Xiaolin Zhong

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor-Green vortex, Shu-Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.

Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations

This article presents a family of very high-order non-uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible Navier-Stokes equations. The high-order compact schemes on non-uniform grids developed in Shukla and Zhong R.K. Shukla, X. Zhong, Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation, J. Comput. Phys. 204 (2005) 404] for linear model equations are extended to the full Navier-Stokes equations in the vorticity and streamfunction formulation. Two methods for the solution of Helmholtz and Poisson equations using high-order compact schemes on non-uniform grids are developed. The schemes are constructed so that they maintain a high-order of accuracy not only in the interior but also at the boundary. Second-order semi-implicit temporal discretization is achieved through an implicit Backward Differentiation scheme for the linear viscous terms and an explicit Adam-Bashforth scheme for the non-linear convective terms. The boundary values of vorticity are determined using an influence matrix technique. The resulting discretized system with boundary closures of the same high-order as the interior is shown to be stable, when applied to the two-dimensional incompressible Navier-Stokes equations, provided enough grid points are clustered at the boundary. The resolution characteristics of the high-order compact finite difference schemes are illustrated through their application to the one-dimensional linear wave equation and the two-dimensional driven cavity flow. Comparisons with the benchmark solutions for the two-dimensional driven cavity flow, thermal convection in a square box and flow past an impulsively started cylinder show that the high-order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary.

Direct Numerical Simulation and the Theory of Receptivity in a Hypersonic Boundary Layer

Direct numerical simulation of receptivity in a boundary layer over a sharp wedge of half-angle 5:3 degrees was carried out with two-dimensional perturbations introduced into the ∞ow by periodic-in-time blowing-suction through a slot. The free stream Mach number was equal to 8. The perturbation ∞ow fleld downstream from the slot was decomposed into normal modes with the help of the biorthogonal eigenfunction system. Filtered-out amplitudes of two discrete normal modes and of the fast acoustic modes are compared with the linear receptivity problem solution. The examples ilustrate how the multimode decomposition technique may serve as a tool for gaining insight into computational results. I. Introduction The progress being made in computational ∞uid dynamics provides an opportunity for reliable simulation of such complex phenomena as laminar-turbulent transition. The dynamics of ∞ow transition depends on the instability of small perturbations excited by external sources. Computational results provide complete information about the ∞ow fleld, which would be impossible to measure in real experiments. However, validation of the results might be a challenging problem. Sometimes, numerical simulations of small perturbations in boundary layers are accompanied by comparisons with results obtained within the scope of the linear stability theory. In principle, this is possible in the case of a ∞ow possessing an unstable mode. Far downstream from the actuator, the perturbations might be dominated by the unstable mode, and one may compare the computational results for the velocity and temperature perturbation proflles and their growth rates with the linear stability theory. This analysis does not work when the amplitude of the unstable mode is comparable to that of other modes, or when one needs to evaluate the amplitude of a decaying mode. Recently, a method of normal mode decomposition was developed for two- and three- dimensional perturbations in compressible and incompressible boundary layers. 1{3 The method is based on the expansion of solutions of linearized Navier{Stokes equations for perturbations of prescribed frequency into the normal modes of discrete and continuous spectra. The instability modes belong to the discrete spectrum, whereas the continuous spectrum is associated with vorticity, entropy, and acoustic modes. Because the problem of perturbations within the scope of the linearized Navier{Stokes equations is not self-adjoint, the eigenfunctions representing the normal modes are not orthogonal. Therefore, the eigenfunctions of the adjoint problem are involved in the computation of the normal modes’ weights. Originally, the method based on the expansion into the normal modes was used for analysis of discrete modes (Tollmien{Schlichting{like modes) only. After clariflcation of uncertainties associated with the continuous spectra in Ref. 1, the method was also applied to the analysis of roughness-induced perturbations. 4{6 In order to flnd the amplitude of a normal mode, one needs proflles of the velocity, temperature, and pressure perturbations, together with some of their streamwise derivatives given at only one station downstream from the disturbance source. Because computational results can provide all the necessary information about the perturbation fleld, the application of the multimode decomposition is straightforward. However, the flrst

High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition

The direct numerical simulation of receptivity, instability and transition of hypersonic boundary layers requires high-order accurate schemes because lower-order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields. The main limiting factor in the application of high-order schemes to practical boundary-layer flow problems is the numerical instability of high-order boundary closure schemes on the wall. This paper presents a family of high-order non-uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary-layer transition. By using an appropriate grid stretching, and clustering grid points near the boundary, high-order schemes with stable boundary closures can be obtained. The order of the schemes ranges from first-order at the lowest, to the global spectral collocation method at the highest. The accuracy and stability of the new high-order numerical schemes is tested by numerical simulations of the linear wave equation and two-dimensional incompressible flat plate boundary layer flows. The high-order non-uniform-grid schemes (up to the 11th-order) are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge. The steady and unsteady results show that the new high-order schemes are stable and are able to produce high accuracy for computations of the nonlinear two-dimensional Navier-Stokes equations for the wall bounded supersonic flow.

A numerical study of compressible turbulent boundary layers

Compressible turbulent boundary layers with free-stream Mach number ranging from 2.5 up to 20 are analyzed by means of direct numerical simulation of the Navier–Stokes equations. The fluid is assumed to be an ideal gas with constant specific heats. The simulation generates its inflow condition using the rescaling-recycling method. The main objective is to study the effect of Mach number on turbulence statistics and near-wall turbulence structures. The present study shows that supersonic/hypersonic boundary layers at zero pressure gradient exhibit close similarities to incompressible boundary layers and that the main turbulence statistics can be correctly described as variable-density extensions of incompressible results. The study also shows that the spanwise streak’s spacing of 100 wall units in the inner region y + 15 still holds for the considered high Mach numbers. The probability density function of the velocity dilatation shows significant variations as the Mach number is increased, but it can also be normalized by accounting for the variable-density effect. The compressible boundary layer also shows an additional similarity to the incompressible boundary layer in the sense that without the linear coupling term, near-wall turbulence cannot be sustained.

A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity

This paper presents a new high-order immersed interface method for elliptic equations with imbedded interface of discontinuity. Compared with the original second-order immersed interface method of [R.J. LeVeque, Z. Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994) 1001-25], the new method achieves arbitrarily high-order accuracy for derivatives at an irregular grid point by imposing only two physical jump conditions together with a wider set of grid stencils. The new interface difference formulas are expressed in a general explicit form so that they can be applied to different multi-dimensional problems without any modification. The new interface algorithms of up to O(h^4) accuracy have been derived and tested on several one and two-dimensional elliptic equations with imbedded interface. Compared to the standard second-order immersed interface method, the test results show that the new fourth-order immersed interface method leads to a significant improvement in accuracy of the numerical solutions. The proposed method has potential advantages in the application to two-phase flow because of its high-order accuracy and simplicity in applications.

A high-order cut-cell method for numerical simulation of hypersonic boundary-layer instability with surface roughness

Laminar-turbulent transition of hypersonic boundary layers can be affected significantly by the existence of surface roughness. Currently many important mechanisms of roughness-induced transition are not well understood. In recent years, direct numerical simulation (DNS) has been extensively applied for investigating instability and transition mechanisms of hypersonic boundary layers. Most of the past DNS studies, however, have been based on body-fitted grids for smooth surfaces without roughness. Due to complex geometry of embedded roughness, the use of body-fitted grids can be very difficult for flow with arbitrary surface roughness. In this paper, we present a new high-order cut-cell method to overcome the natural complexities in grid generation around arbitrary surface of roughness. The new method combines a non-uniform-grid finite-difference method for discrete grid points near the solid boundary and a shock-fitting method for the treatment of the bow shock. The non-uniform-grid finite-difference formulas are expressed in a general explicit form so that they can be applied to different multi-dimensional problems without any modification. The computational accuracy of new algorithms of up to O(h^4) are tested on several one- and two-dimensional elliptic equations in irregular domains. In addition, the new method is applied to the simulation of the receptivity process of a Mach 5.92 flow over a flat plate under the combined effect of an isolated surface roughness element and surface blow and suction. A good agreement is found between the unsteady flow results and those obtained by a Linear Stability Theory (LST).

Effect of wall perturbations on the receptivity of a hypersonic boundary layer

The receptivity of a Mach 5.92 flat plate boundary layer to periodic two-dimensional wall perturbations is studied by numerical simulations and linear stability theory (LST). Free stream flow conditions are the same as the leading edge receptivity experiment of Maslov et al., J. Fluid Mech. 426, 73 (2001). Steady base flow is simulated by solving compressible Navier–Stokes equations with a combination of a fifth-order shock-fitting method and a second-order total variation diminishing scheme. The accuracy of the steady base flow is validated by comparisons with the experimental measurements of Maslov et al. and self-similar boundary-layer solution. In receptivity simulations, streamwise velocity perturbation, blowing suction, and temperature perturbation are introduced to the steady base flow with a forcing slot on flat plate. A model of wall perturbation is proposed based on physical properties of the electric pulse generator used in the experiment of Maslov et al. Stability characteristics of boundary-layer waves are identified and evaluated by comparing the results of LST and numerical simulation. Numerical simulation results show that all three types of wall perturbations eventually result in the same type of instability wave (mode S) in the boundary layer, which indicates that receptivity mechanism of the hypersonic boundary layer to wall perturbation is independent of specific perturbation type. On the other hand, the hypersonic boundary layer is found to be most sensitive to blowing-suction and least sensitive to temperature perturbation.The receptivity of a Mach 5.92 flat plate boundary layer to periodic two-dimensional wall perturbations is studied by numerical simulations and linear stability theory (LST). Free stream flow conditions are the same as the leading edge receptivity experiment of Maslov et al., J. Fluid Mech. 426, 73 (2001). Steady base flow is simulated by solving compressible Navier–Stokes equations with a combination of a fifth-order shock-fitting method and a second-order total variation diminishing scheme. The accuracy of the steady base flow is validated by comparisons with the experimental measurements of Maslov et al. and self-similar boundary-layer solution. In receptivity simulations, streamwise velocity perturbation, blowing suction, and temperature perturbation are introduced to the steady base flow with a forcing slot on flat plate. A model of wall perturbation is proposed based on physical properties of the electric pulse generator used in the experiment of Maslov et al. Stability characteristics of boundary-l...

On high-order shock-fitting and front-tracking schemes for numerical simulation of shock-disturbance interactions

High-order methods that can resolve interactions of flow-disturbances with shock waves are critical for reliable numerical simulation of shock wave and turbulence interaction. Such problems are not well understood due to the limitations of numerical methods. Most of the popular shock-capturing methods are only first-order accurate at the shock and may incur spurious numerical oscillations near the shock. Shock-fitting algorithms have been proposed as an alternative which can achieve uniform high-order accuracy and can avoid possible spurious oscillations incurred in shock-capturing methods by treating shocks as sharp interfaces. We explore two ways for shock-fitting: conventional moving grid set-up and a new fixed grid set-up with front tracking. In the conventional shock-fitting method, a moving grid is fitted to the shock whereas in the newly developed fixed grid set-up the shock front is tracked using Lagrangian points and is free to move across the underlying fixed grid. Different implementations of shock-fitting methods have been published in the literature. However, uniform high-order accuracy of various shock-fitting methods has not been systematically established. In this paper, we carry out a rigorous grid-convergence analysis on different variations of shock-fitting methods with both moving and fixed grids. These shock-fitting methods consist of different combinations of numerical methods for computing flow away from the shock and those for computing the shock movement. Specifically, we consider fifth-order upwind finite-difference scheme and shock-capturing WENO schemes with conventional shock-fitting and show that a fifth-order convergence is indeed achieved for a canonical one-dimensional shock-entropy wave interaction problem. We also show that the method of finding shock velocity from one characteristic relation and Rankine-Hugoniot jump condition performs better than the other methods of computing shock velocities. A high-order front-tracking implementation of shock-fitting is also presented in this paper and nominal rate of convergence is shown. The front-tracking results are validated by comparing to results from the conventional shock-fitting method and a linear-interaction analysis for a two-dimensional shock disturbance interaction problem.

Linear stability of viscous supersonic plane Couette flow

The linear stability of viscous compressible plane Couette flow is not well understood even though the stability of incompressible Couette flow has been studied extensively and has been shown to be stable to linear disturbances. In this paper, the viscous linear stability of supersonic Couette flow for a perfect gas governed by Sutherland viscosity law was studied using two global methods to solve the linear stability equations. The two methods are a fourth-order finite-difference method and a spectral collocation method. Two families of wave modes, modes I and II, were found to be unstable at finite Reynolds numbers, where mode II is the dominant instability among the unstable modes. These two families of wave modes are acoustic modes created by sustained acoustic reflections between a wall and a relative sonic line when the mean flow in the local region is supersonic with respect to the wave velocities. The effects of viscosity on the stability of the two families of acoustic modes were studied by compa...