Kraichnan’s random sweeping hypothesis in homogeneous turbulent convection

Abstract

We report an experimental study of the temperature space-time cross-correlation function, C T (r,τ ), in the central region of turbulent Rayleigh-Bénard convection. The measured C T (r,τ ) is found to have the scaling form C T (r E ,0) in turbulent Rayleigh-Bénard convection, where a fluid layer of thickness H is heated from below and cooled from the top. In the above, δT is the local temperature deviation from the mean and (σ T ) i is its standard deviation at position i. When the Rayleigh number [2] Ra 10 8 , the convective flow becomes turbulent. The flow in the closed convection cell is known to be inhomogeneous with a large-scale circulation (LSC) across the cell height [3]. In the rotation plane of LSC, the flow has a fly-wheel-like structure with a zero mean at the center and a linearly increasing velocity along the radial direction in the bulk region. After reaching its maximum value near the sidewall, the mean vertical velocity starts to drop quickly and becomes zero at the cell wall.As a result, the flow field near the sidewall is similar to that of a channel flow with a mean vertical velocity U 0 and a rms velocity σ v 0.6U 0 [3]. In this case, temperature is a passive scalar and follows the local flow [4,5]. Therefore the energy cascade picture can also be used to describe the spectrum of temperature variance [6], and C T (r,τ ) is expected to have the same functional form as the velocity counterpart C v (r,τ ). It was found [1] that the obtained C T (r,τ ) has the scaling formwith r E being of the elliptical form,where U is a characteristic convection velocity proportional to the local mean velocity U 0 and V is associated with a random sweeping velocity proportional to the local rms velocity σ v . Equation (3) incorporates both Taylor's frozen flow hypothesis [7,8] when V is small and Kraichnan's random sweeping hypothesis [9] for a homogenous and isotropic turbulent flow with a zero mean velocity. The experiment thus verified the theory by He and Zhang [10], in which Eq. (3) was derived for small values of r and τ and both U and V were calculated from the second derivatives of C v (r,τ ). The scaling theory by He and Zhang [10] has important practical implications for a large class of turbulent flows and thus it is essential to test the theory in different flow systems. The above experiment tested the theory in the sidewall region of a convective flow, where there still exists a dominant mean flow but the rms velocity is so large that Taylor's hypothesis [7] does not hold. In the experiment, r was varied only in the longitudinal direction along the mean flow. In this Brief Report, we present new measurements at the center of the convection cell, where the mean flow is zero and velocity fluctuations are approximately homogeneous [3]. Here we vary r in the lateral direction across a large-scale shear imposed by LSC. The experiment further confirms the theory by He and Zhang and demonstrates its applications to homogenous turbulent flows, where Kraichnan's hypothesis [9] holds approximately.The exper...

Space-time correlations in turbulent Rayleigh-Bénard convection

“…(2) and thus C T (r, 0) = C T (r E , 0). The same excellent agreement has also been found for the measured C T (r, τ)'s at the cell center [23]. The measured C T (r, τ) curves with different values of r and τ can all be brought into coincidence with a master curve C T (r E , 0), once r E is used as a scaling length.…”

“…(9), one finds that the ratio of the long axis to the short axis for each ellipse is directly related to the rms velocity V via a/b = V. With this relationship, one can obtain V directly from the iso-correlation contours as shown in Fig. 5b [23]. In general, one can obtain both the values of U and V from the measured C T (r, τ) by using Eq.…”

“…It was first tested in a turbulent RBC experiment using temperature T (x x x, t) as a passive scalar field [22,23]. It was then further verified in the velocity field [24] and applied to the shadowgraph image measurements [25] in RBC.…”

Reynolds numbers and the elliptic approximation near the ultimate state of turbulent Rayleigh–Bénard convection

“…We can use the EA to determine the velocities U and V from multi-point temperature measurements. This method had been tested previously and applied to RBC temperature data with Pr 5 ≃ and Ra 10 10 ≃ for samples with 1 Γ = [36,37], and it had been used for Pr 0.8 ≃ and 0.50 Γ =…”

“…Analogous derivations and self-similarity assumptions can be applied to a passive scalar, for instance to the temperature in the bulk of turbulent RBC (see [35] and references therein), and thus we were able to make velocity determinations from measurements of the temperature space-time correlation functions. This procedure was used before for smaller Ra and various Pr on several occasions [29,[36][37][38][39].In the present paper, after a description of the apparatus and some measurement procedures in section 2, we first discuss in section 3 measurements of the LSC circulation-plane orientation because this orientation influences U. Then, in section 4, we discuss relevant consequences of the EA.…”