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Modern methods for predicting combustion dynamics in high-pressure combustors range from high-fidelity simulations of sub-scale model combustors, mostly for validation purposes or detailed investigations of physics, to linearized, acoustics-based analysis of full-scale practical combustors. Whereas the high-fidelity simulations presumably capture the detailed physics of mixing and heat addition, computational requirements preclude their application for practical design analysis. The linear models that are used during design typically use flame transfer functions that relate the unsteady heat addition q ′ to oscillations in velocity and pressure ( u ′ and p ′ ) that are obtained from the wave equation. These flame transfer functions can be empirically determined from measurements or derived from theory and analysis. This paper describes a hybrid approach that uses high-fidelity simulations to generate flame transfer functions along with nonlinear Euler CFD to predict the combustor flowfield. A model rocket combustor that presented a self-excited combustion instability with pressure oscillations on the order of 10% of mean pressure is used for demonstration. Spatially distributed flame transfer functions are extracted from a high-fidelity simulation of the combustor and then used in a nonlinear Euler CFD model of the combustor to verify the approach. It is shown that the reduced-fidelity model can reproduce the unsteady behavior of the single element combustor that was both measured in the experiment and predicted by a high-fidelity simulation reasonably well.

In this paper, an implicit scheme is presented for a meshless compressible Euler solver based on the Least Square Kinetic Upwind Method (LSKUM). The Jameson and Yoon's split flux Jacobians formulation is very popular in finite volume methodology, which leads to a scalar diagonal dominant matrix for an efficient implicit procedure (Jameson & Yoon, 1987 ). However, this approach leads to a block diagonal matrix when applied to the LSKUM meshless method. The above split flux Jacobian formulation, along with a matrix-free approach, has been adopted to obtain a diagonally dominant, robust and cheap implicit time integration scheme. The efficacy of the scheme is demonstrated by computing 2D flow past a NACA 0012 airfoil under subsonic, transonic and supersonic flow conditions. The results obtained are compared with available experiments and other reliable computational fluid dynamics (CFD) results. The present implicit formulation shows good convergence acceleration over the RK4 explicit procedure. Further, the accuracy and robustness of the scheme in 3D is demonstrated by computing the flow past an ONERA M6 wing and a clipped delta wing with aileron deflection. The computed results show good agreement with wind tunnel experiments and other CFD computations.

To develop high performance computing methods for compressible flow calculation, a GPU-accelerated compressible flow solver is developed with Compute Unified Device Architecture (CUDA). The WENO5 scheme is adopted for spatial discretization, and the third-order Runge-Kutta scheme is used for time discretization. According to the algorithm and programming model, the heterogeneous computing method of the solver is designed. Different kernels are designed to implement different computing functions, and shared memory is used for time-advanced computations. The solver is verified by the one-dimensional shock tube case, and a good acceleration effect is obtained with the increase of the grid size. And the impact of execution configuration on kernel performance was investigated. When the block size is reduced under different grid sizes, the speedup changes in the same way, but the performance parameters change differently.

In this work we propose and demonstrate a Fresnel-lens-inspired method to focus multiple laser-induced shock waves through time-delay superposition at arbitrary locations. While the principle works for any geometry, we demonstrate that this method already achieves focusing with two pairs of photoacoustic shock wave emitters located on a line centred around the acoustic axis ($z$) in a quasi-two-dimensional liquid geometry. Each emitter pair is created by focusing one laser pulse simultaneously at two spots with a spatial light modulator at $z=0\\ \\mathrm {\\mu }$m with $y=\\pm 145\\ \\mathrm {\\mu }$m and $y=\\pm 75\\ \\mathrm {\\mu }$m. The delays between the emitters necessary to vary the location of the focus from $z\\approx 0$ to ${\\sim }206\\ \\mathrm {\\mu }$m are $35$ and $0$ ns, respectively. We find that the location of constructive superposition is significantly closer to the origin than what would be expected for linear waves in homogeneous media. This is confirmed with simulations using an Euler solver that shows the importance of finite-amplitude effects. The simulated dynamics are in reasonable agreement with our measurements. Finally, pressure gains at various locations along the acoustic axis are tested with the response of gaseous microbubbles acting as pressure probes. The measurements agree with calculated pressure ratios at different positions.

An investigation of shock diffraction through a non-quiescent background medium is presented using both experimental and numerical techniques. Unlike diffracting shocks in quiescent media, a spatial distortion of the shock front occurs, producing a region of constant shock angle. An example of this process arises in the exhaust from a pulse-detonation combustor. As the background velocity is increased, such as through the inclusion of a converging nozzle at the exhaust, the spatial distortion becomes more apparent. Numerical simulations using a compressible Euler solver demonstrate that the distortion is not due to the geometrical influence of the nozzle, but rather is a function of the magnitude of the background flow velocity. The distortion is studied using a modified geometrical shock dynamics formulation which includes the background flow and is validated against experiments. A simple model is presented to predict the shock distortion angle in the weak-shock limit. Finally, the axial decay behaviour of the shock is investigated and it is shown that the advection of the shock by the background flow delays the arrival of the head and tail of the expansion characteristic at the centreline. This leads to an increase in the rate of decay of the shock Mach number as the background flow velocity is increased.

This is the second part of our series papers on the deferred correction method for forward backward stochastic differential equations. In this work, we extend our previous work in Tang et al. (Numer Math Theory Methods Appl 10(2):222–242, 2017 ) to solve second-order forward backward stochastic differential equations (2FBSDEs). More precisely, we propose a class of explicit deferred correction schemes for 2FBSDEs. The key feature is that the simple Euler scheme is used as an initialization. Then, by a simple deferred correction iteration scheme, one can obtain an approximated solution with very high accuracy. Yet in each iteration, the computational complexity is always comparable to the Euler solver. Numerical examples are presented to show the effectiveness of the proposed scheme. We believe that the scheme proposed in this work is promising when dealing with 2FBSDEs in moderate dimensions.

Strong viscous interaction and multiple flow regimes exist when vehicles fly at high altitude and high Mach number conditions. The Navier–Stokes(NS) solver is no longer applicable in the above situation. Instead, the direct simulation Monte Carlo (DSMC) method or Boltzmann model equation solvers are usually needed. However, they are computationally more expensive than the NS solver. Therefore, it is of great engineering value to establish the aerodynamic prediction model of vehicles at high altitude and high Mach number conditions. In this paper, the hypersonic aerodynamic characteristics of an X38-like vehicle in typical conditions from 70 km to 110 km are simulated using the unified gas kinetic scheme (UGKS), which is applicable for all flow regimes. The contributions of pressure and viscous stress on the force coefficients are analyzed. The viscous interaction parameters, Mach number, and angle of attack are used as independent variables, and the difference between the force coefficients calculated by UGKS and the Euler solver is used as a dependent variable to establish a nonlinear viscous interaction model between them in the range of 70–110 km. The evaluation of the model is completed using the correlation coefficient and the relative orthogonal distance. The conventional viscous interaction effect and rarefied effect are both taken into account in the model. The model can be used to quickly obtain the hypersonic aerodynamic characteristics of X38-like vehicle in a wide range, which is meaningful for engineering design.

The formation process of reactive materials shaped charge is investigated by X-ray photographs and numerical simulation. In order to study the formation process, a trans-scale discretization method is proposed. A two-dimensional finite element model of shaped charge and reactive material liner is established and the jet formation process, granule size difference induced particle dispersion and granule distribution induced jet particle distribution are analyzed based on Autodyn-2D platform and Euler solver. The result shows that, under shock loading of shaped charge, the Al particle content decreases from the end to the tip of the jet, and increases as the particle size decreases. Besides, the quantity of Al particles at the bottom part of the liner has more prominent influence on the jet head density than that in the other parts, and the Al particle content in the high-speed section of jet shows inversely proportional relationship to the ratio of the particle quantity in the top area to that in the bottom area of liner.

In dredging applications, deep sea mining and land reclamation projects typically large amounts of sediments are transported through pipes in the form of hyper concentrated (40% sediment or more) sediment-water mixtures or slurries. In this paper it is investigated how well a generic Euler-Euler CFD-model is capable to model velocity, concentration profiles and the pressure gradient of sediment above deposition limit velocity in a pipeline. This Euler-Euler solver treats both phases as a continuum with its own momentum and continuity equations. The full kinetic theory for granular flows is accounted for (no algebraic form is used) and is combined with a buoyant turbulence model for the fluid phase. The influence of the mesh size has been checked and grid convergence is achieved. All numerical schemes used are of second-order accuracy in space. The pressure gradient was calibrated by adjusting the specularity coefficient in one calibration case and kept constant afterwards. Simulations were carried out in a wide range of slurry flow parameters, in situ volume concentration (9–42%), pipe diameter (0.05–0.90 m), particle diameter (90–440 μm) and flow velocity of (3–7 m/s). The model shows satisfactory agreement to experimental data from existing literature.

A reduced order modeling method based on algorithm fusion and multifidelity framework for nonlinear unsteady aerodynamics is proposed to obtain a low-cost and high-precision unsteady aerodynamic model. This method integrates the traditional algorithm, intelligent algorithm, and multifidelity data fusion algorithm. In this method, the traditional algorithm is based on separated flow theory, the intelligent algorithm refers to the nonlinear autoregressive (NARX) method, and the multifidelity data fusion algorithm uses different fidelity data for aerodynamic modeling, which can shorten the time cost of data acquisition. In the process of modeling, firstly, a multifidelity model with NARX description provides a general intelligent algorithm framework for unsteady aerodynamics. Then, based on the separated flow theory, the correction equation from low-fidelity model to high-fidelity result is constructed, and the cuckoo algorithm based on chaos optimization is used to identify the parameters. In order to verify the effectiveness of the method, an unsteady aerodynamic model of NACA0012 airfoil is established. Three kinds of data with low, medium, and high fidelity are used for modeling. The low-fidelity and medium-fidelity data is obtained from the CFD-Euler solver and CFD-RANS solver, respectively, while the high-fidelity data comes from the experimental results. Then, the model is established, and its prediction of unsteady aerodynamic coefficients is in good agreement with the CFD results and the experimental data. After that, the model is applied to a two-dimensional aeroelastic system, and the bifurcation and limit cycle response analysis are compared with the experimental results, which further shows that the model can accurately capture the main flow characteristics in the flow range of low speed and high angle of attack. In addition, the convergence of the model is studied; the accuracy and generalization ability as well as applicability scope of the model are compared with other aerodynamic models and finally discussed.