Bernoulli. Water flows under the sluice gate shown. The flow is uniform with constant speed far upstream and far downstream of the gate stations at 1 and 2, with depths H₁ = 10m and H₂ = 2m. A static pressure tap open at the top is placed at point B at depth of 6m in the sluice gate as shown and gives a manometer height, Ah, = 3m. A pitot tube is placed along the centerline at point C at a depth of Im at downstream section 2. Find the values of: (a) V2; (b) static and stagnation gage pressure, Pg and Pog, at the bottom point A in upstream section 1; (c) the flow speed, VB, down along the gate at point B; (d) the pitot tube manometer height, Aher at point C. (take z = 0 as the tank bottom) Ans OM: (a) 10¹ m/s; (b) 10¹ kPa / 10² kPa; (c)10° m/s; (d) 10⁰ m 10 m 3m
Unless stated otherwise take:
Patm = 10^5, P(water) = 1000 kg/m^3 ; P(air) = 1.2 kg/m^3 ; g = 9.8m/s^2
gage pressure Pg = P - Patm
Unless specified otherwise assume uniform velocity profiles in duct locations.
The Bernoulli eq’n (BE), P + ½ρV2 + ρgz = constant
In this formulation the scalar variable z is the vertical height measured opposite the
direction of g (z is always taken up independent of your coordinate system; pgz is potential energy)
This BE is only valid along a streamline in steady state flow, when viscous friction effects can be
ignored, called “ideal” fluid motion, and the density is assumed to be constant
the differential radial component of –?P + ρg = ρa
with the centripetal acceleration , ar = - V^2/r , to get dp/dr = ρgr + ρV^2/r
The Bernoulli eq’n is often divided through by the constant ρg ≡ γ so that each
term has units of length, a height or “head” : (P/ ρg) + (V2/2g) + (z) = constant.
Q = VA = flow rate = constant, as a second eq’n to solve for an unknown velocity.
the difference between stagnation or toal pressure Po = P + pV^2 /2 and the actual
hydrostatic pressure P
The inlet to a stagnation pressure probe , called a pitot tube, is a point where the fluid
speed is brought near to zero as V → 0 then BE gives
P→Po at the probe inlet, which the manometer measures. In the undisturbed flow P is the true fluid pressure. Po is useful
because if an independent static probe measures P then the measured difference
(Po-P) = ρV^2/2 gives the flow speed
the laws of physics from a Lagrangian
coordinate system, f(t), which moves with a fluid element , to an Eulerian coordinate system, f(x,y,z,t), which is fixed in
space and has fluid moving through it. Our concept of steady state applies to the Eulerian system and is defined by the
partial derivative (∂ /∂t)x,y,z = 0 , which means properties do not change with time at any fixed x,y,z . Of course a fluid
particle can change properties as it flows from one point to another, even though the flow field is at steady state in the
Eulerian sense. The material derivative is the transformation from one system to the other given the fluid
velocity field V(x,y,z,t); D( )/Dt = ∂ ( ) /∂t + (V•) ( ).
D ( )/Dt is the Lagrangian rate of change of a property.
the acceleration of a fluid particle is given by a = DV/Dt = ∂V/∂t + (V•)V . Always compute the (V•) scalar operator before you apply it to
Streamlines are lines tangent to the velocity vector. For planar flows, V = ui+vj
Particle trajectories (paths) are obtained by integrating dx/dt = u(x,y,) and
dy/dt = v (x,y,t). For steady state flows particles paths are the same as the streamlines
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