# turbulence forced external mixing ... depth (cm) time (min) interface depth vs time graph (1) graph Post on 15-Nov-2020

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• cm

ωy 1/s

cm

Fig (3b) 60 sec average with interface

ωy 1/s

cm

Fig (3a) 60 sec average with no interface

An average of 60 sec Velocity/Vorticity plot with an interface An average of 60 sec Velocity/Vorticity plot with no interface

Process and Environmental Research Division

Summer Placement 2011

By Mohamed Elmaghrbi Supervisor: Dr Matthew Scase

Turbulence forced external mixing

Introduction: Turbulent/laminar boundaries occur in a vast range of environmental and industrial fluid flows. For example smoke from chimneys and volcanic eruption columns propagating through the atmosphere, fuel ignited in an engine, and dense water flows from the edge of Antarctica which drive the ocean conveyor belt. The interest in the characteristics of the interfacial mixing between the ambient and turbulent layers, has led to numerous detailed laboratory studies aimed at improving the mixing models employed in these situations. A large number of studies have investigated this phenomena using turbulence generated by an oscillating grid, producing zero-mean turbulent flow. The first attempt to do so was by J. S. TURNER in 1968. The experiment consists of two layers of fluid with different density, one of which is stirred by oscillating the grid creating a turbulent motion, and the other layer is quiescent.

Theory: The entrainment rate (the interface velocity) is expressed as a power law of the form: 𝐸 = 𝑘𝑅𝑖−𝑛 (1) Where 𝐸 is the ratio of the entrainment velocity to the turbulent velocity scale, 𝑘 and 𝑛 are constants and are to be determined from the experiments. 𝑅𝑖 is Richardson number and it is a function of the relative density between the fluids, the turbulent length scale ( 𝑙 ) and root mean square velocity scale (𝑢) at the interface and is given by the equation : 𝑅𝑖 =

𝑔∆𝜌𝜌 𝜌𝑢2

(2) The length scale 𝑙 only depends on the distance from the grids origin, where as the velocity scale 𝑢 is dependent on the grid characteristics as well as the distance from the grid. ‘∆𝜌′ is the density difference between the two fluids and ‘𝜌’ is a reference density (density of fresh water) and 𝑔 is gravitational acceleration. 𝑙 = 0.1𝐷 (3) 𝑢 = 0.25𝑓𝑠1.5𝑚0.5𝑧−1 (4) Where 𝐷 is the distance of the interface from the grids origin (m), 𝑓 is the frequency (Hz), 𝑠 is the stroke length (m), 𝑚 is the mesh size (distance between the grids bar centres) (m), and z is the distance from a virtual origin. Detailed experiments by Hopfinger & Toly showed that this virtual origin lies at the grid mid-plane in its equilibrium position (i.e. at the interface 𝑧 = 𝐷)

cm

cm cm cm cm cm cm

Figures (3a) to (3f) are a time sequence with 0.2 sec time step interval. The velocity and vorticity plots here are instantaneous and not averaged unlike in figures (2a) and (2b), and the interface is at depth 6cm from an arbitrary origin. From this time series it can be seen what exactly happens as a jet or an eddy impacts on the interface. In fig(3a) an eddy rotating anti clockwise is impacting on the interface between 4 and 6 cm horizontally. It can be seen that it is directed away from the interface, and at the same time generates another eddy that rotates clockwise between 7 and 9 cm starting from fig(3b). The second eddy propagates through the interface and causes the two layers to mix.

Particle Image Velocimetry (PIV) was used to study the behaviour of the fluid near the interface in order to create an understanding of the mixing procedure. The colour background represents the vorticity (red signifies anti-clockwise motion, and blue is clockwise motion).The arrows show the velocity of the fluid, (an arrow size is proportional to the magnitude of the velocity). Fig (3a) is a 60 sec time average with 200 frames a second. It can be seen the there are no major vortices present away from the grid in the absence of an interface, in fig (3b) where the interface is at depth 20 cm from the grid mid-plane (the origin), it is clear that two rotational motions are present just above the interface, to the right (clockwise) and (anti-clockwise) to the left. This happens because the interface acts like a boundary not allowing the fluid to pass through, and directs the fluid horizontally and then vertically away from the denser bottom layer. As this happens it generates vorticity at the interface opposite to the motion above it, which essentially causes the two fluids to mix.

Interface depth

Results and discussion:

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Time (min)

Interface depth vs Time

Graph (1)

Graph (1) shows the interface height vs time. The initial depth was 14 cm from the grid mid- plane. The range of Richardson numbers for this experiment was : 190.5 < Ri