Unsteady turbulent plume models

Abstract

Four existing integral models of unsteady turbulent plumes are revisited. We demonstrate that none of these published models is ideal for general descriptions of unsteady behaviour and put forward a modified model. We show that the most recent (top-hat) plume model (Scase et al. J. Fluid Mech., vol. 563, 2006, p. 443), and the earlier (Gaussian) plume models (Delichatsios J. Fluid Mech., vol. 93, 1979, p. 241; Yu Trans. ASME, vol. 112, 1990, p.186), are all ill-posed. This ill-posedness arises from the downstream growth of short-scale waves, which have an unbounded downstream growth rate. We show that both the top-hat and the Gaussian (Yu) models can be regularized, rendering them well-posed, by the inclusion of a velocity diffusion term. The effect of including this diffusive mechanism is to include a vertical structure in the model that can be interpreted as representing the vertical extent of an eddy. The effects of this additional mechanism are small for steady applications, and cases where the plume forcing can be considered to follow a power law (both of which have been studied extensively). However, the inclusion of diffusion is shown to be crucial to the general initial-value problem for unsteady models.

Generalised unsteady plume theory

“…The use of a generalised plume theory, resulting in (2.9)-(2.11), circumvents these issues altogether and allows one to understand unsteady plume models in a broader context. Consequently, the theory provides a physics-based means of understanding and remedying the problems associated with existing unsteady plume models (for details of these problems see Scase & Hewitt (2012)). …”

“…To begin, we choose to display the mean buoyancy flux ,f ) display the data/predictions at times t n ≈ 1.95τ s n. The thick solid line corresponds to the DNS data. GPM refers to the Gaussian plume model described in § 6 of this paper, while TPM refers to the top-hat plume model described by Scase & Hewitt (2012).…”

“…DNS data and model prediction of the plume radius r m (a) and the flux-balance parameter Γ (b) at times t n ≈ 1.95τ s n. The thick solid line corresponds to the DNS data. GPM refers to the Gaussian plume model described in § 6 of this paper and TPM refers to the top-hat plume model described by Scase & Hewitt (2012). Note that the profiles in (b) are separated by a distance of 1 unit, as indicated by the scale in the bottom left corner.…”

“…From a modelling perspective it is necessary to understand whether an unsteady plume model is stable with respect to source perturbations. Scase & Hewitt (2012) showed that top-hat unsteady plume models are ill posed because they admit the unbounded (exponential) growth of high-frequency source perturbations, making it impossible to obtain convergence in numerical simulations of the governing integral equations. Using the framework described in this paper, it is possible to adopt a more general approach and analyse the response of the unsteady plume equations without making an assumption about the velocity profile, turbulent transport or pressure.…”

“…A notable virtue of the dispersion closure is that the incorporation of (6.5) does not affect the steady-state solutions. In contrast, the diffusive mixing term for momentum proposed by Scase & Hewitt (2012) was not extended to buoyancy transport, because it would have modified the exponents associated with the steady-state solutions.…”