Exact relations between the moments of dissipation and longitudinal velocity derivatives in turbulent flows

Following an approach by Siggia, we present coefficients C(n) relating the moments of the dissipation of kinetic energy 〈ɛ〉 and the longitudinal velocity gradient 〈∂u(1)/∂x(1)〉 under the assumption of isotropy and continuity. Particularly, we find that the moment 〈ɛ(n)〉 of order n is completely determined by 〈(∂u(1)/∂x(1))(2n)〉 and an order- (and viscosity-) dependent coefficient for all n under the assumption of (local) isotropy. This implies that all theories which specify 〈ɛ(n)〉 also implicitly determine 〈(∂u(1)/∂x(1))(2n)〉 and vice versa. As a corollary to the direct connection between the moments of the dissipation and the longitudinal velocity gradient, the even standardized moments of order 2n of ∂u(1)/∂x(1) (flatness, hyperflatness, and so on) are directly related to the ratio of the moments 〈ɛ(n)〉/〈ɛ〉(n). We compare the theoretical values of the coefficients C(n) up to n=6 with homogeneous isotropic DNS data ranging from Re(λ)=88 to Re(λ)=529.