Parameter estimation for complex thermal-fluid flows using approximate Bayesian computation

Approximate Bayesian computation (ABC) is a data-driven technique that uses many low-cost numerical simulations to estimate unknown physical or model parameters (e.g., boundary conditions and material properties), as well as their uncertainties, given reference data from real-world experiments or higher-fidelity numerical simulations. In this study, ABC is used to estimate unknown parameters in complex thermal-fluid flows, and the technique is demonstrated on a periodically forced high-temperature jet and a steadily forced helium-air plume. In the first case, computational reference data are used to assess the accuracy of ABC when estimating the frequency, amplitude, and mean of the periodic velocity forcing at the jet inlet. These tests show that ABC provides accurate and reasonably certain estimates of inflow parameters even when the model simulations imperfectly represent the physics underlying the reference data. These tests also show that measurements far from the inlet can be used to perform the estimation, and that temperature measurements can be used to infer velocity inflow parameters. For the second case, ABC is used to estimate the inlet Richardson number and its uncertainty given experimental measurements of the Strouhal number within the plume. Once again, the approach is able to accurately estimate unknown parameters with relatively low uncertainty. As a result, ABC is shown to be a versatile technique for estimating unknown physical parameters when knowledge of a real-world system is limited or incomplete.

Figure 1

Schematic of the general ABC approach corresponding to Method D from Marjoram et al. [28] and of the specific ABC implementation used in the present study.

Figure 2

Representative fields of speed $v$ (a) and temperature $T$ (d) from DNS for the periodically forced turbulent buoyant jet. Panels (b) and (c) show reference probability density functions (PDFs) and power spectral densities (PSDs), respectively, of $v$ at heights of 5, 10, 15, and 20 cm. PDFs and PSDs at different heights are shifted vertically for clarity. Panels (e) and (f) show corresponding PDFs and PSDs of $T$. Solid black lines in panels (b), (c), (e), and (f) show DNS reference data and dash-dot blue lines show LES reference data. PSDs are computed using Thomson's multitaper estimate [69].

Figure 3

Top row: Marginal posterior distributions from ABC (visualized using Gaussian kernel estimation) for the frequency (a, b), amplitude (c, d), and mean (e, f) of the periodically forced turbulent buoyant jet using speed (a, c, e) and temperature (b, d, f) reference data from LES. Posteriors are calculated using measurements at heights from 0 to 20 cm (indicated by colors from blue to yellow). True parameter values are shown by vertical black dashed lines. Horizontal dash-dot lines and gray regions show the priors. Bottom row: Vertical profiles of 95% Bayesian confidence intervals, $C_B$, for the marginal posterior distributions in all panels. Blue dash-dot lines and gray shading indicate the 95% Bayesian confidence regions, and the solid red lines show the mean values of the posteriors. True parameter values are shown by vertical black dashed lines.

Figure 4

Vertical profiles of Kullback-Leibler (KL) divergences for the marginal posterior distributions in Fig. 3 obtained using LES reference data. Divergences for the frequency (black solid lines), amplitude (red dash-dot lines), and mean (blue dashed lines) posteriors are shown for speed (a) and temperature (b) reference data. Each KL divergence vertical profile is normalized by its respective maximum value.

Figure 5

Top row: Marginal posterior distributions from ABC (visualized using Gaussian kernel estimation) for the frequency (a, b), amplitude (c, d), and mean (e, f) of the periodically forced turbulent buoyant jet using speed (a, c, e) and temperature (b, d, f) reference data from DNS. Posteriors are calculated using measurements at heights from 0 to 20 cm (indicated by colors from blue to yellow). True parameter values are shown by vertical black dashed lines. Horizontal dash-dot lines and gray regions show the priors. Bottom row: Vertical profiles of 95% Bayesian confidence intervals, $C_B$, for the marginal posterior distributions in all panels. Blue dash-dot lines and gray shading indicate the 95% Bayesian confidence regions, and the solid red lines show the mean values of the posteriors. True parameter values are shown by vertical black dashed lines.

Figure 6

Vertical profiles of Kullback-Leibler (KL) divergences for the marginal posterior distributions in Fig. 3 obtained using DNS reference data. Divergences for the frequency (black solid lines), amplitude (red dash-dot lines), and mean (blue dashed lines) posteriors are shown for speed (a) and temperature (b) reference data. Each KL divergence vertical profile is normalized by its respective maximum value.

Figure 7

Representative 2D fields of density (${\mathrm{kg}/\mathrm{m}}^3$) from LES at six different times (a–f) for the steadily forced plume described in Sec. 4. The fields shown begin at an arbitrary time $t_0$ late in the LES and are separated by a time interval spanning one complete period, $t_P$ ($t_P=1/f=0.22$ s).

Figure 8

Relationship between Strouhal number $\mathrm{St}=fw/v_i$ and inlet Richardson number $\mathrm{Ri}=(1-{\rho }_i/{\rho }_{\infty })gw/v_i^2$ for the steadily forced helium-air plume. Experimental results from Cetegen et al. [73] are indicated by open circles, and the empirical fit $\mathrm{St}=0.55{\mathrm{Ri}}^{0.45}$ is shown as a blue dash-dot line. ABC results are indicated by filled red circles located at the mean $\mathrm{Ri}$ of the posterior distribution. Uncertainty bars show the minimum and maximum values of $\mathrm{Ri}$ in the posteriors.