Statistical properties of four-dimensional turbulence

The energy transfer and small-scale intermittency in decaying turbulence in four dimensions (4D) are studied by direct numerical simulation and by spectral theory in comparison with three dimensions (3D). The energy transfer is more efficient in 4D than in 3D, hence the exponent of energy decay is larger. The Kolmogorov constant is 1.31, which is smaller than 1.72 in 3D. The longitudinal third-order structure function is confirmed to be governed by a 1/2 law, -(1/2)(epsilon) over barr, instead of a 4/5 law in 3D. The intermittency is weaker in 4D for the total dissipation rate nu Sigma(i,j)(partial derivative u(j)/partial derivative x(i))(2) and the associated velocity difference (spherical velocity difference) on scale r than in 3D, while it is slightly stronger for the surrogated dissipation rate nu(partial derivative u(1)/partial derivative x(1))(2) and the associated longitudinal velocity difference. The scaling exponents of the spherical and longitudinal velocity differences are also evaluated, indicating that the spherical velocity difference is less intermittent in 4D, while the longitudinal difference is more intermittent in 4D. The distribution of the eigenvalues of the strain tensor is also examined. It was also found that the normalized variance of the pressure gradient (epsilon) over bar (-3/2)nu(-1/2)<(del p)(2)> in 4D is smaller than in 3D. The roles of the incompressibility condition, the pressure gradient, and the intermittency in d-dimensional turbulence are examined, and the importance of the longitudinal component of turbulent velocity field in the energy transfer toward small scales are discussed. Burgers turbulence as the asymptote of turbulence in large dimension is suggested.