Direct numerical simulation of disturbances generated by periodic suction-blowing in a hypersonic boundary layer

A numerical algorithm and code are developed and applied to direct numerical simulation (DNS) of unsteady two-dimensional flow fields relevant to stability of the hypersonic boundary layer. An implicit second-order finite-volume technique is used for solving the compressible Navier–Stokes equations. Numerical simulation of disturbances generated by a periodic suction-blowing on a flat plate is performed at free-stream Mach number 6. For small forcing amplitudes, the second-mode growth rates predicted by DNS agree well with the growth rates resulted from the linear stability theory (LST) including nonparallel effects. This shows that numerical method allows for simulation of unstable processes despite its dissipative features. Calculations at large forcing amplitudes illustrate nonlinear dynamics of the disturbance flow field. DNS predicts a nonlinear saturation of fundamental harmonic and rapid growth of higher harmonics. These results are consistent with the experimental data of Stetson and Kimmel obtained on a sharp cone at the free-stream Mach number 8.     

三维扰动波的非平行边界层稳定性研究

导出了三维扰动波的原始变量形式的抛物化稳定性方程（PSE）,研究了三维空间模态TS波的非平行边界层稳定性问题.采用了法向四阶紧致格式,以提高计算精度.通过给出不会导致奇性的坐标变换、修改外边界条件以及克服平行流初始值的瞬态影响和推进步长的限制,保证了计算的数值稳定.用补全元素带状矩阵法求解块三对角矩阵,大大提高了速度.计算结果清楚地显示了三维扰动波的演化过程和非平行性对边界层稳定性的影响,特别是,观察到非平行性对三维扰动波的影响,有时会使其稳定性出现逆转的现象.还研究了逆压梯度的作用.算例的结果与其他结果符合良好.     

Spatial Evolution of Nonstationary Perturbations in Hamel Flow

The propagation of small perturbations in longitudinally inhomogeneous flows is investigated. The evolution of the perturbations is studied with reference to the radial flow of a viscous incompressible fluid between plane nonparallel walls, the simplest inhomogeneous flow. Using a generalized method of variation of constants, the corresponding boundary-value problem is reduced to an infinite-dimensional evolutionary system of ordinary differential equations for the complex amplitudes of the eigensolutions of a locally homogeneous problem. Physically, the method can be interpreted as a representation of the perturbation evolution process via two concomitant processes: the independent amplification (attenuation) of normal modes of the locally homogeneous problem and the rescattering of these modes into each other on local inhomogeneities of the base flow. The calculations show that reduced versions of the method are promising for describing the linear stage of laminar-turbulent transition in a boundary layer.     

Mechanism of transition in a hypersonic sharp cone boundary layer with zero angle of attack

Firstly, the steady laminar flow field of a hypersonic sharp cone boundary layer with zero angle of attack was computed.Then,two groups of finite amplitude T-S wave disturbances were introduced at the entrance of the computational field,and the spatial mode transition process was studied by direct numerical simulation (DNS) method. The mechanism of the transition process was analyzed.It was found that the change of the stability characteristics of the mean flow profile was the key issue.Furthermore,the characteristics of evolution for the disturbances of different modes in the hypersonic sharp cone boundary layer were discussed.     

Studies on nonlinear stability of three-dimensional H-type disturbance

The three-dimensional H-type nonlinear evolution process for the problem of boundary layer stability is studied by using a newly developed method called parabolic stability equations (PSE). The key initial conditions for sub-harmonic disturbances are obtained by means of the secondaryinstability theory. The initial solutions of two-dimensional harmonic waves are expressed in Landau expansions. The numerical techniques developed in this paper, including the higher order spectrum method and the more effective algebraic mapping for dealing with the problem of an infinite region, increase the numerical accuracy and the rate of convergence greatly. With the predictor-corrector approach in the marching procedure, the normalization, which is very important for PSE method, is satisfied and the stability of the numerical calculation can be assured. The effects of different pressure gradients, including the favorable and adverse pressure gradients of the basic flow, on the “H-type“ evolution are studied in detail. The results of the three-dimensional nonlinear “H-type“ evolution are given accurately and show good agreement with the data of the experiment and the results of the DNS from the curves of the amplitude variation, disturbance velocity profile and the evolution of velocity.     

超声速平面剪切层声辐射涡模态数值分析

对Mc = 1.2二维超声速空间发展平面自由剪切层, 进行了扰动模态及流动结构的数值分析. 采用时空三阶改进MacCormack格式, 差分求解可压缩扰动Navier-Stokes方程, 直接数值模拟入口不同基频谐波扰动的非线性演化特征. 采用空间线性稳定性理论证明, 计算所促发的扰动波是声辐射涡模态. 扰动参数及特征函数分析显示, 声辐射涡模态是弱色散的快／慢两种外部模态, 在扰动对流Mach数为超声速一侧呈膨胀／压缩状辐射. 单频受迫扰动可无相差地促发多模态混合扰动波, 而在自然扰动条件下, 剪切层的稳定性受慢模态主导.     

Nonlinear equation for amplitude of Tollmien-Schlichting waves in a boundary layer

The nonlinear evolution of a Tollmien-Schlichting wave is analyzed with allowance for the flow being nonparallel in a boundary layer. In contrast to the early work of Zel'man [19], strict allowance is made for the fact that the extent to which the flow is nonparallel is not independent of the Reynolds number — the departure from parallel flow in a boundary layer is small only at large Reynolds numbers. Therefore, an asymptotic theory of Tollmien-Schlichting waves is constructed under the assumption that the Reynolds number tends to infinity.     

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.     

Interactions between pairs of oblique waves in a Bickley jet

We consider the weakly nonlinear spatial evolution of a pair of varicose oblique waves and a pair of sinuous oblique waves superimposed on an inviscid Bickley jet, with each wave being slightly amplified on a linear basis. The two pairs are assumed to both be inclined at the same angle to the plane of the jet. A nonlinear critical layer analysis is employed to derive equations governing the evolution of the instability wave amplitudes, which contain a coupling between the modes. These equations are discussed and solved numerically, and it is shown that, as in related work for other flows, these equations may develop a singularity at a finite distance downstream.     

Prehistory of Instability in a Hypersonic Boundary Layer

The initial phase of hypersonic boundary-layer transition comprising excitation of boundary-layer modes and their downstream evolution from receptivity regions to the unstable region (instability prehistory problem) is considered. The disturbance spectrum reveals the following features: (1) the first and second modes are synchronized with acoustic waves near the leading edge; (2) further downstream, the first mode is synchronized with entropy and vorticity waves; (3) near the lower neutral branch of the Mack second mode, the first mode is synchronized with the second mode. Disturbance behavior in Regions (2) and (3) is studied using the multiple-mode method accounting for interaction between modes due to mean-flow nonparallel effects. Analysis of the disturbance behavior in Region 3) provides the intermodal exchange rule coupling input and output amplitudes of the first and second modes. It is shown that Region (3) includes branch points at which disturbance group velocity and amplitude are singular. These singularities can cause difficulties in stability analyses. In Region (2), vorticity/entropy waves are partially swallowed by the boundary layer. They may effectively generate the Mack second mode near its lower neutral branch. Received 17 July 2000 and accepted 23 March 2001     

高超声速边界层中模态转化的数值研究

在高超声速边界层中,第一模态和第二模态是与转捩有关的两个主要不稳定模态.除了不稳定模态,还存在一类稳定模态,其相速度在前缘接近快声波的相速度称为快模态.在感受性过程中,这类模态对激发边界层中不稳定模态起着很重要的作用.前缘感受性理论解释了边界层外扰动激发边界层中第一模态波的机理.针对高超声速平板边界层,利用相似性解剖面作为基本流,采用线性稳定性理论和直接数值模拟的方法研究了快模态和慢模态的稳定性行为.研究发现模态转化的位置与马赫数有关.根据线性稳定性理论的结果定义了临界频率.当扰动频率高于临界频率,第一模态与第二模态同支;而当扰动频率低于临界频率,第一模态与第二模态的共轭模态同支.借助稳定性方程的伴随方程分析了直接数值模拟的结果.直接数值模拟结果表明不论上游是快模态还是慢模态,当它们经过第二模态的不稳定区,它们都会演化成第二模态. 这可用模态在非平行流中传播的特征来解释.      

Modeling of nonparallel effects in temporal direct numerical simulations of compressible boundary-layer transition

One important alternative to spatial direct numerical simulation (SDNS) of a growing boundary-layer transition is a temporal direct numerical simulation (TDNS), where the flow is assumed to be locally parallel and the transition develops in time. To model nonparallel effects of a growing boundary layer, the TDNS allows the boundary layer to grow in time. This approach has been shown to be effective for an incompressible boundary layer. For a compressible boundary layer, however, a simple application of this approach has been found to be insufficient. To investigate this issue, we first split the variation of the flow field in the streamwise direction into a slowly evolving part and a fast and small-scale fluctuation part. By Taylor-expanding the slowly evolving large-scale part, this study shows that the Navier-Stokes operator can be reformulated as a power series of the perturbation parameter (x–x ₀), yielding one set of equations for each power. Each set of these equations has a periodic solution in the streamwise direction, and therefore a modified TDNS method can be employed to solve these equations. Only the first set of the equations is considered in the applications presented. During the linear stage of transition, the results from this extended formulation show a significant improvement over those from the previous parallel flow formulation, especially for second modes which have short wavelengths. The results are well comparable with those from parabolized stability equations (PSE) and SDNS. A good agreement between this extended formulation and SDNS results is also demonstrated at the nonlinear stage.Part of this work was supported by the Deutsche Forschungsgemeinschaft (DFG).     

Amplitude method of prediction of laminar-turbulent transition on a swept-wing

The amplitudemethod of prediction of laminar-turbulent transition on a swept-wing initiated by the simultaneous action of free-stream turbulence and surface roughness is developed. Generation of growing intrinsic perturbations in the boundary layer is described by the mechanism of distributed generation, i.e., an external perturbation generates an instability mode with the same wavelength and frequency. Generation occurs in a small neighborhood of the neutral point at which the parameters of an external perturbation and the instability modes coincide. The development of proper perturbations in the boundary layer is described by the nonlinear method of parabolized stability equations (PSE). The criterion of laminar-turbulent transition is the combined amplitude. i.e., transition begins at a point at which the sum of amplitudes of steady and traveling modes reaches a critical value.     

Verification of parabolized stability equations for its application to compressible boundary layers

Parabolized stability equations (PSE) were used to study the evolution of disturbances in compressible boundary layers.The results were compared with those ob- tained by direct numerical simulations (DNS),to check if the results from PSE method were reliable or not.The results of comparison showed that no matter for subsonic or supersonic boundary layers,results from both the PSE and DNS method agreed with each other reasonably well,and the agreement between temperatures was better than those between velocities.In addition,linear PSE was used to calculate the neutral curve for small amplitude disturbances in a supersonic boundary layer.Compared with those obtained by linear stability theory (LST),the situation was similar to those for incom- pressible boundary layer.     

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

Selective enhancement of oblique waves caused by finite amplitude second mode in supersonic boundary layer

Nonlinear interactions of the two-dimensional(2D) second mode with oblique modes are studied numerically in a Mach 6.0 flat-plate boundary layer, focusing on its selective enhancement effect on amplification of different oblique waves. Evolution of oblique modes with various frequencies and spanwise wavenumbers in the presence of 2D second mode is simulated successively, using a modified parabolized stability equation(PSE) method, which is able to simulate interaction of two modes with different frequencies efficiently. Numerical results show that oblique modes in a broad band of frequencies and spanwise wavenumbers can be enhanced by the finite amplitude 2D second mode instability wave. The enhancement effect is accomplished by interaction of the 2D second mode, the oblique mode, and a forced mode with difference frequency. Two types of oblique modes are found to be more amplified, i.e., oblique modes with frequency close to that of the 2D second mode and low-frequency first mode oblique waves. Each of them may correspond to one type of transition routes found in transition experiments. The spanwise wavenumber of the oblique wave preferred by the nonlinear interaction is also determined by numerical simulations.     

Spatial direct numerical simulation of high-speed boundary-layer flows part I: Algorithmic considerations and validation

A highly accurate algorithm for the direct numerical simulation (DNS) of spatially evolving high-speed boundary-layer flows is described in detail and is carefully validated. To represent the evolution of instability waves faithfully, the fully explicit scheme relies on non-dissipative high-order compact-difference and spectral collocation methods. Several physical, mathematical, and practical issues relevant to the simulation of high-speed transitional flows are discussed. In particular, careful attention is paid to the implementation of inflow, outflow, and far-field boundary conditions. Four validation cases are presented, in which comparisons are made between DNS results and results obtained from either compressible linear stability theory or from the parabolized stability equation (PSE) method, the latter of which is valid for nonparallel flows and moderately nonlinear disturbance amplitudes. The first three test cases consider the propagation of two-dimensional second-mode disturbances in Mach 4.5 flat-plate boundary-layer flows. The final test case considers the evolution of a pair of oblique second-mode disturbances in a Mach 6.8 flow along a sharp cone. The agreement between the fundamentally different PSE and DNS approaches is remarkable for the test cases presented.     

The upper-branch stability of compressible boundary-layer flows

The upper-branch linear and nonlinear stability of compressible boundary-layer flows is studied using the approach of Smith and Bodonyi (1982) for a similar incompressible problem. Both pressure gradient boundary layers and Blasius flow are considered with and without heat transfer and the neutral eigenrelations incorporating compressibility effects are explicitly obtained. The compressible nonlinear viscous critical-layer equations are derived and solved numerically and the results indicate some solutions with positive phase shift across the critical layer. Various limiting cases are investigated including the case of much larger disturbance amplitudes and this indicates the structure for the strongly nonlinear critical layer of the Benney-Bergeron (1969) type. Finally, we also show how a match with the inviscid neutral inflexional modes arising from the generalized inflexion-point criterion is achieved.J. Cole is grateful to the Science and Engineering Research Council of Great Britain for financial support. J. Gajjar gratefully acknowledges the support of ICOMP, NASA Lewis Research Center, Cleveland, Ohio, where part of this work was done, and he is also grateful to the Computation Center at Iowa State University for a grant which enabled the numerical work in the paper to be completed. The permanent address of the authors is the Mathematics Department, Exeter University, Exeter EX4 4QE, England.     

An accurate model for the thin film flow

This paper deals with the linear stability of a liquid film flowing down an inclined plane. The Navier-Stokes equations were reduced into four evolution equations that describe the development of the film depth, the flow rate, the free surface velocity, and the wall shear stress, using the Karman-Polhausen boundary layer integral method. Thus, we were able to determine the stability threshold and approach well the critical wave number for long waves. The obtained results were found to be in good agreement with the experiments of Liu et al.