  Turbulence is a non-local phenomenon and has multiple-scales. Non-locality
can be addressed either implicitly or explicitly. Implicitly, by subsequent
resolution of all spatio-temporal scales. However, if directly solved for the
temporal or spatially averaged fields, a closure problem arises on account of
missing information between two points. To solve the closure problem in
Reynolds-averaged Navier-Stokes equations (RANS), an eddy-viscosity hypotheses
has been a popular modelling choice, where it follows either a linear or
non-linear stress-strain relationship. Here, a non-constant diffusivity is
introduced. Such a non-constant diffusivity is also characteristic of
non-Fickian diffusion equation addressing anomalous diffusion process. An
alternative approach, is a fractional derivative based diffusion equations.
Thus, in the paper, we formulate a fractional stress-strain relationship using
variable-order Caputo fractional derivative. This provides new opportunities
for future modelling effort. We pedagogically study of our model construction,
starting from one-sided model and followed by two-sided model. Non-locality at
a point is the amalgamation of all the effects, thus we find the two-sided
model is physically consistent. Further, our construction can also addresses
viscous effects, which is a local process. Thus, our fractional model addresses
the amalgamation of local and non-local process. We also show its validity at
infinite Reynolds number limit. This study is further extended to tempered
fractional calculus, where tempering ensures finite jump lengths, this is an
important remark for unbounded flows. Two tempered definitions are introduced
with a smooth and sharp cutoff, by the exponential term and Heaviside function,
respectively and we also define the horizon of non-local interactions. We
further study the equivalence between the two definitions.
