In this paper, we develop an efficient Boltzmann-equation-based mesoscopic approach to simulate three-dimensional (3D) compressible turbulence, using reduced Gauss-Hermite quadrature (GHQ) orders by redefining the second distribution in terms of the total energy in the double distribution function approach. This allows the use of two sets of 3D off-lattice discrete particle velocity models, namely, a 27 discrete velocity model of the seventh-order GHQ accuracy (D3V27A7) combined with a 13 discrete velocity model of the fifth-order GHQ accuracy (D3V13A5), to achieve full consistency with the Navier-Stokes-Fourier system. The source terms in the Boltzmann-Bhatnagar-Gross-Krook system are designed to adjust both the Prandtl number and bulk-to-shear viscosity ratio. Compressible decaying homogeneous isotropic turbulence (DHIT) is simulated at low and moderate turbulent Mach numbers to validate our code. It is observed that the simulation results are in good agreement with those in the existing literatures. Furthermore, the terms in the transport equation of turbulent kinetic energy are analyzed in detail, to illustrate four different transient stages from the initial random flow field to the developed DHIT. It is shown that the transient pressure-dilatation transfer happens rapidly, while the small-scale vortical structures take a longer time to establish physically. Compared to the existing literatures, our approach represents the most efficient mesoscopic scheme for compressible turbulence under the double distribution function formulation.

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In this paper, we implemented the Boltzmann-equation-based mesoscopic model, developed recently by Chen et al. [“Inverse design of mesoscopic models for compressible flow using the Chapman–Enskog analysis,” Adv. Aerodyn. 3, 5 (2021)], to simulate three-dimensional (3D) forced compressible isotropic turbulence. In this model, both the Prandtl number and the ratio of bulk to shear viscosity can be arbitrary prescribed. The statistically stationary turbulent flow is driven by a large-scale momentum forcing in the Fourier space, with the internal heating due to the viscous dissipation at small scales being removed by a thermal cooling function. Under the framework of discrete unified gas kinetic scheme (DUGKS), a 3D direct numerical simulation code has been developed, incorporating a generalized Strang-splitting scheme. The weighted essentially non-oscillatory (WENO) scheme is used to increase local spatial accuracy in the reconstruction of particle distribution functions at the cell interface. A 3D discrete particle velocity model with a ninth-order Gauss–Hermite quadrature accuracy is used to ensure accurate evaluation of viscous stress and heat flux in the continuum regime. We simulate forced compressible isotropic turbulence at both low and high turbulent Mach numbers. A direct comparison is performed with the results obtained from a hybrid compact finite difference-WENO scheme solving directly the Navier–Stokes–Fourier system. The comparison validates our DUGKS code and indicates that DUGKS is a reliable and promising tool for simulating forced compressible isotropic turbulence. The work represents a first study to directly simulate forced compressible turbulence by a mesoscopic method based on the Boltzmann equation.

The relationship between skin friction and the surface optical flow (SOF) in viscous flows is discussed based on the evolution equations of surface temperature, scalar and enstrophy where the SOF is defined as the convection velocity of these quantities. It is found that the SOF is proportional to skin friction, which can be determined by solving the optical flow equation re-cast from these evolution equations. This optical flow method can be applied to surface temperature and mass transfer visualizations to extract skin friction fields in experiments. To examine this method, it is first applied to complex surface enstrophy structures obtained in direct numerical simulation (DNS) data of a turbulent channel flow. Further, it is applied to surface temperature structures obtained in time-resolved temperature sensitive paint (TSP) measurements in a flow over a National Advisory Committee for Aeronautics (NACA) 0015 airfoil model and an impinging jet.

In order to investigate the mechanism of the low‐density solid particle and liquid mixing process, a specialised agitator structure was used. Both computational fluid dynamics simulation and experiments were carried out to study the two‐phase mixing characteristics in the stirred vessel. The mixing process was captured by snapshots. The flow field and solid phase volume fraction evolution were analysed. Experimental and numerical results agreed well with each other. Solid particles floating on the liquid surface were gradually transported to the bottom through the centre of the vessel and the mixing time was predicted and tested. Results indicate that the agitator structure used in this study is able to form an obvious axial circulation in the vessel and then achieve a good performance in low‐density solid and liquid mixing operations. The study provides a valuable reference for the design and optimisation of solid–liquid mixing equipment. © 2011 Canadian Society for Chemical Engineering