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The joint probability density function (pdf) of a scalar and its gradient are investigated. Under isotropic conditions, a relation is derived between this pdf and the joint pdf of a scalar and its x derivative. Physical properties of the scalar derivative are analyzed quantitatively to show that its pdf conditional on a given scalar value is close to a Gaussian distribution. The conditional pdf of the scalar derivative is then expanded according to the Gram–Charlier expansion. The coefficients of this expansion are evaluated using direct numerical simulation (DNS) of stationary turbulent scalar fields. Results obtained from a number of simulations support the assumption of a near-Gaussian distribution for the scalar derivative. Finally, the intermittency of the scalar gradient is discussed.
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Gao et al. (1991) studied this question.
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