shown in figure 23 for the case τ −1 J = 4. The computations are performed with the resolution M = 5. One can see the dependence of ν E on the basic flow configuration. It was found that at τ −1 J < 0.5 the most stable basic flow has U = V = 0 and W = (2E 0 ) 1/2 (cos x + cos y).At τ −1 J > 0.5 the most stable configuration is with b = 0. This provides the possibility of solving the eigenvalue problem (A 11), (A 12) much more accurately. The coefficients of the equations depend only on x in this case and one-dimensional Fourier expansion can be used instead of (A 13).

We describe a noncontact technique for velocity measurement in electrically conducting fluids. The technique, which we term Lorentz force velocimetry (LFV), is based on exposing the fluid to a magnetic field and measuring the drag force acting upon the magnetic field lines. Two series of measurements are reported, one in which the force is determined through the angular velocity of a rotary magnet system and one in which the force on a fixed magnet system is measured directly. Both experiments confirm that the measured signal is a linear function of the flow velocity. We then derive the scaling law that relates the force on a localized distribution of magnetized material to the velocity of an electrically conducting fluid. This law shows that LFV, if properly designed, has a wide range of potential applications in metallurgy, semiconductor crystal growth, and glass manufacturing.

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Direct numerical simulation is applied to investigate instability and transition to turbulence in the flow of an electrically conducting incompressible fluid between two parallel unbounded insulating walls affected by a wall-normal magnetic field (the Hartmann flow). The linear stability analysis of this flow provided unrealistically high critical Reynolds numbers, about two orders of magnitude higher than those observed in experiments. We propose an explanation based on the streak growth and breakdown mechanism described earlier for other shear flows. The mechanism is investigated using a two-step procedure that includes transient growth of two-dimensional optimal perturbations and the subsequent three-dimensional instability of the modulated streaky flow. In agreement with recent experimental investigations the calculations produce a critical range between 350 and 400 for the Hartmann thickness based Reynolds number, where the transition occurs at realistic amplitudes of two-and three-dimensional perturbations.

We report high-resolution local-temperature measurements in the upper boundary layer of turbulent Rayleigh–Bénard (RB) convection with variable Rayleigh number Ra and aspect ratio Γ. The primary purpose of the work is to create a comprehensive data set of temperature profiles against which various phenomenological theories and numerical simulations can be tested. We performed two series of measurements for air (Pr = 0.7) in a cylindrical container, which cover a range from Ra≈109 to Ra≈1012 and from Γ≈1 to Γ≈10. In the first series Γ was varied while the temperature difference was kept constant, whereas in the second series the aspect ratio was set to its lowest possible value, Γ=1.13, and Ra was varied by changing the temperature difference. We present the profiles of the mean temperature, root-mean-square (r.m.s.) temperature fluctuation, skewness and kurtosis as functions of the vertical distance z from the cooling plate. Outside the (very short) linear part of the thermal boundary layer the non-dimensional mean temperature Θ is found to scale as Θ(z)∼zα, the exponent α≈0.5 depending only weakly on Ra and Γ. This result supports neither Prandtl's one-third law nor a logarithmic scaling law for the mean temperature. The r.m.s. temperature fluctuation σ is found to decay with increasing distance from the cooling plate according to σ(z)∼zβ, where the value of β is in the range -0.30>β>-0.42 and depends on both Ra and Γ. Priestley's β=−1/3 law is consistent with this finding but cannot explain the variation in the scaling exponent. In addition to profiles we also present and discuss boundary-layer thicknesses, Nusselt numbers and their scaling with Ra and Γ.