The non-equilibrium part of the inertial range in decaying homogeneous turbulence

Abstract

We use two related non-stationarity functions as measures of the degree of scaleby-scale non-equilibrium in homogeneous isotropic turbulence. The values of these functions indicate significant non-equilibrium at the upper end of the inertial range. Wind tunnel data confirm Lundgren's (2002Lundgren's ( , 2003 prediction that the two-point separation r where the second and third order structure functions are closest to their Kolmogorov scalings is proportional to the Taylor length scale λ, and that both structure functions increasingly distance themselves from their Kolmogorov equilibrium form as r increases away from λ throughout the inertial range. With the upper end of the inertial range in non-equilibrium irrespective of Reynolds number, it is not possible to justify the Taylor-Kolmogorov turbulence dissipation scaling on the basis of Kolmogorov equilibrium.

Energy transfer in turbulent flows behind two side-by-side square cylinders

“… and ) are relatively small, implying that the flow becomes more homogeneous, in agreement with the evolution tendency of the one-point energy transfer. The observation that the maximum is found at is in accord with the case of decaying homogeneous turbulence (Obligado & Vassilicos 2019).

Figure 18.The average scale-by-scale energy transfer in (5.5) along ( a ) , ( b ) and ( c ) at .

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“…Recently, it has emerged that even at very high Reynolds numbers, the energy cascade process is non-equilibrium (Vassilicos 2015) with time lags between and (Goto & Vassilicos 2016) and therefore . In Obligado & Vassilicos (2019), it has been demonstrated in decaying homogeneous isotropic turbulence that the upper end of the inertial range is significantly non-equilibrium and the unsteady term is comparable to the dissipation term. It might be worth stressing that even in a stationary non-equilibrium turbulent flow with high Reynolds number, we should still expect a statistical balance between the temporal-average nonlinear term and the temporal-average dissipation term (e.g.…”

Analysis of Lundgren's matched asymptotic expansion approach to the Kármán-Howarth equation using the EDQNM turbulence closure

“…Obligado & Vassilicos [7] explained why there is no equilibrium cascade at length-scales large enough to justify a Taylor-Kolmogorov dissipation scaling in the case of freely decaying homogeneous isotropic turbulence (HIT) with classical Taylor-Kolmogorov dissipation scaling. The starting point is Von Karman & Howarth [8] equation in the form that it takes when expressed for structure functions as in Landau & Lifschitz [9]:…”

“…The function F is obtained by a normalised integration of the non-stationarity function f which is directly interpretable in terms of equilibrium as it is small at a given scale r and a given time t if ∂S 2 (r,t) ∂t is small compared to ε. The question which was raised by Obligado & Vassilicos [7] concerns the range of scales where one might safely assume F ≈ 0 and this is part of the questions which we now address in terms of EDQNM.…”

“…where S 2 is the second-order moment of the longitudinal velocity increment, S 3 is the third-order moment and f and F are non-stationarity functions [7]. These non-stationarity functions measure the departure from equilibrium scale by scale and appear naturally from the form of the Kármán-Howarth equation (1) given by Landau & Lifschitz [9] (see also [10,29]).…”