An analytical solution to electromagnetically coupled duct flow in MHD

The flow of an electrically conducting fluid in an array of square ducts, separated by arbitrary thickness conducting walls, subject to an applied magnetic field is studied. The analytical solution presented here is valid for thick walls and is based on the homogeneous solution obtained by Shercliff (Math. Proc. Camb. Phil. Soc., vol. 49 (01), 1953, pp. 136-144). Arrangements of ducts arise in a number of applications, most notably in fusion blankets, where liquid metal is used both as coolant and for tritium generation purposes. Analytical solutions, such as those presented here, provide insight into the physics and important benchmarking and validation data for computational magnetohydrodynamics (MHD), as well as providing approximate flow parameters for 1D systems codes. It is well known that arrays of such ducts with conducting walls exhibit varying degrees of coupling, significantly affecting the flow. An important practical example is the so-called Madarame problem (Madarame et al., Fusion Technol., vol. 8, 1985, pp. 264-269). In this work analytical results are derived for the relevant hydrodynamic and magnetic parameters for a single duct with thick walls analogous to the Hunt II case. These results are then extended to an array of such ducts stacked in the direction of the applied magnetic field. It is seen that there is a significant coupling affect, resulting in modifications to pressure drop and velocity profile. In certain circumstances, counter-current flow can occur as a result of the MHD effects, even to the point where the mean flow is reversed. Such phenomena are likely to have significant detrimental effects on both heat and mass transfer in fusion applications. The dependence of this coupling on parameters such as conductivities, wall thickness and Hartmann number is studied

Fixed inducing points online Bayesian calibration for computer models with an application to a scale-resolving CFD simulation

This paper proposes a novel fixed inducing points online Bayesian calibration (FIPO-BC) algorithm to efficiently learn the model parameters using a benchmark database. The standard Bayesian calibration (STD-BC) algorithm provides a statistical method to calibrate the parameters of computationally expensive models. However, the STD-BC algorithm does not scale well with regard to the number of data points and also it lacks an online learning capability. The proposed FIPO-BC algorithm greatly improves the computational efficiency of the algorithm and, in addition, enables online calibration to be performed by executing the calibration on a set of predefined inducing points. To demonstrate the procedure of the FIPO-BC algorithm, two tests are performed, finding the optimal value and exploring the posterior distribution of 1) the parameter in a simple function, and 2) the high-wave number damping factor in a scale-resolving turbulence model (scale adaptive simulation shear-stress transport model/SAS-SST). The results (such as the calibrated model parameter and its posterior distribution) of FIPO-BC with different inducing points are compared to those of STD-BC. It is found that FIPO-BC and STD-BC can provide very similar results, once the predefined set of inducing points in FIPO-BC is sufficiently fine. Given that fewer datapoints are needed in the proposed FIPO-BC algorithm, compared to the STD-BC algorithm, it will be a more computational efficient algorithm. In our demonstration test cases, the proposed FIPO-BC algorithm is at least ten times faster than the STD-BC algorithm. Meanwhile, the online feature of the FIPO-BC allows continuous updating of the calibration outputs and potentially reduces the workload on generating the database

An analytical solution to the heat transfer problem in Shercliff flow

The study of flow in a rectangular duct, subject to a strong transverse magnetic field is of interest in a number of applications. An important application of such flows is in the context of coolants, where the principle issue of interest is convective heat transfer. For fully developed laminar flows, the problem can be characterised in terms of two coupled partial differential equations. In the case of perfectly electrically insulating boundaries, there is a well known analytical solution due to Shercliff, which provides the velocity and induced magnetic field profiles. In this paper, we demonstrate analytical solutions to and heat transfer problems for the Shercliff case in rectangular ducts and obtain temperature profiles and corresponding Nusselt numbers as functions of aspect ratio and Hartmann number

An analytical solution to the heat transfer problem in thick-walled hunt flow

The flow of a liquid metal in a rectangular duct, subject to a strong transverse magnetic field is of interest in a number of applications. An important application of such flows is in the context of coolants in fusion reactors, where heat is transferred to a lead-lithium eutectic. It is vital, therefore, that the heat transfer mechanisms are understood. Forced convection heat transfer is strongly dependent on the flow profile. In the hydrodynamic case, Nusselt numbers and the like, have long been well characterised in duct geometries. In the case of liquid metals in strong magnetic fields (magnetohydrodynamics), the flow profiles are very different and one can expect a concomitant effect on convective heat transfer. For fully developed laminar flows, the magnetohydrodynamic problem can be characterised in terms of two coupled partial differential equations. The problem of heat transfer for perfectly electrically insulating boundaries (Shercliff case) has been studied previously (Bluck et al., 2015). In this paper, we demonstrate corresponding analytical solutions for the case of conducting hartmann walls of arbitrary thickness. The flow is very different from the Shercliff case, exhibiting jets near the side walls and core flow suppression which have profound effects on heat transfer

Quantification of the uncertainty within a SAS-SST simulation caused by the unknown high-wavenumber damping factor

This paper aims to quantify the uncertainty in the SAS-SST simulation of a prism bluff-body flow due to varying the higher-wavenumber damping factor (Cs). Instead of performing the uncertainty quantification on the CFD simulation directly, a surrogate modelling approach is adopted. The mesh sensitivity is first studied and the numerical error due to the mesh is approximated accordingly. The Gaussian processes/Kriging method is used to generate surrogate models for quantities of interest (QoIs). The suitability of the surrogate models is assessed using the leave-one-out cross-validation tests (LOO-CV). The stochastic tests are then performed using the crossvalidated surrogate models to quantify the uncertainty of QoIs by varying Cs. Four prior probability density functions (such as U(0, 1), N ( 0.5, 0.12) , Beta(2, 2) and Beta(5, 1.5)) of Cs are considered. It is demonstrated in this study that the uncertainty of a predicted QoI due to varying Cs is regionally dependent. The flow statistics in the near wake of the prism body are subject to larger variance due to the uncertainty in Cs. The influence of Cs rapidly decays as the location moves downstream. The response of different QoIs to the changing Cs varies greatly. Therefore, the calibration of Cs only using observations of one variable may bias the results. Last but not least, it is important to consider different sources of uncertainties within the numerical model when scrutinising a turbulence model, as ignoring the contributions to the total error may lead to biased conclusions

Do S-type granites commonly sample infracrustal sources? : new results from an integrated O, U-Pb and Hf isotope study of zircon

In contrast to I-type granites, which commonly comprise infracrustal and supracrustal sources, S-type granites typically incorporate predominantly supracrustal sources. The initial aim of this study was to identify the sources of three Scottish Caledonian (~460 Ma) S-type granites (Kemnay, Cove and Nigg Bay) by conducting oxygen, U–Pb and Hf isotope analyses in zircon in order to characterise one potential end-member magma involved in the genesis of the voluminous late Caledonian (~430–400 Ma) I-type granites. Field, whole-rock geochemical and isotopic data are consistent with the generation of the S-type granites by melting their Dalradian Supergroup country rocks. While Hf isotope compositions of magmatic zircon, U–Pb data of inherited zircons, and high mean zircon δ18O values of 9.0 ± 2.7‰ (2SD) and 9.8 ± 2.0‰ for the Kemnay and Cove granites support this model, the Nigg Bay Granite contains zircons with much lower δ18O values (6.8 ± 2.1‰), similar to those found in Scottish I-type granites. This suggests that the Nigg Bay Granite contains low-δ18O material representing either altered supracrustal material, or more likely, an infracrustal source component with mantle-like δ18O. Mixing trends in plots of δ18O vs. εHf for S-type granite zircons indicate involvement of at least two sources in all three granites. This pilot study of Scottish Caledonian S-type granites demonstrates that, while field and whole-rock geochemical data are consistent with local melting of only supracrustal sources, the oxygen isotopic record stored in zircon reveals a much more complex petrogenetic evolution involving two or more magma sources