Chaotic high Reynolds numbers

Turbulent flow — arises at high Reynolds numbers and is characterized by the superposition upon the principal motion of a fluid, countless irregular and chaotic secondary components. 

When a fluid flows rapidly, its flow pattern typically exhibits a subtle mixture of order and chaos, and it is this structured chaotic motion that is referred to as turbulence [27]. Turbulence can be defined as a property of an incompressible fluid flow at a very high Reynolds number given by... 

Figure 7.6 Designs examples for mixing with chaotic advection at high Reynolds numbers (a) obstacles in side wall [28] (b) obstacles in the channel [59, 77] (c) zig-zag shaped channel [60].

The Taylor vortices described above are an example of stable secondary flows. At high shear rates the secondary flows become chaotic and turbulent flow occurs. This happens when the inertial forces exceed the viscous forces in the liquid. The Reynolds number gives the value of this ratio and in general is written in terms of the linear liquid velocity, u, the dimension of the shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid density and the viscosity. For a Couette we have ... 

When flow rates are high, a litjuid will not flow in a laminar fashion, but will become turbulent. The litjuid will start to flow in a chaotic way, forming large and small swirls and eddies. The flow rate at which this happens depends on the geometry of the machine, on the flow rate applied, and on the overall viscosity of the mixture. The transition from laminar to turbulent flow is characterized by the Reynolds number the critical Reynolds number depends on the geometry and the product properties (see Equations (15.9)-(15.10)). 

Note that the inertial range and high Re are not necessary for the existence of this type of Batchelor scaling and it can be also produced by chaotic advection in spatially smooth unsteady large scale flows with relatively small or moderate Reynolds number, or in two-dimensional flows (Jullien et ah, 2000 Pierrehumbert, 2000). 

It is known that a viscoelastic fluid, e.g., a solution with a trace amount of highly deformable polymers, can lead to elastic flow instability at Reynolds number well below the transition number (Re 2,000) for turbulence flow. Such chaotic flow behavior has been referred to as elastic turbulence by Tordella [2]. Indeed, the proper characterization of viscoelastic flows requires an additional nondimensional parameter, namely, the Deborah number, De, which is the ratio of elastic to viscous forces. Viscoelastic fluids, which are non-Newtonian fluids, have a complex internal microstructure which can lead to counterintuitive flow and stress responses. The properties of these complex fluids can be varied through the length scales and timescales of the associated flows [3]. Typically the elastic stress, by shear and/or elongational strains, experienced by these fluids will not immediately become zero with the cessation of fluid motion and driving forces, but will decay with a characteristic time due to its elasticity. 

The flow conditions within a gas can be described through the Knudsen number. Flows through pipes are characterised through the Reynolds number. At relatively high pressures in a rough vacuum, a viscous flow is present which may either be laminar or turbulent. For a viscous flow, a mutual influence between the flowing particles is typical. In the turbulent range, the molecules behave chaotically. The Knudsen number is defined as follows ... 

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