Fluid Dynamics: Flow and Velocity

Flow Categorization

• Constant discharge through a long straight pipe exemplifies a specific flow type.
• Steadily increasing flow through a pipe is another flow category.
• The motion of a river as it navigates around bridge piers is a distinct flow scenario.
• Steadily decreasing flow within a pipe that has a reducing cross-section represents a unique flow type.

Velocity Calculations

• Water velocity in a 200 mm bore pipe is 1 m/s, the mean velocity in a 40 mm diameter jet is 25 m/s.
• Water flows at 10 L/s through a pipe with a 100 mm inner diameter, which reduces to 50 mm, resulting in mean velocities of 1.27 m/s and 5.09 m/s, respectively.
• Water flows at 100 L/s through a nozzle with a 50 mm inner diameter, the height attained when directed vertically upwards is 132 m (in the absence of air resistance).

Tank and Nozzle System Analysis

• Water flows from a tank with no losses, and under 1 bar of atmospheric pressure:
• Gage pressure at point 2 is 0.5 bar
• Velocity of water emerging from the nozzle (v3) is 10 m/s
• Volumetric flow rate (Q3) is 78.5 L/s
• Velocity at point 1 (u1) is 0.004 m/s.
• A 100 mm diameter pipe terminates with a 25 mm diameter nozzle; with a Bourdon gage registering 3 bar at the nozzle entry, the mass flow rate of water through the nozzle is 12 kg/s (assuming no losses).

Laminar Flow Determination

• Oil with a density of 860 kg/m3 and absolute viscosity of 4.2 x 10-3 N s m-2 flows in a 750 mm diameter pipe, the flow most likely ceases to be laminar at a discharge of 6.33 L/s.

Water Tank Dynamics

• A water jet discharges into an open tank:
• The jet's cross-sectional area is 2.4 x 10-3 m2, water velocity is 6 m/s, and the tank's plan area is 4 m2.
• Water leaves the tank through an orifice at 3 x 10-3 m3/s.
• The rate of water mass accumulating in the tank is +11.4 kg/s and the rate of depth increase is +2.85 mm/s, assuming density of water = 1000 kg/m3.

Pitot-Static Tube Measurement

• A Pitot-static tube measures air velocity with a U-tube manometer containing water, and with an air density of 1.2 kg/m3, the velocity when the manometer reads 8mm is 11.44 m/s.

Venturi Meter Application

• A horizontal 200 mm water pipe has a Venturi meter with a 125 mm throat diameter; upstream and throat tappings connect to a mercury U-tube manometer showing a level difference of 878 mm.
• The velocity in the Venturi throat is 16 m/s
• The discharge is 0.2 m3/s, assuming a discharge coefficient CD = 1 and a relative density of mercury to water (ρM/ρ) = 13.6.

Venturi Meter in Pipeline

• A Venturi meter has a 10 cm inlet diameter and 3 cm throat diameter and is fitted to a pipeline conveying liquid with a density of 850 kg/m3.
• A mercury U-tube is fitted between the inlet and throat, with a level difference of 20 cm.
• With a discharge coefficient CD of 0.96, the flow rate of the liquid is 5.23 L/s, assuming a mercury density of 13600 kg/m3.

Fire Hose Nozzle Pressure

• For a water jet issuing from a fire hose at 15 m/s with the nozzle 9 m above the pump and a hose-to-nozzle area ratio of 4:1, the pressure at the pump discharge must be 193.8 kN/m2 or 1.938 b above atmospheric pressure (assuming negligible friction).

Multiple Choice Answers

• Uniform Flow: When flow parameters at any given instant remain the same at every point.
• Reynolds Number: Is dimensionless.
• Viscous and Moving Fluid: The pressure at a point in a fluid will not be the same in all directions when the fluid is viscous and moving.
• Conservation of Mass: The continuity equation is connected with conservation of mass.
• Pitot Tube: An instrument used for measuring the speed of an airplane.
• All of the Above: Venturimeter, orifice plate and nozzle are instruments used to measure flow on the application of Bernoulli's theorem.
• Any Direction: A venturimeter can be installed to measure flow in any direction and in any location.
• Pitot Tube: Measures flow of water in a pipe.
• Zero: In the case of steady flow, the acceleration of any fluid particle is zero.
• Unsteady: During the opening of a valve in a pipe line, the flow is unsteady.
• Streamline Flow: Is when each liquid particle has a definite path and their paths do not cross each other is called.
• Reynolds Number > 4000: Turbulent flow occurs in pipes.
• Energy: Bernoulli's equation deals with the law of conservation of energy.
• Steady Flow: Flow conditions do not change with time at any point
• Equal Velocity: A streamline is defined as the line of equal velocity in a flow
• Velocity: A pitot tube measures the velocity
• Energy is Constant: According to Bernoulli's equation, energy is constant along a stream-line but may vary across streamlines
• Reynolds Number < 2300: Laminar flow in pipes.