  A parallel pseudospectral code for the direct numerical simulation (DNS) of
isotropic turbulence has been developed. The code has been extensively
benchmarked using established results from literature.
  The code has been used to conduct a series of runs for freely-decaying
turbulence. We explore the use of power-law decay of the total energy to
determine an evolved time and compare with the use of dynamic quantities such
as the peak dissipation rate, maximum transport power and velocity derivative
skewness.
  Stationary turbulence has also been investigated, where we ensure that the
energy input rate remains constant for all runs. We present results for
Reynolds numbers up to R{\lambda} = 335 on a 1024^3 lattice. An exploitation of
the pseudospectral technique is used to calculate second and third-order
structure functions from the energy and transfer spectra, with a comparison
presented to the real-space calculation. An alternative to ESS is discussed,
with the second-order exponent found to approach 2/3.
  The dissipation anomaly is considered for forced and free-decay. The
K\'arm\'an-Howarth equation (KHE) is studied and a derivation of a new work
term presented. The balance of energy represented by the KHE is then
investigated. Based on the KHE, we develop a model for the behaviour of the
dimensionless dissipation coefficient that predicts C{\epsilon} =
C{\epsilon}(\infty) + C_L/R_L, with C{\epsilon}(\infty) = 0.47 and C_L = 19.1
obtained from DNS data.
  Theoretical methods based on RG and statistical closures are still being
developed to study turbulence. The dynamic RG procedure used by Forster, Nelson
and Stephen (FNS) is considered in some detail and a disagreement in the
literature is resolved here. The application of statistical closure and
renormalized perturbation theory is discussed and a new two-time model
probability density functional presented.
