Transforming Chaos; the Law of Nature, into Order; The Law of Man

“Big whorls have little whorls, which feed on their velocity. And little whorls have lesser whorls. And so on to viscosity.” - Lewis Fry Richardson, 1922

“A fluid motion is described as turbulent if it is three-dimensional, rotational, intermittent, highly disordered, diffusive and dissipative.”



Complex flow field around a Formula 1 car’s front wing [endplate] in ground effect; lifted from J Pegrum “Experimental Study of the Vortex System Generated by a Formula 1 Front Wing”, Imperial College London, 2006

“Big whorls have little whorls, which feed on their velocity. And little whorls have lesser whorls. And so on to viscosity.” -- Lewis Fry Richardson, 1922



The dimensions of the computational domain must be large enough to encompass/contain the largest turbulence scales Grid resolution must be fine enough to capture the dissipation length scale - specific to that flow regime

(Note: This article will only introduce the concept of turbulent flow modelling as it pertains to single-phase, incompressible flow with a constant viscosity - any equations which have been included are meant purely to inform the information presented; you will not need to fully understand them!!)

Continuity Equation

Reynolds-Averaged Navier-Stokes Equation

Zero Equation Models Both the characteristic velocity and length scale are defined as algebraic equations

One Equation Models The characteristic velocity is defined as the square-root of the turbulent kinetic energy of the flow, with the characteristic length scale defined algebraically

Two Equation Models Use differential equations in order to calculate both the characteristic length and velocity and then use a supplementary “estimation equation” depending on which two equation model being used

Dissipation Rate of Turbulent Kinetic Energy (ε)

Specific Dissipation Rate (⍵)

Length Scale (l)

Time Scale (τ)

Leonard Term - The first two terms combined make up this term, they help to denote interactions between the scales of the flow which have been fully resolved; it can be calculated directly from the LES filtered field and so doesn’t require modeling

Cross Term - The third and fourth term together form this term and it is a measure of how much interaction there is between the large scale flow features and what is “filtered” out as being SGS

Reynolds Term - The final single term denotes how much interaction within the SGS there is with itself

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References:



N.C. Markatos, “The Mathematical Modelling of Turbulent Flows”, Appl. Math. Modell 10, 1986, pp. 190–220



B.E. Launder, “Heat and Mass Transport”, Chapter 6 in Turbulence, Topics in Applied Physics, 1978, pp. 231–287



C.G. Speziale, “Analytical Methods for the Development of Reynolds-Stress Closures in Turbulence”, 1991, pp. 107–157



R.A. Clark, J.H. Ferziger, W.C. Reynolds, “Evaluation of subgrid-scale models using an accurately simulated turbulent flow”, 1979, pp. 1–16