A local scattering theory for the effects of isolated roughness on boundary-layer instability and transition: transmission coefficient as an eigenvalue

2016 ◽  
Vol 794 ◽  
pp. 68-108 ◽  
Author(s):  
Xuesong Wu ◽  
Ming Dong
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Transmission Coefficient ◽  
Roughness Element ◽  
Boundary Value ◽  
S Wave ◽  
Flow Distortion ◽  
Roughness Elements ◽  
Characteristic Wavelength ◽  
The Impact

This paper is concerned with the rather broad issue of the impact of abrupt changes (such as isolated roughness, gaps and local suctions) on boundary-layer transition. To fix the idea, we consider the influence of a two-dimensional localized hump (or indentation) on an oncoming Tollmien–Schlichting (T–S) wave. We show that when the length scale of the former is comparable with the characteristic wavelength of the latter, the key physical mechanism to affect transition is through scattering of T–S waves by the roughness-induced mean-flow distortion. An appropriate mathematical theory, consisting of the boundary-value problem governing the local scattering, is formulated based on triple deck formalism. The transmission coefficient, defined as the ratio of the amplitude of the T–S wave downstream of the roughness to that upstream, serves to characterize the impact on transition. The transmission coefficient appears as the eigenvalue of the discretized boundary-value problem. The latter is solved numerically, and the dependence of the eigenvalue on the height and width of the roughness and the frequency of the T–S wave is investigated. For a roughness element without causing separation, the transmission coefficient is found to be approximately 1.5 for typical frequencies, indicating a moderate but appreciable destabilizing effect. For a roughness causing incipient separation, the transmission coefficient can be as large as $O(10)$, suggesting that immediate transition may take place at the roughness site. A roughness element with a fixed height produces the strongest impact when its width is comparable with the T–S wavelength, in which case the traditional linear stability theory is invalid. The latter however holds approximately when the roughness width is sufficiently large. By studying the two hump case, a criterion when two roughness elements can be regarded as being isolated is suggested. The transmission coefficient can be converted to an equivalent $N$-factor increment, by making use of which the $\text{e}^{N}$-method can be extended to predict transition in the presence of multiple roughness elements.

scholarly journals The impact of dynamic roughness elements on marginally separated boundary layers

2018 ◽  
Vol 855 ◽  
pp. 351-370
Author(s):  
P. Servini ◽  
F. T. Smith ◽  
A. P. Rothmayer
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Separation Bubble ◽  
Roughness Element ◽  
Angle Of Attack ◽  
Dynamic Roughness ◽  
Separation Theory ◽  
Roughness Elements ◽  
Maximum Angle ◽  
The Impact

It has been shown experimentally that dynamic roughness elements – small bumps embedded within a boundary layer, oscillating at a fixed frequency – are able to increase the angle of attack at which a laminar boundary layer will separate from the leading edge of an airfoil (Grager et al., in 6th AIAA Flow Control Conference, 2012, pp. 25–28). In this paper, we attempt to verify that such an increase is possible by considering a two-dimensional dynamic roughness element in the context of marginal separation theory, and suggest the mechanisms through which any increase may come about. We will show that a dynamic roughness element can increase the value of $\unicode[STIX]{x1D6E4}_{c}$ as compared to the clean airfoil case; $\unicode[STIX]{x1D6E4}_{c}$ represents, mathematically, the critical value of the parameter $\unicode[STIX]{x1D6E4}$ below which a solution exists in the governing equations and, physically, the maximum angle of attack possible below which a laminar boundary layer will remain predominantly attached to the surface. Furthermore, we find that the dynamic roughness element impacts on the perturbation pressure gradient in two possible ways: either by decreasing the magnitude of the adverse pressure peak or by increasing the streamwise extent in which favourable pressure perturbations exist. Finally, we discover that the marginal separation bubble does not necessarily have to exist at $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{c}$ in the time-averaged flow and that full breakaway separation can therefore occur as a result of the bursting of transient bubbles existing within the length scale of marginal separation theory.

scholarly journals The impact of static and dynamic roughness elements on flow separation

2017 ◽  
Vol 830 ◽  
pp. 35-62 ◽  
Author(s):  
P. Servini ◽  
F. T. Smith ◽  
A. P. Rothmayer
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Roughness Element ◽  
Separated Flow ◽  
Leading Edge ◽  
Small Region ◽  
Forced Flow ◽  
Dynamic Roughness ◽  
Roughness Elements ◽  
The Impact

The use of static or dynamic roughness elements has been shown in the past to delay the separation of a laminar boundary layer from a solid surface. Here, we examine analytically the effect of such elements on the local and breakaway separation points, corresponding respectively to the position of zero skin friction and presence of a singularity in the roughness region, for flow over a hump embedded within the boundary layer. Two types of roughness elements are studied: the first is small and placed near the point of vanishing skin friction; the second is larger and extends downstream. The forced flow solution is found as a sum of Fourier modes, reflecting the fixed frequency forcing of the dynamic roughness. Solutions for both the static and dynamic roughness show that the presence of the roughness element is able to move the separation points downstream, given an appropriate choice of roughness frequency, height, position and width. This choice is found to be qualitatively similar to that observed for leading-edge separation. Furthermore, for a negative static roughness a small region of separated flow forms at high roughness depth, although there is a critical depth above which boundary-layer breakaway moves suddenly upstream.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

Numerical Method for Singularly Perturbed Parabolic Equations in Unbounded Domains in the Case of Solutions Growing at Infinity

2009 ◽  
Vol 9 (1) ◽  
pp. 100-110
Author(s):  
G. I. Shishkin
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Maximum Norm ◽  
Lateral Part ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.

scholarly journals On the role of acoustic feedback in boundary-layer instability

2014 ◽  
Vol 372 (2020) ◽  
pp. 20130347 ◽  
Author(s):  
Xuesong Wu
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Acoustic Waves ◽  
Feedback Loop ◽  
Peak Frequency ◽  
S Wave ◽  
Flat Plates ◽  
Acoustic Coupling ◽  
Acoustic Feedback ◽  
Roughness Elements

In this paper, the classical triple-deck formalism is employed to investigate two instability problems in which an acoustic feedback loop plays an essential role. The first concerns a subsonic boundary layer over a flat plate on which two well-separated roughness elements are present. A spatially amplifying Tollmien–Schlichting (T–S) wave between the roughness elements is scattered by the downstream roughness to emit a sound wave that propagates upstream and impinges on the upstream roughness to regenerate the T–S wave, thereby forming a closed feedback loop in the streamwise direction. Numerical calculations suggest that, at high Reynolds numbers and for moderate roughness heights, the long-range acoustic coupling may lead to absolute instability, which is characterized by self-sustained oscillations at discrete frequencies. The dominant peak frequency may jump from one value to another as the Reynolds number, or the distance between the roughness elements, is varied gradually. The second problem concerns the supersonic ‘twin boundary layers’ that develop along two well-separated parallel flat plates. The two boundary layers are in mutual interaction through the impinging and reflected acoustic waves. It is found that the interaction leads to a new instability that is absent in the unconfined boundary layer.

Infrared thermography of hypersonic boundary layer transition induced by isolated roughness elements

2021 ◽  
Author(s):  
Shicheng Liu ◽  
Meng Wang ◽  
Hao Dong ◽  
Tianyu Xia ◽  
Lin Chen ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Infrared Thermography ◽  
Roughness Element ◽  
Transition Process ◽  
Boundary Layer Transition ◽  
Hypersonic Boundary Layer ◽  
Layer Transition ◽  
Leading Role ◽  
Roughness Elements ◽  
Element Shape

Roughness element induced hypersonic boundary layer transition on a flat plate is investigated using infrared thermography at Ma = 5 and 6 flow condition. Surface Stanton number is acquired to analyze the effect of roughness element shape and height on the transition process. The correlation between the vortex structure induced by roughness element and the wall heat streaks is established. The results indicate that higher roughness element would induce stronger streamwise heat flux streaks, lead to transition advance in streamwise centerline and increase the width of spanwise wake. Moreover, for low roughness element, the effect of the shape is not obvious, and the height plays a leading role in the transition; for tall roughness element, the effect on accelerating transition for the diamond roughness element is the best, the square is the worst, and the shape plays a leading role in the transition.

scholarly journals Influence of two-dimensional roughness element on boundary layer structure in the favourable pressure gradient region of the swept wing

2019 ◽  
Vol 234 (1) ◽  
pp. 20-27
Author(s):  
Stepan Tolkachev ◽  
Victor Kozlov ◽  
Valeriya Kaprilevskaya
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Layer Structure ◽  
Roughness Element ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Frequency Range ◽  
Swept Wing ◽  
Boundary Layer Structure ◽  
Roughness Elements

In this article, the results of research about stationary and secondary disturbances development behind the localized and two-dimensional roughness elements are presented. It is shown that the two-dimensional roughness element has a destabilizing effect on the disturbances induced by the three-dimensional roughness element lying upstream. In this case, the two-dimensional roughness element causes the appearance of stationary structures, and then secondary perturbations, whose frequency range lies lower than in the case of the stationary vortices excited by a three-dimensional roughness element.

scholarly journals Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks

2011 ◽  
Vol 681 ◽  
pp. 116-153 ◽  
Author(s):  
NICHOLAS J. VAUGHAN ◽  
TAMER A. ZAKI
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Low Frequency ◽  
Finite Amplitude ◽  
Base Flow ◽  
Streamwise Vorticity ◽  
Mean Flow ◽  
S Wave ◽  
Flow Distortion ◽  
Tollmien Schlichting

The secondary instability of a zero-pressure-gradient boundary layer, distorted by unsteady Klebanoff streaks, is investigated. The base profiles for the analysis are computed using direct numerical simulation (DNS) of the boundary-layer response to forcing by individual free-stream modes, which are low frequency and dominated by streamwise vorticity. Therefore, the base profiles take into account the nonlinear development of the streaks and mean flow distortion, upstream of the location chosen for the stability analyses. The two most unstable modes were classified as an inner and an outer instability, with reference to the position of their respective critical layers inside the boundary layer. Their growth rates were reported for a range of frequencies and amplitudes of the base streaks. The inner mode has a connection to the Tollmien–Schlichting (T–S) wave in the limit of vanishing streak amplitude. It is stabilized by the mean flow distortion, but its growth rate is enhanced with increasing amplitude and frequency of the base streaks. The outer mode only exists in the presence of finite amplitude streaks. The analysis of the outer instability extends the results of Andersson et al. (J. Fluid Mech. vol. 428, 2001, p. 29) to unsteady base streaks. It is shown that base-flow unsteadiness promotes instability and, as a result, leads to a lower critical streak amplitude. The results of linear theory are complemented by DNS of the evolution of the inner and outer instabilities in a zero-pressure-gradient boundary layer. Both instabilities lead to breakdown to turbulence and, in the case of the inner mode, transition proceeds via the formation of wave packets with similar structure and wave speeds to those reported by Nagarajan, Lele & Ferziger (J. Fluid Mech., vol. 572, 2007, p. 471).

scholarly journals Boundary Layers in a Two-Point Boundary Value Problem with a Caputo Fractional Derivative

2015 ◽  
Vol 15 (1) ◽  
pp. 79-95 ◽  
Author(s):  
Martin Stynes ◽  
José Luis Gracia
Keyword(s):  
Boundary Layer ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Order Term ◽  
Boundary Value ◽  
Point Boundary ◽  
Leading Term ◽  
Special Case

AbstractA two-point boundary value problem is considered on the interval $[0,1]$, where the leading term in the differential operator is a Caputo fractional derivative of order δ with $1<\delta <2$. Writing u for the solution of the problem, it is known that typically $u^{\prime \prime }(x)$ blows up as $x\rightarrow 0$. A numerical example demonstrates the possibility of a further phenomenon that imposes difficulties on numerical methods: u may exhibit a boundary layer at x = 1 when δ is near 1. The conditions on the data of the problem under which this layer appears are investigated by first solving the constant-coefficient case using Laplace transforms, determining precisely when a layer is present in this special case, then using this information to enlighten our examination of the general variable-coefficient case (in particular, in the construction of a barrier function for u). This analysis proves that usually no boundary layer can occur in the solution u at x = 0, and that the quantity $M = \max _{x\in [0,1]}b(x)$, where b is the coefficient of the first-order term in the differential operator, is critical: when $M<1$, no boundary layer is present when δ is near 1, but when M ≥ 1 then a boundary layer at x = 1 is possible. Numerical results illustrate the sharpness of most of our results.

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