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The accurate prediction of turbulent mixing induced by Rayleigh–Taylor (R–T), Richtmyer–Meshkov (R–M) and Kelvin–Helmholtz (K–H) instabilities is very important in understanding natural phenomena and improving engineering applications. In applications, the prediction of mixing with the Reynolds-averaged Navier–Stokes (RANS) equation remains the most widely used method. The RANS method involves two aspects, i.e. physical modelling and model coefficients. Generally, the latter is determined empirically; thus, there is a lack of universality. In this paper, inspired by the well-known Reynolds decomposition, we propose a methodology to determine the model coefficients with the following three steps: (i) preset a set of analytical RANS solutions by fully using the knowledge of mixing evolutions; (ii) simplify the differential RANS equations to algebraic equations by imposing the preset solutions to RANS equations; (iii) solve the algebraic equations approximately to give the values of the entire model coefficients. The specific application of this methodology in the widely used K–L mixing model shows that, using the same set of model coefficients determined from the current methodology, the K–L model successfully predicts the mixing evolutions in terms of different physical quantities (e.g. temporal scalings and spatial profiles), density ratios and problems (e.g. R–T, R–M, K–H and reshocked R–M mixings). It is possible to extend this methodology to other turbulence models characterised with self-similar evolutions, such as K-$\epsilon$ mixing models.
To explore whether and how group cognitive-behavioural therapy (GCBT) plus medication differs from medication alone for the treatment of generalised anxiety disorder (GAD).
Hundred and seventy patients were randomly assigned to the GCBT plus duloxetine (n=89) or duloxetine group (n=81). The primary outcomes were Hamilton Anxiety Scale (HAMA) response and remission rates. The explorative secondary measures included score reductions from baseline in the HAMA total, psychic, and somatic anxiety subscales (HAMA-PA, HAMA-SA), the Hamilton Depression Scale, the Severity Subscale of Clinical Global Impression Scale, Global Assessment of Functioning, and the 12-item Short-Form Health Survey. Assessments were conducted at baseline, 4-week, 8-week, and 3-month follow-up.
At 4 weeks, HAMA response (GCBT group 57.0% vs. control group 24.4%, p=0.000, Cohen’s d=0.90) and remission rates (GCBT group 21.5% vs. control group 6.2%, p=0.004; d=0.51), and most secondary outcomes (all p<0.05, d=0.36−0.77) showed that the combined therapy was superior. At 8 weeks, all the primary and secondary significant differences found at 4 weeks were maintained with smaller effect sizes (p<0.05, d=0.32−0.48). At 3-month follow-up, the combined therapy was only significantly superior in the HAMA total (p<0.045, d=0.43) and HAMA-PA score reductions (p<0.001, d=0.77). Logistic regression showed superiority of the combined therapy for HAMA response rates [odds ratio (OR)=2.12, 95% confidence interval (CI) 1.02−4.42, p=0.04] and remission rates (OR=2.80, 95% CI 1.27−6.16, p=0.01).
Compared with duloxetine alone, GCBT plus duloxetine showed significant treatment response for GAD over a shorter period of time, particularly for psychic anxiety symptoms, which may suggest that GCBT was effective in changing cognitive style.
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