Ch3 The Bernoulli Equation

Transcription

1 Ch3 The Bernoulli Equation The most used and the most abused equation in fluid mechanics. v 3. Newton s Second Law: F ma In general, most real flows are 3-D, unsteady (x, y, z, t; r,θ, z, t; etc) Let consider a -D motion of flow along streamlines, as shown below. elocity ( v ): Time rate of change of the osition of the article. Streamlines: The lines that are tangent to the velocity vectors throughout the flow field. Note: For steady flows, each article slide along its ath, and its velocity vector is everywhere tangent to the ath. Streamline coordinate: S S(t) ; ds / dt along the streamline can be decided by v and (s) v (the distance R )

2 By chain rule, a s d dt S ds dt S ; a where R is the local radius of curvature of the streamline, and S is the distance measured along the streamline from some arbitrary initial oint. n R Figure 3. Isolation of small fluid article in a flow field. 0 ; R (not straight line) S Forces: Gravity & Pressure (imortant) iscous, Surface tension (negligible) v v 3. F ma along a streamline

3 Figure 3.3 Free-body diagram of a fluid article for which the imortant forces are those due to ressure and gravity.. Consider the small fluid article ( δ S δn ), as show above. If the flow is steady. N. δ FS δm as δm ρδ (3.) S S (Eq. (3.) is valid for both comressible and incomressible fluids) W. δ δw sinθ γδ sinθ ( θ 0 δws 0) W S

4 P. P δs δp S (first term of Taylor series exansion, S because the article is small; P δp P δp ) S S Thus δ FPS : the net ressure force on the article in the streamline direction δf PS ( P δp δp δnδy S S P δ S ) δnδy ( P δp P δsδnδy S S ) δnδy *Note: if ressure gradient is not zero ( P net ressure force. P v P v P S n S n Net Force F δw δf δ S S PS ; C ), then there is a ρ S γ sinθ P S (3.4) Note: Force balancing consideration ρ (mass flux), not ρ or, is a key arameter for fluid mechanics, if ρ constant. Examle 3. Consider the inviscid, incomressible, steady flow along the horizontal streamline A-B in front of the shere of radius a as shown in Fig. E3.a. From a more advanced theory of flow ast a shere, the fluid velocity along this streamline is

5 3 a 0 ( ) 3 x Determinate the ressure variation along the streamline from oint A far in front the shere ( x A and A 0 ) to oint B on the shere ( x B a and B 0)

6 Figure E3. Solution: Since the flow is steady and inviscid, Eq. 3.4 is valid. In addition, since the streamline is horizontal, sin θ sin 0 0 and the equation of motion along the streamline reduces to s s ρ () with the given velocity variation along the streamline, the a 3 0a acceleration term is 0 ( )( ) s x 3 x x 3 a 0 ( ) 3 3 x where we have relaced s by x since the two coordinates are a x

7 identical (within an additive constant) along streamline A-B. It follows that < 0 along the streamline. The fluid slows s down from 0 far ahead of the shere to zero velocity on the nose of the shere ( x a). Thus according to Eq., to roduce the given motion the ressure gradient along the streamline is x 3 3ρa 0 ( a x 4 3 x 3 ) () This variation is indicated in Fig. E3. b. It is seen that the ressure increase in the direction of flow ( > 0) x from oint A to oint B. The maximum ressure gradient 0 a (0.60 ρ ) occurs just slightly ahead of the shere ( x.05a). It is the ressure gradient that slows the fluid down from A 0 to B 0. The ressure distribution along the streamline can be obtained by integrating Eq. from 0 (gage) at x to ressure at location x. The result, lotted in Fig. E3.c, is 6 a 3 ( a x) ρ 0 ( ) (Ans) x The ressure at B, a stagnation oint since 0, is the B

8 highest ressure along the streamline ( B ρ 0 ). As shown in Chater 9, this excess ressure on the front of the shere (i.e. > 0 ) contributes to the net drag force on the B shere. Note that the ressure gradient and ressure are directly roortional to the density of the fluid, a reresentation of the fact that the fluid inertia is roortional to its mass. Since streamline, Thus sin θ dz ds, also d ds n const, so dn 0. d d ds dn ds s n ds Eq. (3.4) dz d d( ) γ ρ ds ds ds along a streamline (s) d( ), and along the ds or d ρ d( ) γdz 0 (3.5) d gz const (3.6) ρ Note: if ρ constant, then ρ f ( ) must be known. The well-known Bernoulli equation:

9 P ρ γz constant Note: 4 assumtions ) μ 0 (inviscid) ) 0 t (steady) 3) ρ C (incomressible) 4) along a streamline ( dn 0 ) Examle 3. Consider the flow of air around a bicyclist moving through still air with velocity 0 as is shown in Fig. E3.. Determine the difference in the ressure between oints () and (). Figure E 3. Solution: In a coordinate system fixed to the bike, it aears as through the air is flowing steadily toward the bicycle with seed 0. If the assumtions of Bernoulli s equation are valid (steady,

10 incomressible, inviscid flow), Eq. 3.7 can be alied as follows along the streamline that asses through () and () ρ γz ρ γz We consider () to be in the free stream so that 0 and () to be at the ti of the bicyclist s nose and assume that z z and 0 (both of which, as is discussed in Section 3.4, are reasonable assumtions). It follows that the ressure at () is greater than at () by an amount ρ ρ0 (Ans) A similar result was obtained in Examle 3. by integrating ressure gradient, which was known because the velocity distribution along the streamline, (s), was known. The Bernoulli equation is a general integration of F ma. To determine, knowledge of the detailed velocity distribution is not needed-only the boundary conditions at () and () are required. Of course, knowledge of the value of along the streamline is needed to determine the seed 0. As discussed in Section 3.5, this is the rincile uon which many

11 velocity measuring devices are based. If the bicyclist were accelerating or decelerating, the flow would unsteady (i.e. 0 constant ) and the above analysis would be in correct since Eq. 3.7 is restricted to steady flow. v v 3.3 F ma normal to a streamline. Examle: The devastating low-ressure region at the center of a tornado can be exlained by alying Newton s nd law across the nearly circular streamlines of the tornado. δm v. δ Fn ρδ δma n (3.8) R R. δ Wn δw cosθ γδ cosθ ( θ 90 ; δwn 0). δ Fn ( P δpn) δsδy ( P δpn) δsδy P δpnδsδy δsδnδy n P δ n P. Thus, δfn δwn δfn ( γ cosθ ) δ (3.9) n Eq. (3.8) & (3.9) & cos θ dz dn dz P γ dn n ρ R (3.0)

12 . if gravity is neglected or if the floor is horizontal dz γ 0 0 dn P n ρ R This ressure difference is needed to be balance the centrifugal acceleration associated with the curved streamlines of the fluid motion.. Read examle 3.3. Integrate across the streamline, using the factor that P n dp dn if S constant dp constant (3.) ρ R dn gz across the streamline v v ( S, n) ; R R( S, n) ρ R if ) t dn γz ) μ 0 C 3) (3.) ρ C

13 Examle 3.3 Shown in Figs. E3.3a, b are two fields with circular streamlines. The velocity distributions are ( r) C r C ( r) r for case (a) for case(b) and where C and C are constant. Determine the ressure distributions, (r), for each given that 0 at r0 r.

14 Figure E 3.3 Solution: We assume the flow are steady, inviscid, and incomressible with streamline in the horizontal lane ( dn 0 dz ) Since the streamlines are circles, the coordinate n oints in a direction oosite of that of the radial coordinate, n r, and the radius of curvature is given by R r. Hence, Eq. 3.0 becomes r ρ r For case (a) this gives ρc r r ρc While for case (b) it gives 3 r r For either the ressure increases as r increase since δ δr > 0 Integration of these equations with resect to r, starting with a known ressure 0 at r r0, gives

15 ρ C ( r r0 ) for case (a) and ρ C ( ) r r (Ans) (Ans) for case (b). These ressure distributions are sketched in Fig E3.3c. The ressure distributions needed to balance the centrifugal accelerations in cases (a) and (b) are not the same because the velocity distributions are different. In fact for case (a) the ressure increase without bound as r, while for case (b) the ressure aroaches a finite value as r. The streamline atterns are the same for each case, however. Physically, case (a) reresents rigid body rotation (as obtained in a can of water on a turntable after it has been sun u ) and case(b) reresents a free vortex (an aroximation to a tornado or the swirl of water in a drain, the bathtub vortex ).

16 3.4 Physical Interretation. Along a streamline: P ρ γ z C Across the streamline: P ρ dn γ z c R If: steady, inviscid, and incomressible (None is exactly true for real flows). Physics: Force balance; Bernoulli s Princile The work done on a article by all forces acting on the article is equal to the change of the kinetic energy of the article. (Newton s second law; first & second laws of thermodynamics). Examle 3.4 Consider the flow of water from the syringe shown in Fig A force alied to the lunger will roduce a ressure greater than atmosheric at oint () within the syringe. The water flows from the needle, oint (), with relatively high velocity and coasts u to oint (3) at the to of its trajectory. Discuss the

17 energy of the fluid at oints (), (), and (3) by using the Bernoulli equation. Figure E 3.4 Solution: If the assumtions (steady, inviscid, incomressible flow) of the Bernoulli equation are aroximately valid, it then follows that the flow can be exlained in terms of the artition of the total energy of the water. According to Eq. 3.3 the sum of the three tyes of energy (kinetic, otential, and ressure) or heads (velocity, elevation, and ressure) must remain constant. The following table indicates the relative magnitude of each of these energies at the three oints shown in the figure.

18 Energy Tye Point Kinetic ρ Potential γ z Pressure Small Zero Large Large Small Zero 3 Zero Large Zero The motion results in (or is due to) a change in the magnitude of each tye of energy as the fluid flows from one location to another. An alternate way to consider this flow is as follows. The ressure gradient between () and () roduces an acceleration to eject the water from the needle. Gravity acting on the article between () and (3) roduces a deceleration to cause the water to come to a momentary sto at the to of its flight. If friction (viscous) effects were imortant, there would be an energy loss between () and (3) and for the given the water would not be able to reach the height indicated in the figure. Such friction may arise in the needle (see chater 8, ie flow) or between the water stream and the surrounding air (see chater

19 9, external flow). Examle 3.5 Consider the inviscid, incomressible, steady flow shown in Fig. E3.5. From section A to B the streamlines are straight, while from C to D they follow circular aths. Describe the ressure variation between oints () and () and oints (3) and (4). Figure E 3.5 Solution: With the above assumtion and the fact that R for the ortion from A to B, Eq. 3.4 becomes γ z constant The constant can be determined by evaluating the known variables at the two locations using P 0 (gage), Z h - to give

20 γ ( z z ) γ h (Ans) Note that since the radius of curvature of the streamline is infinite, the ressure variation in the vertical direction is the same as if the fluid were stationary. However, if we aly Eq. 3.4 between oints (3) band oint (4) we obtain (using dn -dz) Z 4 4 Z ( dz ) ρ γ z R γ z 3 With 4 0 and z 4 -z 3 h 4-3 this becomes Z 4 γ h 4 3 Z dz (Ans) 3 R 3 ρ To evaluate the integral we must know the variation of and R with z. Even without this detailed information we note that the integral has a ositive value. Thus, the ressure at (3) is less than the hydrostatic value, γ h 4 3, by an amount equal to ρ Z Z 3 4 R dz. This lower ressure, caused by the curved streamline, is necessary to accelerate the fluid around the curved ath. Note that we did not aly the Bernoulli equation (Eq. 3.3)

21 across the streamlines from () to () or (3) to (4). Rather we used Eq As is discussed in section 3.6 alication of the Bernoulli equation across streamlines (rather than along) them may lead to serious errors. 3.5 Static, Stagnation (Total), and Dynamic Pressure γz - hydrostatic ressure not a real ressure, but ossible due to otential energy variations of the fluid as a result of elevation changes. - static ressure ρ -dynamic ressure stagnation oint 0 P P ρ stagnation ressure static ressuredynamic ressure (if elevation effect are neglected) Total ressure It reresents the conversion of all of the kinetic energy into a ressure rise.

22 P T γ Z P Total P Hydrostatic P Static P Dynamic P ρ Figure 3.4 Measurement of static and stagnation ressures Pitot-static tube: simle, relatively in exensive.

23 Figure 3.6 The Pitot-static tube Figure 3.7 Tyical Piotot-static tube designs. T ρ γz constant along a streamline center tube: 3 ρ elevation is neglected. outer tube: 4 ρ ( 3 4 ) / ρ 3 4

24 Figure 3.9 Tyical ressure distribution along a Pitot-static tube. The cylinder is rotated until the ressures in the two sideholes are equal, thus indicating that the center hole oints directly ustream-measure stagnation ressure. Side holes ( β 9. 5 measure static ressure. ) Figure 3.0 Cross section of a directional-finding Pitot-static tube.

25 3.6 Examle of Use of the Bernoulli Equation t steady ( 0 ), inviscid ( 0 n and along a streamline ( 0). Between two oints: μ ), incomressible ( constant ρ ) P ρ γz P ρ γz Unknowns: P, P,,, Z, Z (6) If 5 unknowns are given, then determining the remaining one. Free Jets (ertical flows from a tank) Figure 3. ertical flow from a tank () - ():

26 P 0 (gage ressure); P 0 (Free jet) Z h; Z 0; 0; γh ρ γh ρ gh () and (4): atmosheric ressure (5): g( h ) 5 H (3) and (4): P γ h l) ; P 0 ; Z l ; Z 0, ( P3 γ l P4 ρ4 Q 4 gh Recall from Physics or dynamics 自由落體在一真空中 gh ; the same as the liquid leaving from the nozzle. All otential energy kinetic energy (neglecting the viscous effects).

27 Fig 3. Horizontal flow from a tank. ena contracta effect: Fluid can not turn 90 ; Contraction coefficient: C c A j /A h ; A j cross area of jet fluid column; A h cross area of the nozzle exit.

28 Figure 3.4 Tyical flow atterns and contraction coefficients for various round exit configurations Confined flows Figure 3.5 Steady flow into and out of a tank.

29 mass flow rate m ρ Q (kg/s or slugs/s) where Q: volume flow rate (m 3 /s or ft 3 /s) Q A ; m ρ Q ρ A From Fig. 3.5: ρ A ρa Examle 3.7 A stream of diameter d 0.m flows steadily from a tank of diameter D.0m as shown in Fig E3.7a. Determine the flow rate, Q, needed from the inflow ie if the water deth remains constant. h.0m.

30 Figure E 3.7 Solution: For steady, inviscid, incomressible the Bernoulli equation alied between oints () and () is ρ γz ρ γ Z () With the assumtions that 0, z h, and z 0, Eq. becomes gh () Although the water level remains constant (h constant), there is an average velocity,, across section () because of the flow

31 from the tank. From Eq 3.9 for steady incomressible flow, conservation of mass requires Q Q A. Thus, A A, or π D 4 π d 4 d D Hence, ) (3) ( Equation and 3 can be combined to give gh ( d / D) 4 Thus, with the given data (9.8 m/s )(.0 m) 4 (0.m/ m) π m/s 3 and A A (0.m) (6.6m/s) 0.049m /s Q (Ans) In this examle we have not neglected the kinetic energy in the water in the tank ( 0). If the tank diameter is large comared to the jet diameter (D>>>d), Eq. 3 indicates that << and the assumtion that 0 would be reasonable. The error associated with this assumtion can be seen by calculating the ratio of the flowrate assuming 0, denoted Q, to that assuming 0, denoted Q 0. This ratio, written as

32 Q Q gh ( ( d gh D) 4 0 D ( d D) 4 ) is lotted in Fig. E3.7b. With 0< d/d < 0.4 it follows that < Q/Q 0.0, and the error in assuming 0 is less than %. Thus, it is often reasonable to assume 0. Examle 3.9 Water flows through a ie reducer as is shown in Fig. E3.9. The static ressure at () and () are measured by the inverted U-tube manometer containing oil of secific gravity. SG. Less than one. Determine the manometer reading, h. FIGURE E 3.9 Solution: With the assumtion of steady, inviscid, incomressible flow, the Bernoulli equation can be written as

33 z γ ρ z γ ρ The continuity equation (Eq. 3.9) rovides a second relationshi between and, if we assume the velocity rofiles are uniform as those two locations and the fluid incomressible: A A Q By combining these two equations we obtain ] A A [ ρ ) z z ( γ () This ressure difference is measured by the manometer and can be determined by using the ressure-deth ideas develoed in Chater. Thus. l γ h SGγ h γ l γ ) z z ( γ or h SG )γ ( ) z z ( γ () As discussed in Chater, the ressure difference is neither merely γh not γ(hz -z ). Equations and can be combined to give the desired result as follows ] A A [ ρ h SG )γ (

34 or Q/A, thus, h Q A A A. (Ans) g( SG ) The difference in elevation, z -z, was not needed because the change in elevation term in the Bernoulli equation exactly cancels the elevation term in the manometer equation. However, the ressure difference, -, deends on the angle θ, because of the elevation, z -z, in Eq.. Thus, for a given flowrate, the ressure difference, -, as measured by a ressure gage would vary with θ, but the manometer reading, h, would be indeendent of θ. In general, an increase in velocity is accomanied by a decrease in ressure. Air (gases): Comressibility (Ch ) Liquids: Cavitation (Proeller etc.) Flow Rate Measurement Ideal flow meter neglecting viscous and comressible effects. (loss some accuracy)

35 Orifice meter; Nozzle meter:, P ; enturi meter ( P P ) Q A (3.0) A ρ A g( z z ) Sluice gate: Q z b (3.) z z Shar crested weir: Q C Hb gh gh C b The Energy Line and the Hydraulic Grade Line Bernoulli equation is an energy equation with four assumtions: inviscid, incomressible, steady flow, and along a streamline from ts () - (), resectively, and the sum of artition of energy remains constant from ts ()-(). (Head) H z constant along a streamline. γ g

36 FIGURE 3. Reresentation of the energy line and the hydraulic grade line FIGURE 3. The energy line and hydraulic grade line for flow from a tank. 3.8 Restrictions on the use of Bernoulli Equation 3.8. Comressibility Effect

37 Gases: ρ d not a constant A secial case: for comressible flow Given - steady, inviscid, isothermal (T C along the streamline) Sol: RT ρ RT ρ c RT ln d RT ρ d ln g RT z g z g or z g ln g RT z g Comare to the incomressible Bernoulli Equation. ε if Q, << ε then ) ln( ln ε ε z g ) ( g RT z g γ z g γ z g 3.8. Unsteady Effect (s) steady; (s, t) unsteady.

38 a s (local convective accelerations) t s Reeating the stes leading to the Bernoulli Equation. ρ t ds d ρ d ( ) γ dz 0 along a streamline If incomressible, inviscid, S ρ γ z v ρ ds ρ γ S t z Use velocity otential to simlify the roblem (Ch 6). FIGURE 3.5 Oscillation of a liquid column in a U-tube

39 3.8.3 Rotational Effect In general, the Bernoulli constant varies from streamline to streamline. However, under certain restrictions, this constant may be the same throughout the entire flow field, as ex.3.9 illustrates this effect Other restrictions μ 0 (inviscid); Bernoulli Equation: A first integral of Newton s nd laws along a streamline.