Unit C-: List of Subjects The elocity Field The Acceleration Field The Material or Substantial Derivative Steady Flow and Streamlines Fluid Particle in a Flow Field F=ma along a Streamline Bernoulli s Equation F=ma across a Streamline Basic Equations of Fluid
Page of Unit C- The elocity Field u x, y, z, tˆi v x, y, z, tˆj w x, y, z, t kˆ The infinitesimal particles of a fluid are tightly packed together continuum assumption At a given instant in time, a description of any fluid property such as density, pressure, velocity, and acceleration may be given as a function of the fluid s location in 3-D space This representation of fluid parameters as functions of the spatial coordinates is called a field representation of the flow The velocity field in 3-D Cartesian coordinates: u x, y, z, tˆi v x, y, z, tˆj w x, y, z, t kˆ The Convection of a Flow Field The presence of the velocity field is often called the CONECTION of the flow field: the flow field properties mass, momentum, energy, etc. are CONECTED transported from one location to another, within the flow field.
Page of Unit C- The Acceleration Field a A t d dt A t A x y z Taking the time derivative of velocity using the chain rule: da A A dxa A dya A dz A a A t dt t x dt y dt z dt The particle velocity components are given: u dx dt v dy dt w dz dt A A A A A A Hence, the acceleration of particle A is: A A A A aa t ua va wa t x y z A dx dt A A dy dt A A dz dt A
Unit C- Page 3 of The Material or Substantial Derivative u v w t x y z a a t Dt D k j i ˆ ˆ ˆ z y x z w y v x u t Dt D The shorthand notation called material or substantial derivative: Using the gradient or del operator: Operator definition Dot Product: Dt D a z w y v x u t Dt D where, t Dt D k j i ˆ ˆ ˆ z y x where, The acceleration field: Components in x, y, z coordinates: z w y v x u t t a z u w y u v x u u t u a x z v w y v v x v u t v a y z w w y w v x w u t w a z UNIT D UNIT D- SLIDE SLIDE 5 The Material Derivative The Material Derivative 4 4 The shorthand notation called material or substantial derivative: Using the gradient or del operator: Operator definition Dot Product: Hence, the acceleration field is: Dt D a z w y v x u t Dt D where, t Dt D k j i ˆ ˆ ˆ z y x z w y v x u a t Dt D where,
Page 4 of Unit C- Steady Flow and Streamlines sˆ a a ˆ ˆ ˆ ˆ ss ann s n s If the velocity does not change with time at a given location in the flow field: Steady Flow For steady flows, each particle slides along its path and its velocity vector is everywhere tangent to the path the lines that are tangent to the velocity vectors throughout the flow field are called: streamlines Let us define a -D coordinate system that is attached to and moving with a fluid particle along a streamline: the Streamline Coordinate system This is the coordinate system, defined at a specific location on a streamline. If a particle is moving steadily along a streamline, the streamline coordinate system is the coordinate that is attached to this particle and moving together with this particle. The direction along the streamline tangential direction: ŝ The direction across the streamline normal direction: ˆn The normal direction is taken toward the center of radius of curvature.
Page 5 of Unit C- Fluid Particle in a Flow Field Unit C- Consider the free-body diagram of a small fluid particle in a flow field along a streamline: nd Streamlines The important forces are those due to pressure and gravity in aerodynamics, gravity forces are often ignored a this is steady flow, so acceleration is only convective sˆ a a ˆ ˆ ˆ ˆ ss ann s n s as Tangential Acceleration s a ˆ ˆ aˆ n ˆ ss ann s ncentrifugal Acceleration s
Page 6 of Unit C- F=ma along a Streamline sin p s s a s Consider a small fluid particle of size: s n Apply Newton s Second Law along a streamline: p sin s s Equation of Motion a s Euler s Equation A change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle weight along the streamline Textbook Munson, Young, and Okiishi, page 97 In tangential ŝ direction, F = ma becomes, W sin p ps n y p ps n y mas W sin ps ps n y mas note that: ps ps p Dividing by the volume of the particle s n y and taking the limit: s 0, n 0, y 0, also note that W olume specific weight p sin lim as sin p as s0 s s
Page 7 of Unit C- Bernoulli s Equation p z constant along a streamline Equation of motion can be rearranged p sin a s s s Along the streamline: p p p dp dp ds dn dn 0 s n s ds dz d d Also, sin ds s ds ds Hence, along a streamline: dz dp d ds ds ds dp d dz 0 dp Divide by the density, d gdz 0 Assuming constant acceleration of gravity: dp gz constant With additional assumption of constant density incompressible fluid: p z constant along a streamline Bernoulli s Equation
Page 8 of Unit C- Bernoulli s Equation Static Pressure, Dynamic Pressure, Hydrostatic Pressure Each term in the Bernoulli s equation can be interpreted as a form of pressure p z constant along a streamline Static Pressure Dynamic Pressure Hydrostatic Pressure Static Pressure: actual thermodynamic pressure of the fluid as it flows Dynamic Pressure: pressure due to the energy stored in moving fluid Hydrostatic Pressure: pressure due to gravitational effects Recall, UNIT B Bernoulli s equation in conservation of pressure form: p z constant Static Pressure + Dynamic Pressure + Hydrostatic Pressure = Total Pressure Bernoulli s equation in conservation of energy form: p gz constant Flow Energy + Kinetic Energy + Potential Energy = Total Energy Bernoulli s equation in conservation of head form: p z constant 3 g Static Head + Dynamic Head + Head = Total Head
Page 9 of EXAMPLE 3. Textbook Munson, Young, and Okiishi, page 0 Unit C- The Bernoulli Equation Applying the Bernoulli s equation along the same streamline: between point and, p z p z For aerodynamics, usually the hydrostatic pressure is not considered z z. Thus, p p Note that: 0 equal to the velocity of bicycle itself: called the freestream 0 velocity is zero: called the stagnation point Therefore, p p 0
Page 0 of Unit C- F=ma across a Streamline dz dn p n Consider a small fluid particle of size: s n Apply Newton s Second Law Normal to a streamline: dz dn p n Equation of Motion A change in the direction of flow of a fluid particle is accomplished by the appropriate combination of pressure gradient and particle weight normal to the streamline Textbook Munson, Young, and Okiishi, page 97 This is Euler s equation across the streamline the direction normal: ˆn
Page of Class Example Problem Related Subjects... F=ma across a Streamline Unit C- Unit C-C Streamline s a Streamline Unit C-C Applying the equation of motion across the streamline: dz dz p p dn dn n n z where, dz p dz dn, p dp dn, nand = 6 n n dn dp Hence, dn 6 n Integrating this equation yields: n dn n n n n dp ln 6 n 0 dn dn 6 dn dn 6 n 0 n 0 0 0 n n dn ln 6 n ln 6 n ln 6 0 p p n 6 n 6 0 ln 6 ln 6 n ln 6 6 n p p n ln 6 n For point : p 40 kpa and n m => p.0 kpa For point 3: p 40 kpa and n m => p3 0. kpa
Page of Unit C- Basic Equations of Fluid p z constant along a streamline p dnz constant across the streamline Normal to a streamline: dz p dn n Assuming constant acceleration of gravity: dp dn gz constant With additional assumption of constant density incompressible fluid: p dn z constant across the streamline This is Bernoulli s equation across the streamline the direction normal: ˆn The following assumptions were made: The flow is steady The fluid is inviscid The fluid is incompressible A violation of one or more of the above assumptions is a common cause for obtaining an incorrect match between the real world and solutions obtained by use of these equations
ES06 Fluid Mechanics Unit C-: List of Subjects Static and Dynamic Pressure Stagnation Point Pitot-Static Tube Directional Finding Pitot-Static Tube
Page of 7 Unit C- Static and Dynamic Pressure Each term in the Bernoulli s equation can be interpreted as a form of pressure If we apply Bernoulli s equation between points and, assuming = 0 and z = z p p Stagnation Pressure Static Pressure Dynamic Pressure If elevation effects are neglected, stagnation pressure is the largest pressure obtainable along a given streamlinecalled, the total pressure
Page of 7 Unit C- Stagnation Point A point in a flow where the velocity is zero, where any streamline touches a solid surface at an angle: stagnation point p z pt constant along a streamline Bernoulli s equation Textbook Munson, Young, and Okiishi, page 08 NOTE: for aerodynamics, usually the hydrostatic pressure term can be ignored: p p T constant along a streamline Pitot-Static Probe for Airspeed Measurement At the stagnation point, the flow is decelerated to zero speed. This deceleration process increases the pressure static pressure => total pressure by converting kinetic energy dynamic pressure. Hence, the total pressure can be considered as: Total Pressure = Static Pressure + Dynamic Pressure. This total pressure can be sensed at the tip stagnation point of a special device, called the Pitot-Static probe. Pitot-Static probe is also equipped with a side port, called the static port senses the static pressure of the flow. By taking the difference between total pressure and static pressure, the Pitot- Static probe can be used to determine the airspeed of the flow.
Page 3 of 7 Unit C- Pitot-Static Tube Note that: p p3 center tube and p p4 outer tube The center tube measures the stagnation pressure at its open tip p 3 p The outer tube is made with several small holes so that they measure the static pressure p p p 4 Combining these two equations: p 3 p 4 p 3 p4 / Textbook Munson, Young, and Okiishi, page 09
Page 4 of 7 Unit C- Pitot-Static Tube Pitot-Static Probe Pitot-static probe can determine airspeed, by sensing the pressure difference between total and static pressures p pt ps: Usually this can be measured by differential U-tube manometers. Then the airspeed can be determined by: p
Page 5 of 7 Directional Finding Pitot-Static Tube Unit C- p p / Three pressure taps are drilled into a small circular cylinder, fitted with small tube, and connected to three pressure transducers The side holes are located at a specific angle = 9.5 The speed is then obtained from: p p / Preview of aerodynamics Based on the ideal flow analysis aero, at the angle of β = 30º, the pressure on the surface of the circular cylinder becomes equal to the freestream static pressure hence, static ports can be established by making the holes at these locations. Some Important Points for Pitot-Static Probe Design An accurate measurement of static pressure requires that none of the fluid s kinetic energy be converted into a pressure rise at the point of measurement This requires a smooth hole with absolutely no burrs or imperfections Textbook Munson, Young, and Okiishi, page It is important that the pressure taps be properly located to ensure that the pressure measured is actually the static pressure Textbook Munson, Young, and Okiishi, page
Page 6 of 7 EXAMPLE 3.6 Textbook Munson, Young, and Okiishi, page 0 Unit C- Pitot-Static Tube a The pressure at point far ahead of the airplane From the textbook table C. in Appendix C page 764: 3 3 p 0.08 lb/in psi.756 0 slugs/ft b The pressure at the stagnation point on the nose of the airplane, point Applying Bernoulli s equation assuming steady, incompressible, and inviscid flow with no elevation change: p p 88 ft/s where, 00 mph 46.667 ft/s 60 mph and 0 stagnation point 3 3 in.7560 slugs/ft 46.667 ft/s Hence, p p 0.08 lb/in ft =,474.44 lb/ft 3 = 0.39 psi c The pressure difference indicated by a Pitot-static probe attached to the fuselage The pitot-static tube measures the difference between freestream static pressure p and stagnation pressure p : p p 0.39 psi 0.08 psi = 0.3 psi
Page 7 of 7 Class Example Problem Related Subjects... Pitot-Static Tube Unit C- Pitot-static tube measures the difference between static and stagnation pressure: At point, the tip of the probe: p p3 stagnation pressure At point, the static port of the probe: p p4 patm 0 gage pressure Applying Bernoulli s equation yields: p3 p4 or, p3 p4 For indicated also called, the equivalent airspeed e : Sea-Level p3 p4 e eqn. For true airspeed true : 0,000 fttrue p3 p4 eqn. Combining eqns. & for the same pressure difference p3 p4: Sea-Level 0,000 ft e true 3 3 Sea-Level.377 0 slugs/ft Therefore, true e 0 knots = 30.33 knots 3 3 0,000 ft.67 0 slugs/ft
ES06 Fluid Mechanics Unit C-3: List of Subjects Bernoulli s Equation Free Jets Confined Flows Cavitation Flowrate Measurement
Page of 9 Unit C-3 Bernoulli s Equation p z p z Between any two points, and, on a streamline in steady, inviscid, and incompressible flow ideal flow: p Free Jets z Confined Flows Flowrate Measurement p z Bernoulli s Equation in 3 Forms p z p z Pressure form p p gz gz Energy per unit mass form p p z z g g Head form
Page of 9 Unit C-3 Free Jets p p4 0 Between and : h h gh g h H p p4 Textbook Munson, Young, and Okiishi, page 0 Free Jet elocity The magnitude of free jet velocity depends solely on the depth of liquid, relative to the location of the free jet flow. This can be seen as a velocity of falling baseball conversion of potential energy into kinetic energy.
Page 3 of 9 Unit C-3 Confined Flows Nozzles and pipes of variable diameter for which the fluid velocity changes because the flow area is different from one section to another Need to use the concept of conservation of mass the continuity equation along with the Bernoulli s equation The rate at which the fluid flows into the volume must equal the rate at which it flows out of the volume otherwise, mass would not be conserved The mass flowrate: The volume flowrate: m Q Q To conserve mass, the inflow rate must be equal to the outflow rate: m m The continuity equation: If the density is constant incompressible fluid: A A Q Q A A A
Page 4 of 9 EXAMPLE 3.7 Textbook Munson, Young, and Okiishi, page 5 Unit C-3 Flow from a Tank Gravity Assuming a steady, incompressible, and inviscid flow, the Bernoulli s equation can be applied between points and : p z p z where, p p patm 0 gage pressure with z h and z 0 Hence, h => gh eqn. In order to maintain constant water depth h.0 m, the flowrate in and out of the tank must be equal: Q Q. Since the flowrate is defined as: Q A, A A this is called, the conservation of mass. d Hence, D d => eqn. 4 4 D Combining eqns. & yields: gh 9.8 m/s.0 m 4 4 = 6.6 m/s dd 0. m m Q A d 0.0 m 6.6 m/s = 0.049 m 3 /s 4 4
Page 5 of 9 Class Example Problem Related Subjects... Confined Flows Unit C-3 Assuming a steady, incompressible, and inviscid flow, the Bernoulli s equation can be applied between points 0 and 4 as: 0 4 p0 z0 p4 z4 where, p0 p4 patm 0 gage pressure with z0 5 ft and z4 0. Also, 0 0 4 Therefore, z0 => 4 z0 gz0 3. ft/s 5 ft = 7.94 ft/s The flowrate can be determined as: Q A 4 4 0. ft 7.94 ft/s = 0.4 ft 3 /s 4 Pressure at point can be determined by applying the Bernoulli s equation as: 4 p z p4 z4 where, p4 patm 0 gage pressure with z 8 ft and z4 0. Also, Q constant conservation of mass, so: 4 since A A4 3 Hence, p z 6.4 lb/ft 8 ft = 499 lb/ft gage pressure 3 Similarly, p p3 z z3 6.4 lb/ft 5 ft = 3 lb/ft gage pressure
Page 6 of 9 Unit C-3 Cavitation Cavitation occurs when the pressure is reduced to the vapor pressure Textbook Munson, Young, and Okiishi, page 9 Cavitation at Siphon During a siphon, it is possible to establish a cavitation, when the lowest pressure at highest elevation during the siphon becomes vapor pressure of the liquid under the given temperature.
Page 7 of 9 EXAMPLE 3.0 Textbook Munson, Young, and Okiishi, page 0 Unit C-3 Siphon and Cavitation Assuming a steady, incompressible, and inviscid flow, the Bernoulli s equation can be 3 applied between points and 3 as: p z p3 z3 where, p p3 patm 0 gage pressure with z 5 ft and z3 5 ft. Also, 0 Therefore, z 3 z3 or, 3 z z3 g z z3 3. ft/s 5 ft 5 ft = 35.9 ft/s Also, Q constant conservation of mass, so: 3 since A A3 Now, let us apply the Bernoulli s equation between points and as: p z p z where, p patm 0 gage pressure with z 5 ft and z H. Also, 0 and 3 35.9 ft/s. At incipient of cavitation, p pvapor =.60 psi absolute =.60 in 4.7 = 4.47 lb/in,083.68 lb/ft gage ft Therefore, z p z 3.94 slugs/ft 35.9 ft/s 3 3 or, 6.4 lb/ft 5 ft,083.68 lb/ft 6.4 lb/ft H Hence, H 8.36 ft
Page 8 of 9 Unit C-3 Flowrate Measurement Assuming horizontal z = z, steady, inviscid, and incompressible flow between points and : p p If velocity profiles are assumed to be uniform at sections and : Q A A Combining these equations: Q A p [ A p A ] Textbook Munson, Young, and Okiishi, page
Page 9 of 9 EXAMPLE 3. Textbook Munson, Young, and Okiishi, page Unit C-3 enturi Meter Combining the Bernoulli s equation and conservation of mass: p p Q A A A Rearranging this equation provides: Q A A p p A 3 where, Q 0.005 0.050 m /s A A D D 0.06 m 0.0 m = 0.36 3 Also, from SG 0.85, SG HO 0.85,000 kg/m = 850 kg/m 3 3 Therefore, for Q 0.005 m /s : 0.36 p 3 3 p 0.005 m /s 850 kg/m =,60 N/m.6 kpa 40.06 m 3 For Q 0.05 m /s : 0.36 p 3 3 p 0.05 m /s 850 kg/m = 60 3 N/m 6 kpa 40.06 m