Sring 018 ME 3560 Fluid Mechanic Chater III. Elementary Fluid Dynamic The Bernoulli Equation 1
Sring 018 3.1 Newton Second Law A fluid article can exerience acceleration or deceleration a it move from one location to another. Thi motion i ruled by Newton' econd law of motion: F ma Auming that the flow i invicid under a teady tate roce, the governing equation for a article of fluid immered in the flow will be deducted next. 3. F = ma along a Streamline Streamline are line that are tangent to the velocity vector everywhere in the flow.
dx u dy v dz w Sring 018 V d 0 ( uiˆ vj ˆ wkˆ) ( dxiˆ dyj ˆ dzkˆ) 0 ( vdz wdy)ˆ i ( wdx udz) ˆj ( udy vdx) kˆ 0 Conider an infiniteimally mall fluid article of ize δδnδy. Unit vector along and normal to the treamline are ŝ and ň. Steady flow, Newton'econdlaw along the treamline direction i F F ma V mv V V Differential Equation Rereenting Streamline in 3D.
The weight on the article i δw = δv, (= g). The comonent of the weight in the direction of the treamline i W W in V In general, for teady flow, =(, n). If the reure at the center of the article i. It average value on the two end face erendicular to the treamline are +δ and δ. F n y in V Sring 018 The 1 t term in a Taylor erie exanion for i: If δf i the net reure force on the article in direction: F ( ) n y ( ) n y n y
The net force acting in the treamline direction on the article i F W F in V Thu, from the equation obtained reviouly: And the reviou exreion: V in V a Thi equation can be rearranged and integrated a follow. Notice that in θ = dz/d and that VdV/d = ½ d(v )/d. On the treamline n i cont. (dn =0): d=( / )d+( / n)dn =( / ) d. Hence, along a given treamline: (, n)=() and / = d/d. F V Sring 018 V
Thu the equation: in V Can be reented a: dz d 1 d ( V ) 1 ; d d ( V d d d Sring 018 d V gz C (along a treamline) In general it i not oible to integrate the reure term becaue the denity may not be contant. To comlete the integral the relation between and mut be known. For teady, invicid, incomreible flow, the Bernoulli Equation i obtained: 1 V z Thi equation i valid when: (1) vicou effect are aumed negligible, ()theflowiaumedtobeteady,(3)theflowiaumedtobe incomreible, (4) the equation i alicable along a treamline. V ) dz contant a 0 (on a treamline) along treamline
3.3 F = ma Normal to a Streamline In thi ection we will conider alication of Newton' econd law in a direction normal to the treamline. Conider the force balance on the fluid article hown: Conidering comonent in the normal direction, n, and writing Newton' econd law in thi direction a mv R VV R F n Sring 018
Sring 018 Auming teady tate flow, normal acceleration a n = V /R, wherer i the local radiu of curvature of the treamline. a n i due to the change in direction of the article' velocity a it move along a curved ath. Aume that the only force of imortance are reure and gravity. The comonent of the weight (gravity force) in the normal direction i W n W co V If the reure at the center of the article i, then uing a Taylor erie exanion: F n n y n n y n co F n V n
Thu, the net force acting in the normal direction on the article i: F n W F Then, by combining the reviou eqn. with n Sring 018 V The equation of motion along the normal direction i found (notice that co θ = dz/dn): dz dn n That i, a change in the direction of flow of a fluid article (i.e., a curved ath, R< ) i accomlihed by the aroriate combination of reure gradient and article weight normal to the treamline. A larger eed or denity or a maller radiu of curvature of the motion require a larger force unbalance to roduce the motion. n co mv R R n VV F n V R
Sring 018 Neglecting gravity (commonly done for ga flow) or if the flow i in a horizontal (dz/dn = 0) lane, the reviou eqn. become V - n R Thi indicate that the reure increae with ditance away from the center of curvature ( / n i negative ince V /R i oitive the oitive n direction oint toward the inide of the curved treamline). Multily the reviou eqn. by dn, and noticing that / n = d/dn if i contant, the integral acro the treamline (in the n direction) i: d V R dn gz cont. acro treamline To comlete the indicated integration, it i neceary to know how the denity varie with reure and how the fluid eed and radiu of curvature vary with n. For incomreible flow = cont. Still it i neceary to know the relation between V and R with n[v = V(, n) and R= R(, n)]. Thu, the final form of Newton' econd law alied acro the treamline for teady, invicid, incomreible flow i V dn z contant acro the treamline R the
Sring 018 3.5 Static, Stagnation, Dynamic, and Total Preure From Bernoulli Equation: 1 V z contant along treamline, i the actual thermodynamic reure (tatic reure) of the fluid a it flow. To meaure it value, one could move along with the fluid, thu being tatic relative to the moving fluid. Or by drilling a hole in a flat urface and faten a iezometer tube a indicated by the location of oint (3). The reure in the flowing fluid at (1) i 1 = h 3-1 + 3, the ame a if the fluid were tatic. From the manometer relation: 3 = h 4-3. Thu, ince h 3-1 + h 4-3 = h it follow that 1 = h.
From Bernoulli Equation: 1 V z contant along treamline The term z, i the hydrotatic reure. It i not actually a reure but doe rereent the change in reure oible due to otential energy variation of the fluid a a reult of elevation change. Sring 018 The term V /, i the dynamic reure. It interretation can be een in the figure by conidering the reure at the end of a mall tube inerted into the flow and ointing utream. After the initial tranient motion ha died out, the liquid will fill the tube to a height of H a hown. The fluid in the tube, including that at it ti, (), will be tationary. That i, V = 0, or oint () i a tagnation oint.
Alying the Bernoulli equation between oint (1) and (), (V =0) and z 1 = z : 1 1 V 1 Sring 018 The reure at the tagnation oint i greater than the tatic reure, 1, by an amount V /, the dynamic reure. There i a tagnation oint on any tationary body that i laced into a flowing fluid. Some of the fluid flow over and ome under the object. The dividing line (or urface for two-dimenional flow) i termed the tagnation treamline and terminate at the tagnation oint on the body. For ymmetrical object (uch a a baeball) the tagnation oint i clearly at the ti or front of the object.
Sring 018 3.6 Examle of Ue of the Bernoulli Equation Between any two oint, (1) and (), on a treamline in teady, invicid, incomreible flow the Bernoulli equation can be alied in the form 1 1 1 V1 z1 V z If 5 of the 6 variable are known, the remaining one can be determined. Frequently, it i neceary to introduce other equation (continuity, etc.) 3.6.1 Free Jet When a jet of liquid of diameter d flow from the nozzle with velocity V. Bernoulli equation give: h 1 V The fluid leave a a free jet ( = 0) V gh
3.6.3 Flowrate Meaurement Several device uing rincile involved in the Bernoulli equation have been develoed to meaure fluid velocitie and flowrate. The Pitot-tatic tube i an examle. Other examle are flowrate meter, for ie and channel. The flowrate in a ie can be meaured uing: the orifice meter, the nozzle meter, and the Venturi meter. Alication of Bernoulli equation reult in: 1 Which when combined with the ma conervation equation: Q V1 A1 V A Yield: ( 1 ) Q A [1 ( A / A ) ] 1 1 V 1 Sring 018 1 V