Stability of Supersonic Boundary Layers on a Cone at an Angle of Attack
Stability of Supersonic Boundary Layers on a Cone at an Angle of Attack
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DOI：
10.2514/6.2009-3555
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发表时间：
2009-06
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影响因子：
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通讯作者：
P. Balakumar
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作者：
P. Balakumar
本文用数值方法研究了在来流马赫数为3.5和两种高雷诺数（0.25和0.50 × 10 6/inch）下，攻角为4.2 °，绕7 °尖锥的三维超声速边界层的稳定性和感受性。稳态横流涡的产生和演化也进行了研究，通过与位于圆锥表面的三维粗糙元进行模拟。通过求解柱坐标系下的Navier-Stokes方程，空间离散采用五阶精度加权基本无振荡（WENO）格式，时间积分采用三阶TVD龙格库塔（Runge-Kutta）格式，得到了有无粗糙元的流场.稳定性计算表明，对于放大最大的行波扰动，方位波数在m~25 - 50之间，对于定常扰动，方位波数在m~40 - 70之间。N因子计算预测，与迎风面和背风面附近的过渡锋相比，过渡将发生在锥体中部更靠前的位置。模拟结果表明，从机头区域产生的横流涡向背风面传播。在锥体的下部没有观察到扰动。I.当无粘流线沿展向弯曲时，存在三维边界层。当无粘流线弯曲时，在垂直于无粘流线的方向上存在压力梯度。在边界层内，由于粘性效应，速度小于无粘区的速度。因此，这种压力梯度在边界层内产生一个速度分量，称为横流速度，它垂直于无粘速度矢量。这种横流速度在其轮廓中包含一个拐点，并导致一种新的不稳定性，称为横流不稳定性。横流不稳定性对三维行波扰动和定常扰动都是不稳定的。定常扰动来源于孤立的粗糙元，表现为同转涡。除强湍流环境外，在大多数情况下，定常横流涡主导转捩过程。这种现象在几种不可压缩和可压缩流中观察到，包括后掠翼、旋转盘、旋转锥体和有攻角的锥体。Gergory等（1955）、Dehyle和Bippes（1996）、Saric等（1998）和Malik等（1994）的开创性工作很好地解释了不可压缩流动中的线性和非线性横流不稳定性。Saric等人（2003）在一篇综述论文中总结了不可压缩流中线性和非线性横流不稳定性的主要发现。早期对超音速边界层稳定性特性的研究6 - 8揭示了一个重要的发现，即超音速边界层中的不稳定扰动是三维的。他们还发现，在边缘马赫数为3.5的边界层中，最大放大扰动的波角与无粘流线倾斜约60 - 65度。Balakumar和Reed（1991）用数值方法研究了旋转锥体轴对称三维可压缩边界层的线性不稳定性。他们的计算表明，与非旋转锥体上的二维流动相比，由于横流的存在，行进扰动的增长率增加了2~4倍，这种增加随马赫数的增加而减小。
The stability and receptivity of three-dimensional supersonic boundary layers over a 7ϒ sharp tipped straight cone at an angle of attack of 4.2ϒ is numerically investigated at a free stream Mach number of 3.5 and at two high Reynolds numbers, 0.25 and 0.50*10 6 /inch. The generation and evolution of stationary crossflow vortices are also investigated by performing simulations with three-dimensional roughness elements located on the surface of the cone. The flow fields with and without the roughness elements are obtained by solving the full Navier-Stokes equations in cylindrical coordinates using the fifth-order accurate weighted essentially non-oscillatory (WENO) scheme for spatial discretization and using the thirdorder total-variation-diminishing (TVD) Runge-Kutta scheme for temporal integration. Stability computations reveal that the azimuthal wavenumbers are in the range of m ~ 25-50 for the most amplified traveling disturbances and in the range of m ~ 40-70 for the stationary disturbances. The N-Factor computations predicted that transition would occur further forward in the middle of the cone compared to the transition fronts near the windward and the leeward planes. The simulations revealed that the crossflow vortices originating from the nose region propagate towards the leeward plane. No perturbations were observed in the lower part of the cone. I. Introduction Three-dimensional boundary layers exist when the inviscid streamlines are curved in the spanwise direction. When the inviscid streamlines are curved, there exists a pressure gradient in the direction normal to the inviscid streamlines. Inside the boundary layer, due to the viscous effect, the velocity is smaller than that in the inviscid region. Hence, this pressure gradient causes a velocity component, called crossflow velocity, inside the boundary layer that is perpendicular to the inviscid-velocity vector. This crossflow velocity contains an inflection point in its profile and causes a new instability called crossflow instability. The crossflow instability is unstable to three-dimensional traveling and stationary disturbances. The stationary disturbances originate from isolated roughness elements and appear as corotating vortices. The stationary crossflow vortices dominate the transition process in most of the cases except in high turbulence environments. This phenomenon is observed in several incompressible and compressible flows including swept wings, rotating disks, rotating cones and cones at angles of attack. The linear and nonlinear crossflow instability in incompressible flow is well explained by the pioneering work of Gergory et al. (1955), Dehyle and Bippes (1996), Saric et al. (1998) and Malik et al. (1994). The major findings about the linear and nonlinear crossflow instability in incompressible flows are summarized in a review paper by Saric et al. (2003). Early investigations of the stability characteristics of supersonic boundary layers 6-8 revealed the important finding that the unstable disturbances in supersonic boundary layers are three-dimensional. They also found that the wave angles of the most amplified disturbances are inclined around 60-65 degrees from the inviscid streamlines in a boundary layer with an edge Mach number of 3.5. The linear instability of axi-symmetric three-dimensional compressible boundary layers for a rotating cone was numerically investigated by Balakumar and Reed (1991). Their calculations showed that the growth rate of the traveling disturbances is increased by a factor of 2 to 4 due to the presence of the crossflow compared with the two-dimensional flow over a non-rotating cone and this increase decreases with increasing Mach number.