Fluid Mechanics Example Classes Example Questions (Set IV), Summaries of Fluid Mechanics

Download Fluid Mechanics Example Classes Example Questions (Set IV) and more Summaries Fluid Mechanics in PDF only on Docsity! ES2A7 - Fluid Mechanics Example Classes Example Questions (Set IV) Question 1: Dimensional analysis a) It is observed that the velocity ‘V’ of a liquid leaving a nozzle depends upon the pressure drop ‘P’ and its density ρ. Show that the relationship between them is of the form V C P ρ= b) It is observed that the frequency of oscillation of a guitar string ‘f’ depends upon the mass ‘m’, the length ‘l’ and tension ‘F’. Show that the relationship between then is f C F ml= c) Find the dimension of the bulk modulus K, knowing its relationship with the speed of the sound ‘a’ in a liquid and the density ‘ρ’: a C K ρ= a) Dimension: [ ] [ ] [ ]2 3V L T, P M LT , M Lρ= = = The relationship should verify: [ ] [ ] [ ]a b c a b 3c b c 2b a V P 1 L M T 1 ρ − − + − − = = This leads to the following system: a b 3c 0 2b b 3b 0 a 1 b c 0 b c b 12 2b a 0 2b a c 12   − − = − − + = =        + = =− =−      − − = =− =       The relation is thus verified b) Dimension: [ ] [ ] [ ] [ ] 2f 1 T, m M, l L, T ML T = = = = The relationship should verify: [ ] [ ] [ ] [ ]a b c d c d b d a 2d f m l T 1 L M T 1+ + − − = = This leads to the following system: a 1 c d 0 d c b 12 b d 0 d b c 12 a 2d 0 2d a d 12  = + = =−     =   + = =−     =  − − = =−      =− The relation is thus verified c) From the relation, we have : 2K a Cρ= so [ ] [ ][ ] 2 2 3 2 2 M L M K a L T LT ρ= = = Question 2: Dimensional analysis Water flows through a 2cm diameter pipe at 1.6m/s. Calculate the Reynolds number and find also the velocity required to give the same Reynolds number when the pipe is transporting air. For the water the kinematic viscosity was 1.31 10-6 m2/s and the density was 1000 kg/m3. For air those quantities were 15.1 10-6 m2/s and 1.19kg/m3. Kinematic viscosity is dynamic viscosity over density = ν = µ/ρ. The Reynolds number is : Ud Ud Re ρ µ ν = = Reynolds number when carrying water: water 6 16 0.02 Re 24427 1.31 10− × = = × To calculate Reair we know: air waterRe Re= air 6 1 air U 0.02 24427 15 10 U 18.44m.s − − × ⇔ = × ⇔ = Question 3: Momentum conservation The figure below shows a smooth curved vane attached to a rigid foundation. The jet of water, rectangular in section, 75mm wide and 25mm thick, strike the vane with a velocity of 25m/s. Calculate the vertical and horizontal components of the force exerted on the vane and indicate in which direction these components act. From the question: 3 2 1 1 1a 1.875 10 m , u 25m.s− −= × = Since the jet section is constant a1=a2, the velocity is also constant u1=u2 (Mass conservation in the case of an incompressible flow) The bulk flow is thus 3 1 1 1 2 2Q a .u a .u 0.0469m .s−= = = Calculation of the total force using the momentum equation: 2 1F Q(V V )ρ= −


-4 -2 0 2 4 0 50 100 R h Question 5: Mass conservation + Bernoulli A water clock is an axisymmetric vessel with a small exit pipe in the bottom. Find the shape for which the water level falls equal heights in equal intervals of time. Continuity equation: ( ) 0ɺ ɺout in d m m dt ρυ + − = There is no inflow and the fluid is incompressible. Let ‘a’ be the cross-section of the exit pipe: ( )( ) ( )ɺ out d t m aU t dt υ ρ ρ= − = − , ( ) ( ) 0 t t a U t dtυ = − ∫ Bernoulli equation between the upper free surface and the exit section: ( ) ( )2U t gh t 0 0 2 ρ ρ + = + ( ) ( )U t 2gh t= So using the first relation: ( ) ( ) ( ) ( ) 0 0 2 2 t t t a gh t dt t a g h t dt υ υ = − = − ∫ ∫ From the question, we know that the variation of h is linear in time: ( ) 0h t h xt= − So ( ) ( ) ( ) ( ) ( )( ) 0 0 3 2 0 3 2 2 2 2 3 2 2 3 t t a g h xtdt a t g h xt x a t g h t x υ υ υ = − − = − = ∫ The volume of fluid can always be written as ( ) ( )( ) ( ) ( )2 2 0 0 0 0 R z h t h t t rdrd dz R z dz π υ θ π= =∫ ∫ ∫ ∫ So ( ) ( ) ( )( ) ( )( ) ( )( )3 2 2 1 22 0 2 2 2 3 h t a a R z dz g h t R h t g h t x xπ π = ⇒ =∫ Question 6: Momentum conservation Because the fluid is contracted at the nozzle forces are induced in the nozzle. Anything holding the nozzle (e.g. a fireman) must be strong enough to withstand these forces. Determine these forces. The analysis takes the following procedure: 1) Draw a control volume 2) Decide on co-ordinate axis system 3) Calculate the total force 4) Calculate the pressure force 5) Calculate the body force 6) Calculate the resultant force 1 & 2 Control volume and Co-ordinate axis have been done for you and are shown in the figure below. Notice how this is a one dimensional system which greatly simplifies matters. 3) Calculation of the total force: ( )x 2 1F F Q u uρ= = − From the continuity equation, the bulk flow is : 1 1 2 2Q A u A u= = so 2 2 1 1 1 F Q A A ρ   = −    4) Calculation of the pressure force 1 2P P PF F F= +


x 1 2P P P PF F F F= = − We use the Bernoulli equation to calculate the pressure 2 2 1 1 2 2 1 2 P u P u z z g 2g g 2gρ ρ + + = + + The nozzle is horizontal ( z1=z2 ) and the pressure outside is atmospheric (P2=0). With continuity, it gives : 2 1 2 2 2 1 Q 1 1 p 2 A A ρ   = −    22 1 P 2 2 AQ F 1 2 A ρ   = −    5) Calculation of the body force The only body force is the weight due to gravity in the y-direction - but we need not consider this as the only forces we are considering are in the x-direction. 6) Calculation of the resultant force R P 22 2 1 R P 2 2 1 2 22 1 R 2 2 2 1 F F F A1 1 Q F F F Q 1 A A 2 A AQ 2 2 F 1 2 A A A ρ ρ ρ = +       = − = − − −           = − + −    So the fireman must be able to resist the force of 22 1 R 2 2 2 1 AQ 2 2 R F 1 2 A A A ρ   = − = − − +    This force increases when A1 increase or A2 decrease, which correspond to the expected behaviour. Question 7: Momentum conservation Consider a rocket of mass mr traveling at a speed ur as measured from the ground. Exhaust gases leave the engine nozzle (area Ae) at a speed Ue relative to the nozzle of the rocket, and with a pressure that is higher than local atmospheric pressure by an amount p e. The aerodynamic drag force on the rocket is D. Derive an equation for the acceleration of the rocket.