Traveling disturbance appearing in boundary layer transition in a Yawed cylinder

Boundary layers that develop over a body in fluid flow are in most cases three-dimensional owing to the spin, yaw, or surface curvature of the body. Therefore, the study of three-dimensional (3D) boundary-layer transition is essential to work in practical aerodynamics. The present investigation is concerned with the problem of 3D boundary layers over a yawed body. A yawed cylinder model that represents the leading edge portion of a swept wing and the mechanism of crossflow instability are investigated in detail using hot-wire velocimetry and a flow visualization technique. As a result, traveling disturbances having frequencies f₁ and f₂, which differ by about one order of magnitude, are detected in the transition region. The phase velocities and directions of travel of those disturbances are measured. Results for the low-frequency disturbance f₁ show qualitative coincidence with results numerically predicted for a crossflow unsteady disturbance. Nameley, F₁ travels nearly spanwise to the yawed cylinder and very close to the cylinder wall. The results for the high-frequency disturbance f₂ good agreement with the existing experimental results. The ₂ disturbance is found to be the high-frequency inflectional secondary instability that appears in 3D boundary layer transition in general. A two-stage transition process, where stationary crossflow vortices appear as the primary instability and a traveling inflectional disturbance is generated as a secondary instability, was observed. Secondary instability seems to play a major role in turbulent transition.     

Spatially developing secondary instabilities in compressible swept airfoil boundary layers

Two-dimensional eigenvalue analysis is used on a massive scale to study the spatial instabilities of compressible shear flows with two inhomogeneous directions. The main focus of the study is crossflow dominated swept-wing boundary layers although the methodology can also be applied to study other types of flows, such as the attachment-line flow. Certain unique aspects of formulating a spatial, two-dimensional eigenvalue problem for the secondary instability of finite amplitude crossflow vortices are discussed, namely, fixing the spatial growth direction unambiguously through a non-orthogonal formulation of the linearized disturbance equations. A primary test case used for parameter study corresponds to Numerical results are presented for the low-speed, NLF-0415(b) airfoil configuration as tested in the ASU Unsteady Wind Tunnel, wherein a spanwise periodic array of roughness elements was placed near the leading edge in order to excite stationary crossflow modes with a specified fundamental wavelength. The two classes of flow conditions selected for this analysis include those for which the roughness array spacing corresponds to either the naturally dominant crossflow wavelength, or a subcritical wavelength that serves to reduce the growth of the naturally excited dominant crossflow modes. Numerical predictions are compared with the measured database, both as indirect validation for the spatial instability analysis and to provide a basis for comparison with a higher Reynolds number, supersonic swept-wing configuration. Application of the eigenvalue analysis to the supersonic configuration reveals that a broad spectrum of stationary crossflow modes can sustain sufficiently strong secondary instabilities as to potentially cause transition over this configuration. In particular, the control mode itself, if initiated with too large an amplitude, may lead to an earlier transition.     

Evolution of disturbances in a laminarized supersonic boundary layer on a swept wing

Experimental data on stability of a three-dimensional supersonic boundary layer on a swept wing are presented. The experiments are performed on a swept wing model with a lenticular profile with a 40° sweep angle of the leading edge at a zero angle of attack. The supersonic boundary layer on the swept wing was laminarized with the use of distributed roughness. A pioneering study of interaction of traveling and stationary disturbances is performed. Some specific features of this interaction are identified. The main reason for turbulence emergence in a supersonic boundary layer on a swept wing is demonstrated to be secondary crossflow instability. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 40–46, March–April, 2008.     

曲率对机翼边界层二次失稳影响

采用非线性抛物化扰动方程(NPSE)计算了层流机翼NLF0415(2)在几个工况下横流涡的非线性发展, 应用Floquet理论分析了横流涡的二次失稳. 较系统地分析了后掠机翼在多种参数下表面曲率对其流动稳定性的影响. Haynes证明曲率对于横流在线性稳定性计算(LST)和非线性抛物化扰动方程计算中(NPSE)都起着非常明显的稳定作用; 然而, 该文计算结果表明, 曲率在二次失稳计算中影响不大.      

Experimental study of nonlinear processes in a swept-wing boundary layer at the mach number M=2

Results of experiments aimed at studying the linear and nonlinear stages of the development of natural disturbances in the boundary layer on a swept wing at supersonic velocities are presented. The experiments are performed on a swept wing model with a lens-shaped airfoil, leading-edge sweep angle of 45°, and relative thickness of 3%. The disturbances in the flow are recorded by a constant-temperature hot-wire anemometer. For determining the nonlinear interaction of disturbances, the kurtosis and skewness are estimated for experimentally obtained distributions of the oscillating signal over the streamwise coordinate or along the normal to the surface. The disturbances are found to increase in the frequency range from 8 to 35 kHz in the region of their linear development, whereas enhancement of high-frequency disturbances is observed in the region of their nonlinear evolution. It is demonstrated that the growth of disturbances in the high-frequency spectral range (f > 35 kHz) is caused by the secondary instability.     

Subcritical spatial transition of swept Hiemenz flow

It is known from experimental investigations that the leading-edge boundary layer of a swept wing exhibits transition to turbulence at subcritical Reynolds numbers, i.e. at Reynolds numbers which lie below the critical Reynolds number predicted by linear stability theory. In the present work, we investigate this subcritical transition process by direct numerical simulations of a swept Hiemenz flow in a spatial setting. The laminar base flow is perturbed upstream by a pair of stationary counter-rotating vortex-like disturbances. This perturbation generates high- and low-speed streaks by a non-modal growth mechanism. Further downstream, these streaky structures exhibit a strong instability to secondary perturbations which leads to a breakdown to turbulence.The observed transition mechanism has strong similarities to by-pass transition mechanisms found for two-dimensional boundary layers. It can be shown that transition strongly depends on the amplitude of the primary perturbation as well as on the frequency of the secondary perturbation.     

Linear stability of three-dimensional boundary layers

Stability of compressible three-dimensional boundary layers on a swept wing model is studied within the framework of the linear theory. The analysis based on the approximation of local self-similarity of the mean flow was performed within the Falkner-Skan-Cooke solution extended to compressible flows. The calculated characteristics of stability for a subsonic boundary layer are found to agree well with the measured results. In the case of a supersonic boundary layer, the results calculated for a Mach number M = 2 are also in good agreement with the measured spanwise scales of nonstationary vortices of the secondary flow. The calculated growth rates of disturbances, however, are substantially different from the measured values. This difference can be attributed to a high initial amplitude of disturbances generated in the experiment, which does not allow the linear stability theory to be applied. The evolution of natural disturbances with moderate amplitudes is fairly well predicted by the theory. The effect of compressibility on crossflow instability modes is demonstrated to be insignificant. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 3–14, March–April, 2008.     

Applications of EPSE method for predicting crossflow instability in swept-wing boundary layers

The nth-order expansion of the parabolized stability equation(EPSEn) is obtained from the Taylor expansion of the linear parabolized stability equation(LPSE) in the streamwise direction. The EPSE together with the homogeneous boundary conditions forms a local eigenvalue problem, in which the streamwise variations of the mean flow and the disturbance shape function are considered. The first-order EPSE(EPSE1) and the second-order EPSE(EPSE2) are used to study the crossflow instability in the swept NLF(2)-0415 wing boundary layer. The non-parallelism degree of the boundary layer is strong. Compared with the growth rates predicted by the linear stability theory(LST),the results given by the EPSE1 and EPSE2 agree well with those given by the LPSE.In particular, the results given by the EPSE2 are almost the same as those given by the LPSE. The prediction of the EPSE1 is more accurate than the prediction of the LST, and is more efficient than the predictions of the EPSE2 and LPSE. Therefore, the EPSE1 is an efficient e~N prediction tool for the crossflow instability in swept-wing boundary-layer flows.     

A theoretical investigation of the development of stationary crossflow vortices in the boundary layer on a swept wing

Crossflow instability plays very important role in the transition of the boundary layer on a swept wing, typical in the engineering applications. Experiments revealed that the linear stability theory well predicted the form of the crossflow vortices, but usually much overpredicted their growth rate. Using nonlinear theory of hydrodynamic stability, combined with some other considerations, we were able to obtain the growth rate in good agreement with experimental observations. The project supported by the National Natural Science Foundation of China, Grant No. 19572048     

可压缩横流失稳及其控制

边界层流动转捩的预测与控制一直是流体力学研究中的一个重要问题. 三维边界层流动工程中十分常见, 而横流失稳是导致三维边界层流动转捩的主要原因. 本文综述了近些年来三维边界层失稳和转捩方面的研究概况. 从机理上讨论了横流扰动的感受性、首次失稳、二次失稳和转捩控制等方面的研究进展. 在数值计算方面, 简要概述了线性稳定性理论、非线性稳定性理论和直接数值模拟方法在横流失稳和转捩方面的应用.本文对横流失稳研究当前存在的问题进行了讨论, 对今后研究的发展趋势作了相应展望.     

Optimum Suction Distribution for Transition Control

The optimum suction distribution which gives the longest laminar region for a given total suction is computed. The goal here is to provide the designer with a method of finding the best suction distribution subject to some overall constraints applied to the suction. We formulate the problem using the Lagrangian multiplier method with constraints. The resulting nonlinear system of equations is solved using the Newton–Raphson technique. The computations are performed for a Blasius boundary layer on flat-plate and crossflow cases. For the Blasius boundary layer, the optimum suction distribution peaks upstream of the maximum growth rate region and remains flat in the middle before it decreases to zero at the end of the transition point. For the stationary and travelling crossflow instability, the optimum suction peaks upstream of the maximum growth rate region and decreases gradually to zero. Received 8 May 1997 and accepted 5 November 1998     

Transition control in a three-dimensional boundary layer by direct attenuation of nonlinear crossflow vortices using plasma actuators

Direct numerical simulations are used to probe the potential of plasma actuators to attenuate nonlinear steady crossflow vortices (CFVs). The investigated base flow mimics the three-dimensional boundary-layer flow of a swept wing. The plasma actuators are positioned at selected spanwise positions to weaken oncoming CFVs and thus the associated (secondary) instability. It is shown that both volume forcing against or in the direction of the crossflow (CF) can be effective, and a significant transition delay can be achieved. The spanwise position of the actuators should be such that the actuation-induced downdraft inhibits the CFV. The forcing in the direction of the CF does not reduce the mean CF, and an unfavourable spanwise position of the actuator may directly increase the strength of the CFV and eventually promote turbulence onset. The forcing against the CF never turned out to promote turbulence onset for all investigated positions, because of the favourable reduction of the mean CF. Adding then a second or third actuator downstream at appropriate spanwise positions can yield enhanced transition delay.     

Active Control of Wave Instabilities in Three-Dimensional Compressible Flows

In many fluid flows of practical importance transition is caused by the linear growth of wave instabilities, such as Tollmien–Schlichting waves, which eventually grow to a finite size at which stage secondary instabilities come into play. If transition is to be delayed or even avoided in such flows, then the linear growth of the disturbances must be prevented since control in the nonlinear regime would be a considerably more difficult task. Here a strategy for active control of two-dimensional incompressible and compressible Tollmien–Schlichting waves and its use in controlling the more practically relevant problem of crossflow instability which arises in swept-wing flows is discussed. The control is through an active suction/blowing distribution at the wall though the same result could be achieved by variable wall heating. In order to control the instability it is assumed that the wall shear stress and pressure are known from measurements. It is shown that, certainly at finite Reynolds numbers, it is sufficient to know the flow properties at a finite number of points along the wall. The cases of high and finite Reynolds numbers are discussed using asymptotic and numerical methods respectively. It is shown that a control strategy can be developed to stop the growth of all two-dimensional Tollmien–Schlichting waves at finite and large Reynolds numbers. Some discussion of nonlinear effects in the presence of active control is given and the possible control of other instability mechanisms investigated. Received 1 May 1998 and accepted 24 September 1998     

Nonparallel stability of the flow in an axially rotating pipe

The linear stability of the developing flow in an axially rotating pipe is analyzed using parabolized stability equations (PSE). The results are compared with those obtained from a near-parallel stability approximation that only takes into account the axial variation of the basic flow. Though the PSE results obviously coincide with the near-parallel ones far downstream, when the flow has reached a Hagen-Poiseuille axial velocity profile with superimposed solid-body rotation, they differ significantly in the developing region. Therefore, the onset of instability strongly depends on the axial evolution of the perturbations. The PSE results are also compared with experimental data from Imao et al. [Exp. Fluids 12 (1992) 277], showing a good agreement in the frequencies and wavelengths of the unstable disturbances, that take the form of spiral waves. Finally, a simple method for detecting one of the conditions to characterize the onset of absolute instability using PSE is given.     

The axisymmetric boundary layer beneath a Rankine-like vortex

 An experimental investigation of the three-dimensional boundary layer induced by a Rankine-like vortex with its axis normal to a stationary disk is described. The velocity field through the boundary layer was measured for Reynolds number Re (based on the tangential velocity and radius at the disk edge) ranging from 10 000 to 25 000 at various radial distances by means of a 4-beam, 2-component Laser Doppler Anemometer. Our results show that the nature of the boundary layer is affected by two factors: an inflexional instability caused by the crossflow velocity profile and a stability factor caused by the favorable pressure gradient. At lower Reynolds number, the radial pressure gradient has a very strong stabilizing effect on the boundary layer and acts to revert it to its laminar state upstream of the effusing core. At higher Re the inflexional instability caused by the crossflow velocity dominates while the stabilizing influence of the favorable pressure gradient recedes. As such, laminar reversion likely occurs closer to the effusion core. Thus, the point of laminar reversion moves closer to the effusion core as the Reynolds number is increased. Received 23 May 1996 / Accepted 29 July 1996     

Streamwise streaks and secondary instability in a two-dimensional bent channel

Streamwise streaks generated from a pair of oblique waves and secondary instability of the streaks are studied in a two-dimensional bent channel. Nonlinear parabolized stability equations (NPSE) are employed to investigate streamwise streaks and vortices. A pair of oblique waves from linear stability analysis is imposed as initial disturbances. Generation of streamwise streaks and vortices and subsequent development are described in detail. The case of plane channel is also studied to provide comparable data. Through comparison, the effect of bent region is clearly highlighted. Results of parametric studies to examine the effect of Reynolds number, radius of curvature, and bent angle are also given and discussed in detail. Secondary instability analysis for the modified mean flow due to the streamwise streaks is carried out by solving a two-dimensional eigenvalue problem. Several unstable modes which can be classified into fundamental and subharmonic mode of secondary instability are identified. Among several unstable modes, two modes are turned out to be dominant modes. Details on these two modes including generation mechanism, typical pattern, and dependency on wave number and streak amplitude are discussed. It is found that the presence of bent channel can lead to early oblique-mode breakdown via strong growth of the streamwise streaks due to the curved section. Such large amplitude of streaks and its secondary instability eventually could trigger transition even for small amplitude oblique waves at subcritical channel Reynolds numbers.     

Electrohydrodynamic instability of a dielectric compressible liquid sheet streaming into an ambient stationary compressible gas

The effect of compressibility of fluids on the linear electrohydrodynamic instability of a dielectric liquid sheet issued from a nozzle into an ambient dielectric stationary gas in the presence of a horizontal electric field is investigated. It is found that increasing the Mach number from subsonic to transonic causes the maximum growth rate and the dominant wavenumber of the disturbances to increase, and the increase is higher in the presence of the electric field. Liquid compressibility has been found to have a minimal effect on instability. At constant wavenumber and electric field values, the growth rate of disturbances increases as the gas Mach number tends to 1, and then begins to decrease with further increase in the gas Mach number. At small values of wavenumber, antisymmetrical disturbances grow faster than symmetrical ones, while the growth rate of both types of disturbances approach each other at large wavenumbers, which increases by increasing the electric field values. At small Weber numbers, antisymmetrical disturbances exhibit a higher maximum growth rate and a lower dominant wavenumber than symmetrical disturbances. However, the maximum growth rate and dominant wavenumber of the two types of disturbances are almost identical when both Weber number and electric field values become large. An increase in the gas to liquid density ratio enhances the instability, and this effect is enhanced for high electric field values. Surface tension and electric fields always oppose and increase the development of instability, respectively; and they have opposite effects for long wavelengths and high Weber numbers.     

Instability due to second normal stress jump in two-layer shear flow of the Giesekus fluid

The two-layer Couette flow of superposed Giesekus liquids is examined. In order to emphasize the effect of a jump in the second normal stress difference, the analysis is focused on flows where the shear rate and first normal stress difference are continuous across the interface. In this case, the flow is neutrally stable to streamwise disturbances, but can be unstable for spanwise disturbances driven by a jump in the second normal stress difference. Whether the long and order one waves are stable or not depends on the sign of this difference. Short waves are unstable. In the case of order one wave instability, the mode of maximum growth rate gives rise to stationary ripples perpendicular to the flow. The eigenvalue problem for purely spanwise wave vectors can in principle be solved analytically, although, in general, the analytical solution is too complicated to obtain. In most cases, however, a simplifying assumption can be made which makes analytical solutions feasible. We present such solutions and compare them with purely numerical solutions.     

On the formation of the chain of ring-like vortices in a transitional boundary layer

One mechanism by which laminar flow becomes turbulence is through the instability of vortex interaction. The nonlinear evolution of the interactions between the Λ-vortex and the secondary closed vortex has been investigated. This nonlinear instability process leads to some fundamental changes in the topology of flows. Experimental observations are presented here showing the formation of a chain of ring-like vortices, which generate high-frequency spikes on time traces in a transitional boundary layer. Received: 13 March 2000/Accepted: 5 July 2000     

PSE as applied to problems of secondary instability in supersonic boundary layers

Parabolized stability equations (PSE) approach is used to investigate prob-lems of secondary instability in supersonic boundary layers. The results show that the mechanism of secondary instability does work, whether the 2-D fundamental disturbance is of the first mode or second mode T-S wave. The variation of the growth rates of the 3-D sub-harmonic wave against its span-wise wave number and the amplitude of the 2-D fundamental wave is found to be similar to those found in incompressible boundary layers. But even as the amplitude of the 2-D wave is as large as the order 2%, the maximum growth rate of the 3-D sub-harmonic is still much smaller than the growth rate of the most unstable second mode 2-D T-S wave. Consequently, secondary instability is unlikely the main cause leading to transition in supersonic boundary layers.