day08_flow_2 - Euler’s& Bernoulli’s equations F=ma...

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Unformatted text preview: Euler’s & Bernoulli’s equations F=ma …for inviscid, incompressible flow of course! ∂ (p + γz ) = ρaL − ∂L Along a streamline , s (tangent to the velocity) for a steady flow 2 ∂v ∂ ∂ ⎛v ⎞ − (p + γz ) = ρa s = ρv s s = ρ ⎜ s ⎟ ⎜2⎟ ∂s ∂s ∂s ⎝ ⎠ 2 ⎞ ∂⎛ ⎜ p + γz + ρv s ⎟ = 0 2⎟ ∂s ⎜ ⎠ ⎝ 2 Integrate along the streamline: p + γz + Bernoulli’s equation Volume and mass flow rates Continuity Cavitation (maybe Tuesday) ρv s = constant 2 Lagrangian vs. Eulerian descriptions Lagrangian: follow particle Eulerian: measure local velocities, etc. Bernoulli – a simple application ?? ?? (a) (b) ?? ?? (c) Bernoulli example 2 In the Venturi section shown, 3 piezometric tubes are connected, two to the regions of largest areas, one to the region of smallest area. h1=3m; z1=1m; h2= 1m; z2 = 2m v1 = 1 m/s. (a) What are the pressures at points 1 and 2? (b) What are the differences in pressure head and piezometric pressure head at points 1 and 2? h3 (c) What is the h1 velocity at h2 point 2? An analogous example: Holes in a soda bottle; flow from base of dam Now, how do we calaculate the velocity of the fluid as it leaves the tank? (or soda bottle)? 1 Bernoulli illustrated Bernoulli example 3 (4.51 8th ed 4.67 9th ed) The flow‐metering device shown consists of a stagnation probe at station 2 and a static pressure tap at station 1. The velocity at station 2 is twice that at station 1. Air with a density of 1.2 kg/m3 flows through the duct. A water manometer is connected between the stagnation probe and the pressure tap, and a deflection of 10 cm is measured. What is the velocity at station 2? p + γz + ρv s = constant 22 v p + z + s = constant γ 2g 2 Lagrangian vs. Eulerian descriptions Lagrangian: follow particle: Bernoulli (follows streamline) Eulerian: measure local velocities, etc.: Continuity …before continuity Volume and mass flow rates and mass flow rates Volume and mass discharge Imagine a wire ring With cross section A m2 A Vector calculus--review Force and velocity are vectors and important quantities in fluid mechanics dot product between two vectors b a θ a ⋅ b = AB cos θ a = ( ai, aj, ak ) 2 A ≡ a = ai2 + a2 + ak j Place this ring in so that it area is normal to a fluid flow of average speed V m/s V Q = v ⋅ nA = VN cos θA = V cos θA Fluid Volume passing through the ring per second If the ring is placed at an angle to the flow v n v Magnitude Q = VA m3/s n Unit outward normal Vector with unit magnitude (N =1) in direction normal to a surface A Q = v ⋅ nA = VN cos θA = V cos θA dot product Mass Rate crossing the surface is m = ρQ v ⋅ n = VN cos θ = V cos θ So dot-product gives Magnitude of vector u in direction of outward normal 2 5.3 (8th ed) 5.7 (9th ed) A pipe with a 2 m diameter carries water having a velocity of 4 m/s. What is the discharge in cubic meters per second and in cubic feet per second? 3 ...
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