Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and first explored by Deardorff (1970). LES is currently applied in a wide variety of engineering applications, including combustion, acoustics, and simulations of the atmospheric boundary layer. The simulation of turbulent flows by numerically solving the Navier–Stokes equations requires resolving a very wide range of time and length scales, all of which affect the flow field. Such a resolution can be achieved with direct numerical simulation (DNS), but DNS is computationally expensive, and its cost prohibits simulation of practical engineering systems with complex geometry or flow configurations, such as turbulent jets, pumps, vehicles, and landing gear. The principal idea behind LES is to reduce the computational cost by ignoring the smallest length scales, which are the most computationally expensive to resolve, via low-pass filtering of the Navier–Stokes equations. Such a low-pass filtering, which can be viewed as a time- and spatial-averaging, effectively removes small-scale information from the numerical solution. This information is not irrelevant, however, and its effect on the flow field must be modelled, a task which is an active area of research for problems in which small-scales can play an important role, such as near-wall flows, reacting flows, and multiphase flows. Filter definition and properties An LES filter can be applied to a spatial and temporal field and perform a spatial filtering operation, a temporal filtering operation, or both. The filtered field, denoted with a bar, is defined as: where is the filter convolution kernel. This can also be written as: The filter kernel has an associated cutoff length scale and cutoff time scale . Scales smaller than these are eliminated from . Using the above filter definition, any field may be split up into a filtered and sub-filtered (denoted with a prime) portion, as It is important to note that the large eddy simulation filtering operation does not satisfy the properties of a Reynolds operator. Filtered governing equations The governing equations of LES are obtained by filtering the partial differential equations governing the flow field . There are differences between the incompressible and compressible LES governing equations, which lead to the definition of a new filtering operation. Incompressible flow For incompressible flow, the continuity equation and Navier–Stokes equations are filtered, yielding the filtered incompressible continuity equation, and the