Numerical simulation of compressible turbulent flow via improved gas‐kinetic BGK scheme

In order to solve compressible turbulent flow problems, this study focuses on incorporating the Spalart–Allmaras turbulence model into gas‐kinetic BGK (Bhatnagar–Gross–Krook) scheme. The Spalart–Allmaras turbulence model is solved using finite difference discretization. The variables on the cell interface are interpolated via the van Leer limiter in the reconstruction stage. Simulation of subsonic and transonic flow over a NACA0012 airfoil has been implemented using two‐dimensional body‐fitted grids. The numerical results obtained appear in good agreement with the AGARD results, demonstrating the effectiveness and usefulness of the strategy of coupling the Spalart–Allmaras turbulence model with the BGK scheme for compressible turbulent flow simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

Compact finite difference schemes on non‐uniform meshes. Application to direct numerical simulations of compressible flows

In this paper, the development of a fourth‐ (respectively third‐) order compact scheme for the approximation of first (respectively second) derivatives on non‐uniform meshes is studied. A full inclusion of metrics in the coefficients of the compact scheme is proposed, instead of methods using Jacobian transformation. In the second part, an analysis of the numerical scheme is presented. A numerical analysis of truncation errors, a Fourier analysis completed by stability calculations in terms of both semi‐ and fully discrete eigenvalue problems are presented. In those eigenvalue problems, the pure convection equation for the first derivative, and the pure diffusion equation for the second derivative are considered. The last part of this paper is dedicated to an application of the numerical method to the simulation of a compressible flow requiring variable mesh size: the direct numerical simulation of compressible turbulent channel flow. Present results are compared with both experimental and other numerical (DNS) data in the literature. The effects of compressibility and acoustic waves on the turbulent flow structure are discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Residual‐based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods

In this paper, we present a finite element method with a residual‐based artificial viscosity for simulation of turbulent compressible flow, with adaptive mesh refinement based on a posteriori error estimation with sensitivity information from an associated dual problem. The artificial viscosity acts as a numerical stabilization, as shock capturing, and as turbulence capturing for large eddy simulation of turbulent flow. The adaptive method resolves parts of the flow indicated by the a posteriori error estimates but leaves shocks and turbulence under‐resolved in a large eddy simulation. The method is tested for examples in 2D and 3D and is validated against experimental data. Copyright © 2012 John Wiley & Sons, Ltd.     

An attempt to improve accuracy of higher‐order statistics and spectra in direct numerical simulation of incompressible wall turbulence by using the compact schemes for viscous terms

We attempt to improve accuracy in the high‐wavenumber region in DNS of incompressible wall turbulence such as found in fully developed turbulent channel flow. In particular, it is shown that the improvement of accuracy of viscous terms in the Navier–Stokes equations leads to the improvement of accuracy of higher‐order statistics and various spectra. It is emphasized that increase in required computational cost will not be crucial when incompressible flow is simulated, because the introduction of a higher‐order scheme into the viscous terms does not increase computational cost for solving the Poisson equation. We introduced fourth‐order and eighth‐order central compact schemes for discretizing the viscous terms in DNS of a fully developed turbulent channel flow. The results are compared with those using second‐order and fourth‐order central‐difference schemes applied to the viscous terms and those obtained by the spectral method. The results show that accuracy improvement of the viscous terms improve accuracy of higher‐order statistics (i.e., skewness and flatness factors of streamwise velocity fluctuation) and various spectra of velocity and pressure fluctuations in the high‐wavenumber region. Copyright © 2013 John Wiley & Sons, Ltd.     

Pressure‐based unified solver for gas‐liquid two‐phase flows where compressible and incompressible flows coexist

We propose a pressure‐based unified solver for gas‐liquid two‐phase flows where compressible and incompressible flows coexist. Unlike the original thermo–Cubic Interpolated Propagation Combined Unified Procedure (CIP‐CUP) method proposed by Himeno et al (Transactions of the Japan Society of Mechanical Engineers, Series B, 2003), we split the advection term of the governing equations into a conservation part and into the rest. The splitting of advection term has two advantages. One is the high degree of freedom in choosing discretization schemes such as central‐difference schemes, upwind schemes, and Total Variation Diminishing (TVD) schemes. The other is the ease of implementation on unstructured grids. The advantages enable the analyses of various flows such as turbulent and supersonic ones in actual complicated boundaries. Therefore, the solver is useful for practical analyses. The solver was validated on the following test cases: subsonic single‐phase flows, incompressible single‐phase turbulent flows, and incompressible gas‐liquid two‐phase flows. With unstructured grids, we obtained the equivalent results as the ones with structured grids. After the validations, subsonic jet impinging on a water pool was calculated and compared with experimental results. It was confirmed that the calculated results were consistent with the experimental ones.     

Optimized group velocity control scheme and DNS of decaying compressible turbulence of relative high turbulent Mach number

To overcome the difficulty in the DNS of compressible turbulence at high turbulent Mach number, a new difference scheme called GVC8 is developed. We have succeeded in the direct numerical simulation of decaying compressible turbulence up to turbulent Mach number 0.95. The statistical quantities thus obtained at lower turbulent Mach number agree well with those from previous authors with the same initial conditions, but they are limited to simulate at lower turbulent Mach numbers due to the so‐called start‐up problem. The energy spectrum and coherent structure of compressible turbulent flow are analysed. The scaling law of compressible turbulence is studied. The computed results indicate that the extended self‐similarity holds in decaying compressible turbulence despite the occurrence of shocklets, and compressibility has little effects on relative scaling exponents when turbulent Mach number is not very high. Copyright © 2005 John Wiley & Sons, Ltd.     

A high-order scheme based on lattice Boltzmann flux solver for viscous compressible flow simulations

Applied Mathematics and Mechanics - In this paper, a high-order scheme based on the lattice Boltzmann flux solver (LBFS) is proposed to simulate viscous compressible flows. The flux reconstruction...     

Direct simulation of breakdown to turbulence following oblique instability waves in a supersonic boundary layer

The late stages of transition to turbulence in a Mach two boundary layer are investigated by direct numerical simulation of the compressible Navier-Stokes equations. The primary instability at this Mach number consists of oblique waves, which are known to form a pattern of quasi-streamwise vortices. It is found that breakdown does not follow immediately from these vortices, which decay in intensity. The generation of new vortices is observed by following the evolution of the pressure and vorticity in the simulation, and analysed by consideration of vorticity stretching. It is found that the slight inclined and skewed nature of the quasi-streamwise vortices leads to a production of oppositely signed streamwise vorticity, which serves as a strong localised forcing of the shear layer alongside the original vortices, formed by convection and stretching of spanwise vorticity. The shear layer rolls up into many new vortices, and is followed by a sharp increase in the energy of higher frequencies and in the skin friction.     

A γ‐model BGK scheme for compressible multifluids

We present a γ‐model BGK scheme for the numerical simulation of compressible multifluids. The scheme is based on the incorporation of a conservative γ‐model scheme given in (J. Comput. Phys. 1996; 125 :150–160) into the gas kinetic BGK scheme (J. Comput. Phys. 1993; 109 :53–66, J. Comput. Phys. 1994; 114 :9–17), and is simple to implement. Several numerical examples presented in this paper validate the scheme in the application of compressible multimaterial flows. Copyright © 2004 John Wiley & Sons, Ltd.     

A gas‐kinetic scheme for the modified Baer–Nunziato model of compressible two‐phase flow

Numerical methods for the Baer–Nunziato model of compressible two‐phase flow have attracted much attention in recent years. In this paper, a two‐phase Bhatnagar–Gross–Krook (BGK) model is constructed in which the non‐conservative terms in the Baer–Nunziato model are considered as the external forces and the collisions both with particles of their phases and other phases are taken into consideration. On the basis of this BGK model, the so‐called modified Baer–Nunziato model is derived and a gas‐kinetic scheme for this modified model is presented. The distribution functions are constructed at the cell interface based on the integral solutions of the BGK equations for both phases. Then, numerical fluxes can be obtained by taking moments of the distribution functions, and non‐conservative terms are explicitly introduced into the construction of numerical fluxes. In this method, not only the iterative processes in the exact Riemann solvers are eliminated but also the collisions with the particles of other phases are taken into account. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparison of variants of the bi-conjugate gradient method for compressible Navier-Stokes solver with second-moment closure

Variants of the bi-conjugate gradient (Bi-CG) method are used to resolve the problem of slow convergence in CFD when it is applied to complex flow field simulation using higher-order turbulence models. In this study the Navier-Stokes and Reynolds stress transport equations are discretized with an implicit, total variation diminishing (TVD), finite volume formulation. The preconditioning technique of incomplete lower-upper (ILU) factorization is incorporated into the conjugate gradient square (CGS), bi-conjugate gradient stable (Bi-CGSTAB) and transpose-free quasi-minimal residual (TFQMR) algorithms to accelerate convergence of the overall itertive methods. Computations have been carried out for separated flow fields over transonic bumps, supersonic bases and supersonic compression corners. By comparisons of the convergence rate with each other and with the conventional approximate factorization (AF) method it is shown that the Bi-CGSTAB method gives the most efficient convergence rate among these methods and can speed up the CPU time by a factor of 2·4–6·5 as compared with the AF method. Moreover, the AF method may yield somewhat different results from variants of the Bi-CG method owing to the factorization error which introduces a higher level of convergence criterion.     

Numerical flux functions for Reynolds‐averaged Navier–Stokes and kω turbulence model computations with a line‐preconditioned p‐multigrid discontinuous Galerkin solver

We present an eigen‐decomposition of the quasi‐linear convective flux formulation of the completely coupled Reynolds‐averaged Navier–Stokes and kω turbulence model equations. Based on these results, we formulate different approximate Riemann solvers that can be used as numerical flux functions in a DG discretization. The effect of the different strategies on the solution accuracy is investigated with numerical examples. The actual computations are performed using a p‐multigrid algorithm. To this end, we formulate a framework with a backward‐Euler smoother in which the linear systems are solved with a general preconditioned Krylov method. We present matrix‐free implementations and memory‐lean line‐Jacobi preconditioners and compare the effects of some parameter choices. In particular, p‐multigrid is found to be less efficient than might be expected from recent findings by other authors. This might be due to the consideration of turbulent flow. Copyright © 2012 John Wiley & Sons, Ltd.     

Applications of a higher‐order bounded numerical scheme to turbulent flows

This paper applies the higher‐order bounded numerical scheme Weighted Average Coefficients Ensuring Boundedness (WACEB) to simulate two‐ and three‐dimensional turbulent flows. In the scheme, a weighted average formulation is used for interpolating the variables at cell faces and the weighted average coefficients are determined from a normalized variable formulation and total variation diminishing (TVD) constraints to ensure the boundedness of the solution. The scheme is applied to two turbulent flow problems: (1) two‐dimensional turbulent flow around a blunt plate; and (2) three‐dimensional turbulent flow inside a mildly curved U‐bend. In the present study, turbulence is evaluated by using a low‐Reynolds number version of the k–ω model. For the flow simulation, the QUICK scheme is applied to the momentum equations while either the WACEB scheme (Method 1) or the UPWIND scheme (Method 2) is used for the turbulence equations. The present study shows that the WACEB scheme has at least second‐order accuracy while ensuring boundedness of the solutions. The present numerical study for a pure convection problem shows that the ‘TVD’ slope ranges from 2 to 4. For the turbulent recirculating flow, two different mixed procedures (Method 1 and Method 2) produce a substantial difference for the mean velocities as well as for the turbulence kinetic energy. Method 1 predicts better results than Method 2 does, comparing the analytical solution and the experimental data. For the turbulent flow inside the mildly curved U‐bend, although the predictions of velocity distributions with two procedures are very close, a noticeable difference of turbulence kinetic energy is exhibited. It is noticed that the discrepancy exists between numerical results and the experimental data. The reason is the limit of the two‐equation turbulence model to such complex turbulent flows with extra strain‐rates. Copyright © 2001 John Wiley & Sons, Ltd.     

Direct numerical simulation of preferential particle concentration in decaying turbulence under the influence of magnetic field

This paper presents a direct numerical simulation of an isotropic turbulence at Reynolds number R_λ=71.4. The statistics of heavy particles in such turbulence has been investigated. The numerical results indicate the merit of method in solving the magnetohydrodynamic turbulence. The revision of the paper was made based on the reviewers' comments. The paper was improved significantly. Hence, review suggests accepting the submission in current form. Copyright © 2009 John Wiley & Sons, Ltd.     

Implicit unfactored implementation of two-equation turbulence models in compressible Navier–Stokes methods

An implicit unfactored method for the coupled solution of the compressible Navier–Stokes equations with two-equation turbulence models is presented. Both fluid-flow and turbulence transport equations are discretized by a characteristics-based scheme. The implicit unfactored method combines Newton subiterations and point-by-point Gauss–Seidel subrelaxation. Implicit-coupled and -decoupled strategies are compared for their efficiency in the solution of the Navier–Stokes equations in conjunction with low-Re two-equation turbulence models. Computations have been carried out for the flow over an axisymmetric bump using the k–ϵ and k–ω models. Comparisons have been obtained with experimental data and other numerical solutions. The present study reveals that the implicit unfactored implementation of the two-equation turbulence models reduces the computing time and improves the robustness of the CFD code in turbulent compressible flows. © 1998 John Wiley & Sons, Ltd.     

Finite‐difference 4th‐order compact scheme for the direct numerical simulation of instabilities of shear layers

An algorithm based on the 4th‐order finite‐difference compact scheme is developed and applied in the direct numerical simulations of instabilities of channel flow. The algorithm is illustrated in the context of stream function formulation that leads to field equation involving 4th‐order spatial derivatives. The finite‐difference discretization in the wall‐normal direction uses five arbitrarily spaced points. The discretization coefficients are determined numerically, providing a large degree of flexibility for grid selection. The Fourier expansions are used in the streamwise direction. A hybrid Runge–Kutta/Crank–Nicholson low‐storage scheme is applied for the time discretization. Accuracy tests demonstrate that the algorithm does deliver the 4th‐order accuracy. The algorithm has been used to simulate the natural instability processes in channel flow as well as processes occurring when the flow is spatially modulated using wall transpiration. Extensions to three‐dimensional situations are suggested. Copyright © 2005 John Wiley & Sons, Ltd.     

An improved ghost‐cell immersed boundary method for compressible flow simulations

This study presents an improved ghost‐cell immersed boundary approach to represent a solid body in compressible flow simulations. In contrast to the commonly used approaches, in the present work, ghost cells are mirrored through the boundary described using a level‐set method to farther image points, incorporating a higher‐order extra/interpolation scheme for the ghost‐cell values. A sensor is introduced to deal with image points near the discontinuities in the flow field. Adaptive mesh refinement is used to improve the representation of the geometry efficiently in the Cartesian grid system. The improved ghost‐cell method is validated against four test cases: (a) double Mach reflections on a ramp, (b) smooth Prandtl–Meyer expansion flows, (c) supersonic flows in a wind tunnel with a forward‐facing step, and (d) supersonic flows over a circular cylinder. It is demonstrated that the improved ghost‐cell method can reach the accuracy of second order in L¹ norm and higher than first order in L^∞ norm. Direct comparisons against the cut‐cell method demonstrate that the improved ghost‐cell method is almost equally accurate with better efficiency for boundary representation in high‐fidelity compressible flow simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

Flow modulation of a planar free shear layer with large bubbles—direct numerical simulations

The flow of a planar free shear layer with cylindrical bubbles is simulated using a finite difference/front tracking scheme. This approach allows direct numerical simulation of the multiphase flow by wholly incorporating the local bubble flow field in conjunction with the large scale vortical structures of the liquid. The role of large bubbles in modifying low Reynolds number ( 250) shear flow structures is investigated, specifically for bubbles whose diameter approaches the scale of the largest liquid eddies. The results indicate that duration of eddy crossing is the main mechanism for flow modulation, which is typically characterized by decreased vortex coherency and size, modified fluctuation statistics and significant variations in pairing/merging phenomena. The comparison of fluctuating statistics and flow field visualization also allowed qualitative discrimination between the modulation of the non-linear eddy dynamics and fluctuations due simply to the random bubble induced perturbations.