A comprehensive review of all relevant experimental data was completed, including recent data for the drag coefficient for a sphere in supersonic and hypersonic flows. The primary characterization parameter included the relative Mach, Knudsen, and Reynolds numbers based on the relative velocity, the sphere diameter, and other parameters. This review of data showed that the previously proposed nexus at a Reynolds number below 45 was not strictly met, and it instead included a weak transonic bump, which was identified numerically for the first time with the present simulations. New continuum-gas and rarefied-gas simulations were conducted and were combined with the expanded experimental dataset to improve the quantitative description of the drag coefficient in this region. The results indicated that a quasi nexus bridges the rarefaction regime and the compressible flow regimes. The comprehensive dataset was then used to develop new empirical models for the drag coefficient that showed improved robustness and accuracy as compared to previous models. These models are limited by the critical Reynolds number associated with boundary-layer transition on the sphere, which was found to increase substantially with the sphere Mach number.

In this study, analysis of flow properties around a sphere and its aerodynamic coefficients in the high-Mach-and-low-Reynolds-numbers conditions is carried out by direct numerical simulations solving the three-dimensional compressible Navier–Stokes equations. The calculation is performed on a boundary-fitted coordinate system with a high-order scheme of sufficient accuracy. The analysis is conducted by assuming a rigid sphere with a Reynolds number of between 50 and 300, based on the diameter of the sphere and the freestream velocity and a freestream Mach number of between 0.3 and 2.0, together with the adiabatic wall boundary condition. The calculation shows the following yields: (1) unsteady fluctuation of hydrodynamic forces become smaller as the Mach number increases under the same Reynolds number condition, (2) the drag coefficient increases with the Mach number due to an increase in the pressure drag by the shock wave, and (3) an accurate prediction of the drag coefficient in the supersonic regime using traditional models might be difficult.

In this paper, the sparse sensor placement problem for least-squares estimation is considered, and the previous novel approach of the sparse sensor selection algorithm is extended. The maximization of the determinant of the matrix which appears in pseudo-inverse matrix operations is employed as an objective function of the problem in the present extended approach. The procedure for the maximization of the determinant of the corresponding matrix is proved to be mathematically the same as that of the previously proposed QR method when the number of sensors is less than that of state variables (undersampling). On the other hand, the authors have developed a new algorithm for when the number of sensors is greater than that of state variables (oversampling). Then, a unified formulation of the two algorithms is derived, and the lower bound of the objective function given by this algorithm is shown using the monotone submodularity of the objective function. The effectiveness of the proposed algorithm on the problem using real datasets is demonstrated by comparing with the results of other algorithms. The numerical results show that the proposed algorithm improves the estimation error by approximately 10% compared with the conventional methods in the oversampling case, where the estimation error is defined as the ratio of the difference between the reconstructed data and the full observation data to the full observation. For the NOAA-SST sensor problem, which has more than ten thousand sensor candidate points, the proposed algorithm selects the sensor positions in few seconds, which required several hours with the other algorithms in the oversampling case on a 3.40 GHz computer.

The problem of selecting an optimal set of sensors estimating a high-dimensional data is considered. Objective functions based on D-, A-, and E-optimality criteria of optimal design are adopted to greedy methods, that maximize the determinant, minimize the trace of the inverse, and maximize the minimum eigenvalue of the Fisher information matrix, respectively. First, the Fisher information matrix is derived depending on the numbers of latent state variables and sensors. Then, a unified formulation of the objective function based on A-optimality is introduced and proved to be submodular, which provides the lower bound on the performance of the greedy method. Next, greedy methods based on D-, A-, and E-optimality are applied to randomly generated systems and a practical dataset concerning the global climate; these correspond to an almost ideal and a practical case in terms of statistics, respectively. The D- and A-optimality-based greedy methods select better sensors. The E-optimality-based greedy method does not select better sensors in terms of the index of E-optimality in the oversample case, while the A-optimality-based greedy method unexpectedly does so in terms of the index of E-optimality. The poor performance of the E-optimality-based greedy method is due to the lack of submodularity in the E-optimality index and the better performance of the A-optimality-based greedy method is due to the relation between A- and E-optimality. Indices of D- and A-optimality seem to be important in the ideal case where the statistics for the system are well known, and therefore, the D- and A-optimality-based greedy methods are suitable for accurate reconstruction. On the other hand, the index of E-optimality seems to be critical in the more practical case where the statistics for the system are not well known, and therefore, the A-optimality-based greedy method performs best because of its superiority in terms of the index of E-optimality.

In the present study, a compressible low-Reynolds-number flow over a circular cylinder was investigated using a low-density wind tunnel with time-resolved schlieren visualizations and pressure and force measurements. The Reynolds number ( effect on the drag coefficient can be characterized by the maximum width of the recirculation region and the Prandtl–Glauert transformation.

The present study proposes a sensor selection method based on the proximal splitting algorithm and the A-optimal design of experiment using the alternating direction method of multipliers (ADMM) algorithm. The performance of the proposed method was evaluated with a random sensor problem and compared with previously proposed methods, such as the greedy and convex relaxation methods. The performance of the proposed method is better than the existing greedy and convex relaxation methods in terms of the A-optimality criterion. Although, the proposed method requires a longer computational time than the greedy method, it is quite shorter than that of convex relaxation method in large-scale problems. Then the proposed method was applied to the data-driven sparse-sensor-selection problem. The dataset adopted was the National Oceanic and Atmospheric Administration optimum interpolation sea surface temperature dataset. At a number of sensors larger than that of the latent variables, the proposed method showed similar and better performance compared with previously proposed methods in terms of the A-optimality criterion and reconstruction error.