentm Ths la states that the rate of change of momentm n the control olme s eqal to the net forces actng on the control olme Snce the ater nder consderaton s mong, t s acted pon b eternal forces hch ll lead to the Neton s second la d F Vρd dt c.. c. s. VρV. d Sm of forces on the C.V. Momentm stored thn the C.V Momentm flo across the C. S. Q t Q g g( S o S f ) 0 Modle 6

18 Modle 6 pplcatons of dfferent forms of momentm eqaton Knematc ae: hen grat forces and frcton forces balance each other (steep slope channels th no back ater effects) Dffson ae: hen pressre forces are mportant n addton to grat and frctonal forces Dnamc ae: hen both nertal and pressre forces are mportant and backater effects are not neglgble (mld slope channels th donstream control)

19 ppromatons to the fll dnamc eqatons The three most common appromatons or smplfcatons are: Knematc Dffson Qas-stead models Knematc ae rotng: ssmes that the moton of the hdrograph along the channel s controlled b grat and frcton forces. Therefore, nform flo s assmed to take place. Then momentm eqaton becomes a ae eqaton: Q c t here Q s the dscharge, t the tme, the dstance along the channel, and c the celert of the ae (speed). Q 0 knematc ae traels donstream th speed c thot eperencng an attenaton or change n shape. Therefore, dffson s absent.

20 Modle 6 Dffson ae rotng The dffson ae appromaton ncldes the pressre dfferental term bt stll consders the nertal terms neglgble; ths constttes an mproement oer the knematc ae appromaton. S f S 0 The pressre dfferental term allos for dffson (attenaton) of the flood ae and the nclson of a donstream bondar condton hch can accont for backater effects. Ths s approprate for most natral, slo-rsng flood aes bt ma lead to problems for flash flood or dam break aes

21 Modle 6 Qas-Stead Dnamc Wae Rotng It ncorporates the conecte acceleraton term bt not the local acceleraton term, as ndcated belo: S f S 0 VV ( ) ( ) g In channel rotng calclatons, the conecte acceleraton term and local acceleraton term are opposte n sgn and ths tend to negate each other. If onl one term s sed, an error reslts hch s greater n magntde than the error created f both terms ere eclded (Brnner, 99). Therefore, the qas-stead appromaton s not sed n channel rotng.