The investigation of supersonic channel flow with periodic oscillatory backpressures at the outlet of the channel was performed using large-eddy simulation for the inlet free-stream Mach number 4 and the Reynolds number approximately $5.2 \times {10^4}$ based on the height of the channel. Results have been validated carefully against our experimental data. Three typical backpressures are considered for constant backpressure and both periodic oscillatory backpressures with low and high frequency. The oscillatory backpressure can obviously influence the flow features occurring up to the middle region of the channel for the low frequency case and the downstream region for the high frequency case. Obvious differences of phase-averaged quantities at different phases are observed for the low frequency backpressure while the differences are relatively small for the high frequency backpressure. The spectral analysis reveals that the flow field experiences a periodic-like evolution of flow structures including shocks and vortices for the low frequency backpressure, resulting in the enhancement of turbulence fluctuations due to the complicated interaction of shocks and vortices.

The stability of the flow under the magnetic force is one of the classical problems in fluid mechanics. In this paper, the flow in a rectangular duct with different Hartmann $(Ha)$ number is simulated. The finite volume method and the SIMPLE algorithm are used to solve a system of equations and the energy gradient theory is then used to study the (associated) stability of magnetohydrodynamics (MHD). According to the energy gradient theory, $K$ represents the ratio of energy gradient in transverse direction and the energy loss due to viscosity in streamline direction. Position with large $K$ will lose its stability earlier than that with small $K$. The flow stability of MHD flow for different Hartmann $(Ha)$ number, from $Ha=1$ to 40, at the fixed Reynolds number, $Re=190$ are investigated. The simulation is validated firstly against the simulation in literature. The results show that, with the increasing Ha number, the centerline velocity of the rectangular duct with MHD flow decreases and the absolute value of the gradient of total mechanical energy along the streamwise direction increases. The maximum of $K$ appears near the wall in both coordinate axis of the duct. According to the energy gradient theory, this position of the maximum of $K$ would initiate flow instability (if any) than the other positions. The higher the Hartmann number is, the smaller the $K$ value becomes, which means that the fluid becomes more stable in the presence of higher magnetic force. As the Hartmann number increases, the $K$ value in the parallel layer decreases more significantly than in the Hartmann layer. The most dangerous position of instability tends to migrate towards wall of the duct as the Hartmann number increases. Thus, with the energy gradient theory, the stability or instability in the rectangular duct can be controlled by modulating the magnetic force.

Nonlinear normal modes and a numerical iterative approach are applied to study the parametric vibrations of pipes conveying pulsating fluid as an example of gyroscopic continua. The nonlinear non-autonomous governing equations are transformed into a set of pseudo-autonomous ones by employing the harmonic balance method. The nonlinear normal modes are constructed by the invariant manifold method on the state space and a numerical iterative approach is adopted to obtain numerical solutions, in which two types of initial conditions for the modal coefficients are employed. The results show that both initial conditions can lead to fast convergence. The frequency-amplitude responses with some modal motions in phase space are obtained by the present iterative method. Quadrature phase difference and traveling waves are found in the time-domain complex modal analysis.

The Degasperis-Procesi (DP) equation is split into a system of a hyperbolic equation and an elliptic equation. For the hyperbolic equation, we use an optimized finite difference weighted essentially non-oscillatory (OWENO) scheme. New smoothness measurement is presented to approximate the typical shockpeakon structure in the solution to the DP equation, which evidently reduces the dissipation arising from discontinuities simultaneously removing nonphysical oscillations. For the elliptic equation, the Fourier pseudospectral method (FPM) is employed to discretize the high order derivative. Due to the combination of the WENO reconstruction and FPM, the splitting method shows an excellent performance in capturing the formation and propagation of shockpeakon solutions. The numerical simulations for different solutions of the DP equation are conducted to illustrate the high accuracy and capability of the method.

This paper proposes a stable and efficient implicit block Lower-Upper Symmetric-Gauss-Seidel (LU-SGS) algorithm-based lattice Boltzmann flux solver (LBFS) for simulation of hypersonic flows. In this method, the finite volume method (FVM) is applied to discretize the Navier-Stokes equations, and the LBFS is utilized to evaluate the numerical flux at the cell interface. In LBFS, the local solution of discrete velocity Boltzmann equation (DVBE) with the non-free parameter D1Q4 lattice Boltzmann model is adopted to reconstruct the inviscid flux across the cell interface, and the viscous flux is approximated by conventional smooth function approach. In order to improve the robustness and convergence rate of the simulation for hypersonic flows, especially for problems with complex geometry, the implicit block LU-SGS algorithm is introduced to solve resultant discrete governing equations. A double cone model at Mach number of $Ma=9.86$ is firstly simulated to validate the proposed scheme, and a hypersonic flight vehicle with wings and rudders at Mach number of $Ma=5.56$ is then calculated to extend the application in practical engineering problems. Numerical results show that the proposed scheme could offer a more accurate and effective prediction for hypersonic flows.

Data center plays an increasingly important role in everyday life. As data center is becoming more and more powerful, energy consumption is also increasing dramatically. The air conditioning system occupies at least 50 percent of the total energy consumption. Therefore, delicate analysis on the air conditioning system could help to reduce energy consumption in data center. An advanced Finite Volume Method with RNG $k-\varepsilon$ model and convective heat exchange model is used in this paper to study the airflow and the temperature distribution of modular data center under different arrangements. Specifically, the calculation formula of convective heat transfer coefficient for plate flow is adopted to simplify analysis; and fans on the back of racks are simplified to be walls with a certain pressure jump. Simulations reveal that, in the case where air conditioners are arranged face-to-face, the temperature distribution on the back of racks is not uniform, and local high temperature points emerge near the side wall of air conditioners. By analyzing the distribution of air flow and temperature, geometric model is optimized by using a diagonal rack arrangement and drilling holes on the side wall. In the same energy consumption situation, the overall maximum temperature of the optimized model is reduced by 2.3$^\circ$C compared with that of the original one, and the maximum temperature on the server surface is reduced by 1$^\circ$C. Based on the optimized model, the effect of the hot aisle distance on the temperature distribution is studied. By simulating four different cases with various distances of hot aisle of 100cm, 120cm, 130cm and 150cm, it is found that the temperature is generally lower and distributed more evenly in the case with 120cm hot aisle distance. This demonstrates that the distance of hot aisle has an effect on temperature.

The aim of this paper is to propose a fast meshless numerical scheme for the simulation of non-linear Schrödinger equations. In the proposed scheme, the implicit-Euler scheme is used for the temporal discretization and the localized method of approximate particular solution (LMAPS) is utilized for the spatial discretization. The multiple-scale technique is introduced to obtain the shape parameters of the multi-quadric radial basis function for 2D problems and the Gaussian radial basis function for 3D problems. Six numerical examples are carried out to verify the accuracy and efficiency of the proposed scheme. Compared with well-known techniques, numerical results illustrate that the proposed scheme is of merits being easy-to-program, high accuracy, and rapid convergence even for long-term problems. These results also indicate that the proposed scheme has great potential in large scale problems and real-world applications.

We propose an accurate and robust Roe-type scheme applied to the compressible Euler system at all Mach numbers. To study the occurrence of unstable modes during the shock wave computation, a shock instability analysis of several Roe-type schemes is carried out. This analysis approach allows proposing a simple and effective modification to eliminate shock instability of the Roe method for hypersonic flows. A desirable feature of this modification is that it does not resort to any additional numerical dissipation on linear degenerate waves to suppress the shock instability. With an all Mach correction strategy, the modified Roe-type scheme is further extended to solve flow problems at all Mach numbers. Numerical results that are obtained for various test cases indicate that the new scheme has a good performance in terms of accuracy and robustness.

This paper focuses on the development and application of a three-dimensional gas-kinetic Bhatnagar-Gross-Krook (BGK) method for the viscous flows in rotating machinery. For such flows, a rotating frame of reference is usually used in formulating the Navier-Stokes (N-S) equations, and there are two major concerns in constructing the corresponding BGK model. One is the change of the convective velocities in the N-S equations, which can be reflected through modification of the gas streaming velocity. The other one is the necessity to account for the effect of the additional Coriolis and centrifugal forces. Here, a specifically-designed acceleration term is added into the modified Boltzmann equation so that the source effects can be naturally included into the gas evolution process and the resulted fluxes. Under the finite-volume framework, the constructed BGK model is locally solved at each cell interface and then the numerical fluxes can be evaluated. When employing the BGK scheme, it is sometimes found that the calculated spatial derivatives of the initial and equilibrium distribution functions are sensitive to the mesh quality especially in complex rotating flow applications, which may significantly influence flux evaluation. Therefore, an improved approach for computing these slopes is adopted, through which the modeling capability for viscous flows is enhanced. For validation, several numerical examples are presented. The computed results show that the present method can be well applied to a wide range of flows in rotating machinery with favorable accuracy.

In this paper, we firstly present a novel simple method based on a Picard integral type formulation for the nonlinear multi-dimensional variable coefficient fourth-order advection-dispersion equation with the time fractional derivative order $\alpha\in (1,2)$. A new unknown function $v(\mathbf{x},t)=\partial u(\mathbf{x},t)/\partial t$ is introduced and $u(\mathbf{x},t)$ is recovered using the trapezoidal formula. As a result of the variable $v(\mathbf{x},t)$ is introduced in each time step, the constraints of traditional plans considering the non-integer time situation of $u(\mathbf{x},t)$ are no longer considered. The stability and solvability are proved with  detailed  proofs and the precise description of error estimates is derived. Further, Chebyshev spectral collocation method supports accurate and efficient variable coefficient model with variable coefficients. Several numerical results are obtained and analyzed in multi-dimensional spatial domains and numerical convergence order is consistent with the theoretical value $3-\alpha$ order for different $\alpha$ under infinite norm.

In this paper, a bilinear Petrov-Galerkin finite element method is introduced to solve the variable matrix coefficient elliptic equation with interfaces using non-body-fitted grid. Different cases the interface cut the cell are discussed. The condition number of the large sparse linear system is studied. Numerical results demonstrate that the method is nearly second order accurate in the $L^\infty$ norm and $L^2$ norm, and is first order accurate in the $H^1$ norm.

This paper is a generalization of some recent results concerned with the lower bound property of eigenvalues produced by both the enriched rotated $Q_1$ and Crouzeix-Raviart elements of the Stokes eigenvalue problem. The main ingredient is a novel and sharp $L^2$ error estimate of discrete eigenfunctions, and a new error analysis of nonconforming finite element methods.