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# Turbulence scales

Although a huge number of theoretical, experimental and computational investigations on turbulence have been performed, the structure and description of turbulence still remain open issues. The motivation behind the study of turbulent flows is due to the fact that the vast majority of flows is turbulent and turbulence significantly increases the rates of transport and mixing of matter, momentum and heat in the flows (PopeSB:2000). Turbulence itself is a very complex phenomenon, involving different time and length scales. In the well-known energy-cascade theory, the kinetic energy enters the turbulence at the largest scales of motion through the production mechanism. This energy is transferred from the largest scale (integral scale) to smaller and smaller scales until it is dissipated by viscous action at the smallest scale, the so-called Kolmogorov length scale. The wide range of length-scales and time-scales, the production, the dissipation and the energy transfer between different length-scales make the modelling of turbulence one of the greatest scientific challenge. The level of difficulty in investigating premixed flames is greatly enhanced when also turbulence comes into play, because of the complex interactions between the turbulence itself and the combustion.

The transfer rate of kinetic energy per mass unit of fluid over any particular scale between the macro structure and the micro structure obeys the relationship (this is known as Kolmogorov’s law), e.g., (BatchelorGK:1993):

\varepsilon =-\frac{d}{dt} U_{}^{2} =A\frac{U_{}^{3} }{L} \ \ \ \ \ \ (1)

where A is a constant in the order of unity. Together with the root-mean-square of the instantaneous velocity fluctuations, this expression can be used to characterise the macro structure in terms of a macro velocity scale, a macro length scale and a macro time scale:

The microstructure can be characterised by the Kolmogorov velocity, length and time scale:

where \nu denotes the kinematic viscosity. It should be noted that equation (1) describes the kinetic energy transfer on the larger scales as well as the viscous dissipation on the micro structure despite the apparent absence of the molecular viscosity.

A combustion zone may also be characterised in terms of velocity, length and time scales. If the laminar burning velocity, the laminar flame thickness, and the chemical time scale:

are taken as the fundamental scales of the combustion ((PetersN:1991), (BorghiR:1988)), the structure of a turbulent flame may be considered by relating these combustion scales to those of the turbulent flow field. The occurrence of a particular combustion regime is determined by the value of the Damkohler number:

Da=\frac{L/U}{\tau _{c} } =\frac{\ell _{t} /u_{rms}^{'} }{\delta _{L} /S_{u L} }

When the scales of the micro structure of the turbulent flow field are substituted for L and U, it constitutes the reciprocal Karlovitz number:

Ka^{-1} =\frac{L/U}{\tau _{c} } =\frac{\ell _{K} /u_{K} }{\delta _{L} /S_{u L} }

The Damkohler and Karlovitz number can be used to identify different combustion regimes [14, 15] by employing two dimensionless parameters,

\left(\frac{u_{rms}^{,} }{S_{u L} } \right) and \left(\frac{\ell _{t}^{} }{\delta _{L}^{} } \right)

and by rewriting their definitions into

\frac{u_{rms}^{,} }{S_{u L} } = Da_{}^{-1} \left(\frac{\ell _{t}^{} }{\delta _{L}^{} } \right) (2)

and

\frac{u_{rms}^{,} }{S_{u L} } = Ka_{}^{2/3} \left(\frac{\ell _{t}^{} }{\delta _{L}^{} } \right)_{}^{1/3} (3)

Since the expressions for the laminar burning velocity and the laminar flame thickness (DahoeAE:2003),

\frac{S_{u L} }{S_{u L}^{o}} =\sqrt{\frac{\lambda (T_{u} )}{\lambda (T_{u 0} )} } \left(\frac{T_{u} }{T_{u 0}} \right)\left(\frac{P_{}^{} }{P_{0}^{} } \right)_{}^{\frac{n-2}{2} } \left(\frac{T_{f} }{T_{f}^{o} } \right)^{-\frac{n}{2} } \left(\frac{T_{f} }{T_{f}^{o} } \right)^{\frac{m}{2} } \exp \left[-\frac{Ea}{R} \left(\frac{1}{T_{f}} -\frac{1}{T_{f}^{o} } \right)\right]

\frac{\delta _{L}^{} }{\delta _{L}^{o} } =\sqrt{\frac{\lambda (T_{u})}{\lambda (T_{u 0})} } \left(\frac{P }{P_{0}} \right)^{-\frac{n}{2} } \left(\frac{T_{f}}{T_{f}^{o} } \right)_{}^{\frac{n}{2} } \left(\frac{T_{f}^{} }{T_{f}^{o} } \right)_{}^{-\frac{m}{2} } \exp \left[+\frac{Ea}{R} \left(\frac{1}{T_{f}^{} } -\frac{1}{T_{f}^{o} } \right)\right]

in conjunction with the definition of the thermal conductivity and the viscosity in terms of molecular properties imply that

S_{u L} \delta _{L}^{} =\frac{\lambda }{\hat{C}_{P}^{} } =\frac{\mu }{\rho }

a third relationship may be derived as

\frac{u_{rms}^{,} }{S_{u L} } = Re\left(\frac{\ell _{t}^{} }{\delta _{L}^{} } \right)_{}^{-1} (4)

Equations (2), (3) and (4) may then be used to construct a figure which is known as the Borghi diagram.

Pope S.B. (2000) Turbulent flows. Cambridge University Press, Cambridge, United Kingdom.(BibTeX)
Batchelor G.K. (1993) The theory of homogeneous turbulence. Cambridge University Press.(BibTeX)
Peters N. (1991) Length scales in laminar and turbulent flames. Progress in Astronautics and Aeronautics, 135:155-182.(BibTeX)
Borghi R. (1988) Turbulent Combustion Modelling. Progress in Energy and Combustion Science, 14:245-292.(BibTeX)