Fluid Mechanics 3
A series of podcasts summarising the lectures for the course Fluid Mechanics 3
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In a steady turbulent flow the turbulence level is equal to the rms velocity fluctuation divided by the mean velocity, usually expressed as a percentage. By Taylor’s hypothesis the integral length and time scales L and T , obtained by integrating under the normalised correlation curves, are related by L = Mean velocity x T. In Laser Doppler Anemometry the instantaneous Doppler frequency is equal to the flow velocity divided by the fringe spacing. In the whole field technique of Particle Image Velocimetry, cross-correlation of the intensity across successive images is used to determine the particle displacements and hence the velocity vectors.
In deep water the celerity increases as the square root of wavelength and the group velocity is equal to half the celerity. Water particles trace out circular paths and the crest and trough velocities are equal to half the wave height times the angular frequency. Water particle accelerations are towards the centres of the circles. In shallow water the celerity is equal to the square root of g times the water depth. Here water particle velocities are approximately the same at the surface as near the bed. The power in a wave is half potential and half kinetic. Froude scaling is used to match wave tank tests to prototype. Velocities scale as the square root of scale ratio and accelerations are the same in model and prototype.
The relative importance of viscosity is determined by the Reynolds Number. Flow in a pipe is laminar at Reynolds below about 2000. Flows with extremely small Reynolds numbers, about one or less, are knows as creeping flows and display the characteristic of reversibility i.e. if a boundary is displaced and then moved back to its original position the fluid particles will return to their starting points. When the gap between two parallel plates is filled with fluid and one plate moves at constant speed relative to the other the velocity distribution between the plates is linear. The shear stress is then the dynamic viscosity times the velocity gradient. For flow between parallel fixed plates or through a circular pipe the velocity distribution is parabolic, the velocities being proportional to the velocity gradient. They only remain parabolic at low Womersley numbers.
This podcast is about three types of turbines, the Francis turbine, the Kaplan turbine and the Pelton Wheel. The Francis and Kaplan are known as reaction turbines and the Pelton Wheel as an impulse turbine. The Euler head is the net head available to the turbine for doing work. Multiplying this by the flow rate, fluid density, acceleration of gravity and the efficiency gives the power output. The Euler equation relates the Euler head to the runner velocity and the whirl velocities at inlet and outlet. The performance of different machines can be compared by their Type Numbers. For example the Pelton wheel has a low type number and is most suitable for low flow rates and high heads.
The topics covered in the Fluid Mechanics 3 course will be (a) Turbomachinery (b) Dynamic similarity and scaling (c) Viscous flow modelling (d) Aerodynamics and compressible flows (e) water waves and (f) flow measurement. There will be 18 lectures plus two revision classes. In two practical assignments (for which 20% of the marks will be allocated) you will measure the lift and drag on an aerofoil and periodic and random waves in a flume. Before starting the course it is important to check that you are familiar with the material covered in FM2, particularly the Bernoulli equation.