FLUID MECHANICS. Chapter 3 Elementary Fluid Dynamics - The Bernoulli Equation

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1 FLUID MECHANICS Chapter 3 Elementary Fluid Dynamics - The Bernoulli Equation

2 CHAP 3. ELEMENTARY FLUID DYNAMICS - THE BERNOULLI EQUATION CONTENTS 3. Newton s Second Law 3. F = ma along a Streamline 3.3 F = ma Normal to a Streamline 3.4 Physical Interpretation 3.5 Static, Stagnation, Dynamic, and Total Pressure 3.6 Examples of Use of the Bernoulli Equation 3.7 The Energy Line and the Hydraulic Grade Line 3.8 Restrictions on Use of the Bernoulli Equation

3 3. Newton s Second Law Newton s Second Law F d ( m v) dt Fma Streamline The lines that are tangent to the velocity vector throughout the flow field a dv dt V (3.) as V, a V n s R where, R : Local radius of curvature of the streamline (a) Flow in the x z plane. (b) Flow in terms of streamline and normal coordinates.

4 3. Newton s Second Law Isolation of a small fluid particle in a flow field. (Photo courtesy of Diana Sailplanes.) For force, Important force: gravity, pressure Negligible force: viscous force, surface tension effects

5 3. F = ma along a Streamline Sum of s components of forces acting on the particle (3.) Gravity force (Weight) F V V s mas mv V V s s Ws V W W sin V sin s Pressure force p p s s s F ( p p ) n y ( p p ) n y ps s s ps n y p s n y s p V s Freebody diagram of a fluid particle for which the important forces are those due to pressure and gravity.

6 3. F = ma along a Streamline Net force in the streamline dir. p (3.3) Fs Ws Fps sin V s (3.4) Along the streamline, sin dz / ds dz dp d( V ) V d( V ) V ds ds ds s ds (3.5) dp d( V ) dz 0 (along a streamline) integrated to dp (3.6) V gz C (along a streamline) For steady, inviscid, incompressible flow p sin V V a s s const. s (3.7) p V z const. along the streamline Bernoulli Equation

7 Example 3. GIVEN Inviscid, incompressible, steady flow along the horizontal streamline A B Radius of the sphere: a 3 Fluid velocity along this streamline: 0 a V V x3 FIND Determine the pressure variation along the streamline from point A to point B on the sphere x and V V A A x a and V 0 B B 0 SOL) () () (Ans) p V V s s V V a 3Va V V V s x x x 3V x3 x p 3 a V0 ( a x ) x x4 p V a x 3 a a ( a/ x) 6

8 3.3 F = ma Normal to a Streamline Sum of normal components of forces acting on the particle (3.8) mv R Fn Gravity force (Weight) V V R W W cos V cos n Pressure force F ( p p ) s y ( p p ) s y p s y pn n n n p p s n y V n n Net force in the streamline dir. p (3.9) Fn Wn Fpn cos V n dz p V (3.0a) dn n R p V (3.0b) n R Freebody diagram of a fluid particle for which the important forces are those due to pressure and gravity.

9 3.3 F = ma Normal to a Streamline Across the streamline p dp If s const., n dn integrate across the streamline dp V (3.) dn gz const. (across the streamline) R For steady, inviscid, incompressible flow const. (3.) p V dnz const. (across the streamline) R Final form of Newton s law (Bernoulli Equation)

10 Example 3.3 GIVEN Shown in Figure E3.3 are two flow fields with circular streamlines Velocity distributions are V ( r) ( V0/ r0) r for case (a) where V0:velocity at r r0 V ( r) ( V r ) / r for case (b) FIND Determine the pressure distributions, p = p(r), for each, given that p = p 0 at r = r 0. SOL) 00 p V r r p (a) ( / ) r ( Vr p 00) (b) r r3 V 0 r0 r Integrate these equations with repect to r (Ans) p p ( V / )[( r / r ) ] for case (a) (Ans) p p ( V / )[ ( r / r) ] for case (b) (a) Rigid body rotation (b) Free vortex (Bathtube vortex)

11 3.4 Physical Interpretation Basic assumption (3.3) (3.4) p V z ) steady ) inviscid 3) incompressible const. along the streamline p V dnz const. across the streamline R Work-energy principle The work done on a particle by all forces acting on the particle is equal to the change of the kinetic energy of the particle. Bernoulli Equation p V z Pressure Head g Kinetic Elevation Energy Head (Velocity Head) const. on a streamline Work done by pressure & gravitational force

12 z 3.5 Static, Stagnation, Dynamic, and Total Pressure From the Bernoulli equation, p V z const. Static pressure Dynamic pressure ; Hydrostatic pressure ; ; Actual thermodynamic pressure ; Move along with the fluid Static relative to the moving fluid ; p h p p 3 3 h 3 43 V Measurement of static and stagnation pressures. Total pressure = Static pressure + Dynamic pressure + Hydrostatic pressure = const. Stagnation point ; V 0 For z z, p p V If elevation effect are neglected, the stagnation pressure( p V /) is the largest pressure along the streamline. All of the kinetic energy converse into a pressure rise.

13 3.6 Examples of Use of the Bernoulli Equation Free jets Bernoulli equation between () and () (3.7) p V z p V z p p, p p p a 4 V 0, V V z h, z 0 h V (3.8) V h gh a Vertical flow from a tank. Bernoulli equation between () and (5) Bernoulli equation between (3) and (4) p V z p V z p V z p V z p p p, V 0, z h H, z 0 p p, V 0, z l, z 0 5 a 5 ( h H) V V g( h H) a p l v ( hl) 3

14 3.6 Examples of Use of the Bernoulli Equation For horizontal nozzle, V V V 3 In general, d h we can use Average velocity Vena contracta effect If the exit is NOT a smooth, well-contoured nozzle, dj dh d j : areas of the jet where, d : area of the hole h Contraction coefficient Cc Aj Ah where, Aj : areas of the jet at the vena contracta A : area of the hole h Horizontal flow from a tank. Vena contracta effect for a sharp-edged orifice.

15 3.6 Examples of Use of the Bernoulli Equation Confined flows Nozzles Pipes of variable diameter Conservation of mass AV A V If density const., Continuity equation (3.9) A V A V or Q Q (a) Flow through a syringe. (b) Steady flow into and out of a volume.

16 Example 3.7 GIVEN A stream of refreshing beverage of diameter d = 0.0 m flows steadily from the cooler of diameter D = 0.0 m as shown in Figs. E3.7 FIND Determine the flowrate, Q, from the bottle into the cooler if the depth of beverage in the cooler is to remain constant at h = 0.0 m SOL) () p V z p V z p p 0 z h, z 0 () V gh V Q Q, where Q AV A V D d V V 4 4 (3) V d V D V gh (9.8)(0.0).98 m/s 4 4 ( d/ D) (0.0/ 0.0) Q AV A V (0.0) (.98) 4 (Ans) Q m /s

17 Example 3.9 GIVEN Water flows through a pipe reducer as is shown in Fig. E3.9. The static pressures at () and () are measured by the inverted U-tube manometer containing oil of specific gravity, SG, less than one. FIND Determine the manometer reading, h. SOL) p V z p V z Q AV A V () p p ( z z ) V [ ( A / A ) ] p p ( z z ) ( SG) h ( SG) h V [ ( A / A ) ] V [ ( A / A ) ] h ( SG) Q AV (Ans) ( Q / A ) [ ( A / A h ) ] g(- SG)

18 3.6 Examples of Use of the Bernoulli Equation Flowrate measurement To measure fluid velocities and flowrates using Bernoulli equation. Orifice meter Nozzle meter Venturi meter If flow is horizontal ( z z ) p V p V If the velocity profiles are uniform at () and (), Continuity equation is Q A V A V Therefore, (3.0) Q A ( p p ) A A [ ( ) ] Typical devices for measuring flowrate in pipes

19 3.6 Examples of Use of the Bernoulli Equation. Open channel Sluice gates Bernoulli equation p V z p V z If the gate is the same width ( A bz and A bz ) Continuity equation If Q AV bvz AV bvz (3.) Q z b g( z z ) ( z z) z z, Q z b gz Vena contracta effect Contraction coefficient Cc z a Weir Q C Hb gh C b gh 3/ Sluice gate geometry. (Photograph courtesy of Plasti- Fab, Inc.) Rectangular, sharp-crested weir geometry.

20 3.7 The Energy Line and the Hydraulic Grade Line For steady, inviscid, incompressible flow, the Bernoulli equation (3.) p V g p z p V z constant on a streamline H g z ; Energy Line (EL) ; Hydraulic Grade Line (HGL) Representation of the energy line and the hydraulic grade line.

21 3.8 Restrictions on Use of the Bernoulli Equation Compressibility effects For a simple case of compressible flow with the temperature of a perfect gas, p RT, where T const. For steady, inviscid, isothermal flow, dp RT V gz constant p V RT p V (3.3) z ln z g g p g p p p where, p p ln( ) for small RT V z z V g g g For steady, isentropic flow, k p/ C where, k : specific heat ratio / k / k C p dp V gz constant / k p / k / k k ( k )/ k ( k )/ k C p dp C [ p p ] p k (3.4) k p k k p V k p V gz gz k k p

22 3.8 Restrictions on Use of the Bernoulli Equation For stagnation point flow, ) Compressible flow If z z, V 0, Ma V / c, c krt (3.5) k/( k) p p k Ma (compressible) p ) Incompressible flow Bernoulli equation V p p p p V V ( p RT) p p RT (3.6) Ma p p k p (incompressible)

23 3.8 Restrictions on Use of the Bernoulli Equation Unsteady effects Bernoulli equation including the unsteady effect ( V/ t 0) V ds dp d ( V ) dz 0 (along a streamline) t For incompressible flow, s V (3.7) p V z ds p s V z t (along a streamline) Other effects Rotational effects Viscous effects Pump or turbine

24 Example 3.6 GIVEN An incompressible, inviscid liquid is placed in a vertical, constant diameter U-tube as indicated in Fig. E3.6. When released from the non-equilibrium position shown, the liquid column will oscillate at a specific frequency. FIND Determine this frequency. SOL) p p 0 z z, z z where, z z( t) V V V V ds dv ds l dv where, l: length of liquid column t dt dt s s s s ( z) l dv z dt d z g z 0 V dz, g dt l dt solution : g (Ans) l z( t) C sin( g / lt) C cos( g / lt) Period of this oscillation: t l / g 0

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