High-order accurate, low numerical diffusion methods for aerodynamics

A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two-dimensional unsteady convection-diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter μ and fourth order accurate in space. For 0.5 ≤ μ ≤ 1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection-diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection-diffusion problem and three flows of varying complexities governed by the two-dimensional incompressible Navier-Stokes equations

A class of higher order compact schemes for the unsteady two‐dimensional convection–diffusion equation with variable convection coefficients

﻿The direct numerical simulation of boundary layer transition over a 5d half-cone-angle blunt cone is performed. The free-stream Mach number is 6 and the angle of attack is 1d. Random wall blow-and-suction perturbations are used to trigger the transition. Different from the authorsr previous work ﻿[Li et al., AIAA J. 46, 2899(2008)], the whole boundary layer flow over the cone is simulated (while in the authorrs previous work, only two 45d regions around the leeward and the windward sections are simulated). The transition location on the cone surface is determined through the rapid increase in skin fraction coefficient (Cf). The transition line on the cone surface shows a nonmonotonic curve and the transition is delayed in the range of 0d l t l 30d (t = 0d is the leeward section). The mechanism of the delayed transition is studied by using joint frequency spectrum analysis and linear stability theory (LST). It is shown that the growth rates of unstable waves of the second mode are suppressed in the range of 20d l t l 30d, which leads to the delayed transition location. Very low frequency waves VLFWs� are found in the time series recorded just before the transition location, and the periodic times of VLFWs are about one order larger than those of ordinary Mack second mode waves. Band-pass filter is used to analyze the low frequency waves, and they are deemed as the effect of large scale nonlinear perturbations triggered by LST waves when they are strong enough.The direct numerical simulation of boundary layer transition over a 5d half-cone-angle blunt cone is performed. The free-stream Mach number is 6 and the angle of attack is 1d. Random wall blow-and-suction perturbations are used to trigger the transition. Different from the authorsr previous work [ Li et al., AIAA J. 46, 2899 (2008) ], the whole boundary layer flow over the cone is simulated (while in the authorrs previous work, only two 45d regions around the leeward and the windward sections are simulated). The transition location on the cone surface is determined through the rapid increase in skin fraction coefficient (Cf). The transition line on the cone surface shows a nonmonotonic curve and the transition is delayed in the range of 20d l t l 30d (t = 0d is the leeward section). The mechanism of the delayed transition is studied by using joint frequency spectrum analysis and linear stability theory (LST). It is shown that the growth rates of unstable waves of the second mode are suppressed in the range of 20d l t l 30d, which leads to the delayed transition location. Very low frequency waves (VLFWs) are found in the time series recorded just before the transition location, and the periodic times of VLFWs are about one order larger than those of ordinary Mack second mode waves. Band-pass filter is used to analyze the low frequency waves, and they are deemed as the effect of large scale nonlinear perturbations triggered by LST waves when they are strong enough.

Direct numerical simulation of hypersonic boundary layer transition over a blunt cone with a small angle of attack

A generalized Reynolds analogy (GRA) is proposed for compressible wall-bounded turbulent flows (CWTFs) and validated by direct numerical simulations. By introducing a general recovery factor, a similarity between the Reynolds-averaged momentum and energy equations is established for the canonical CWTFs (i.e. pipes, channels, and flat-plate boundary layers that meet the quasi-one-dimensional flow approximation), independent of Prandtl number, wall temperature, Mach number, Reynolds number, and pressure gradient. This similarity and the relationships between temperature and velocity fields constitute the GRA. The GRA relationship between the mean temperature and the mean velocity takes the same quadratic form as Walz’s equation, with the adiabatic recovery factor replaced by the general recovery factor, and extends the validity of the latter to diabatic compressible turbulent boundary layers and channel/pipe flows. It also derives Duan & Martin’s (J. Fluid Mech., vol. 684, 2011, pp. 25–59) empirical relation for flows at different physical conditions (wall temperature, Mach number, enthalpy condition, surface catalysis, etc.). Several key parameters besides the general recovery factor emerge in the GRA. An effective turbulent Prandtl number is shown to be the reason for the parabolic profile of mean temperature versus mean velocity, and it approximates unity in the fully turbulent region. A dimensionless wall temperature, that we call the diabatic parameter, characterizes the wall-temperature effects in diabatic flows. The GRA also extends the analysis to the fluctuation fields. It recovers the modified strong Reynolds analogy proposed by Huang, Coleman & Bradshaw (J. Fluid Mech., vol. 305, 1995, pp. 185–218) and explains the variation of the temperature–velocity correlation coefficient with wall temperature. Thus, the GRA unveils a generalized similarity principle behind the complex nonlinear coupling between the thermal and velocity fields of CWTFs.

A generalized Reynolds analogy for compressible wall-bounded turbulent flows

A higher order compact (HOC) finite difference solution procedure has been proposed for the steady two-dimensional (2D) convection–diffusion equation on non-uniform orthogonal Cartesian grids involving no transformation from the physical space to the computational space. Effectiveness of the method is seen from the fact that for the first time, an HOC algorithm on non-uniform grid has been extended to the Navier–Stokes (N–S) equations. Apart from avoiding usual computational complexities associated with conventional transformation techniques, the method produces very accurate solutions for difficult test cases. Besides including the good features of ordinary HOC schemes, the method has the advantage of better scale resolution with smaller number of grid points, with resultant saving of memory and CPU time. Gain in time however may not be proportional to the decrease in the number of grid points as grid non-uniformity imparts asymmetry to some of the associated matrices which otherwise would have been symmetric. The solution procedure is also highly robust as it computes complex flows such as that in the lid-driven square cavity at high Reynolds numbers (Re), for which no HOC results have so far been seen. Copyright © 2004 John Wiley & Sons, Ltd.

A transformation‐free HOC scheme for steady convection–diffusion on non‐uniform grids

Direct numerical simulations of a spatially evolving supersonic flat-plate turbulent boundary layer flow with free Mach number M∞ = 2.25 and Reynolds number Re = 365000/in are performed. The transition process from laminar to turbulent flow is obtained by solving the three-dimensional compressible Navier–Stokes equations, using high-order accurate difference schemes. The obtained statistical results agree well with the experimental and theoretical data. From the numerical results it can be seen that the transition process under the considered conditions is the process which skips the Tollmien–Schlichting instability and the second instability through the instability of high gradient shear layer and becomes of laminar flow breakdown. This means that the transition process is a bypass-type transition process. The spanwise asymmetry of the disturbance locally upstream imposed is important to induce the bypass-type transition. Furthermore, with increasing the time disturbance frequency the transition will delay. When the time disturbance frequency is large enough, the transition will disappear.

https://iopscience.iop.org/article/10.1088/0256-307X/22/7/041/pdf

Direct Numerical Simulation of Supersonic Turbulent Boundary Layer Flow

Direct numerical simulation is carried out for a spatially evolving supersonic turbulent boundary layer at free-stream Mach number 6. To overcome numerical instability, the seventh-order WENO scheme is used for the convection terms of Navier–Stokes equations, and fine mesh is adopted to minimize numerical dissipation. Compressibility effects on the near-wall turbulent kinetic energy budget are studied. The cross-stream extended self-similarity and scaling exponents including the near-wall region are studied. In high Mach number flows, the coherence vortex structures are arranged to be smoother and streamwised, and the hair-pin vortices are less likely to occur.

Direct Numerical Simulation of a Spatially Evolving Supersonic Turbulent Boundary Layer at Ma = 6

A new finite difference method for the discretization of the incompressible Navier-Stokes equations is presented. The scheme is constructed on a staggered-mesh grid system. The convection terms are discretized with a fifth-order-accurate upwind compact difference approximation, the viscous terms are discretized with a sixth-order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth-order difference approximation on a cell-centered mesh. Time advancement uses a three-stage Runge-Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented.

Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

Direct numerical simulation (DNS) of decaying compressible isotropic turbulence at turbulence Mach numbers of M t = 0.2—0.7 and Taylor Reynolds numbers of 72 and 153 is performed by using the 7th order upwind-biased difference and 8th order center difference schemes. Results show that proper upwind-biased difference schemes can release the limit of “start-up” problem to Mach numbers. Compressibility effects on the statistics of turbulent flow as well as the mechanics of shocklets in compressible turbulence are also studied, and the conclusion is drawn that high Mach number leads to more dissipation. Scaling laws in compressible turbulence are also analyzed. Evidence is obtained that scaling laws and extended self similarity (ESS) hold in the compressible turbulent flow in spite of the presence of shocklets, and compressibility has little effect on scaling exponents.

Direct numerical simulation of compressible isotropic turbulence

An approach which combines direct numerical simulation (DNS) with the Lighthill acoustic analogy theory is used to study the potential noise sources during the transition process of a Mach 2.25 flat plate boundary layer. The quadrupole sound sources due to the now fluctuations and the dipole sound sources due to the fluctuating surface stress are obtained. Numerical results suggest that formation of the high shear layers leads to a dramatic amplification of amplitude of the fluctuating quadrupole sound sources. Compared with the quadrupole sound source, the energy of dipole sound source is concentrated in the relatively low frequency range.

https://iopscience.iop.org/article/10.1088/0256-307X/26/9/094701/pdf

Ma Yanwen

Papers

Direct Numerical Simulation of Supersonic Turbulent Boundary Layer Flow

Direct Numerical Simulation of a Spatially Evolving Supersonic Turbulent Boundary Layer at Ma = 6

Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

Direct numerical simulation of compressible isotropic turbulence

Acoustic Calculation for Supersonic Turbulent Boundary Layer Flow