The governing equations for the base flow

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The magnifying glass

Once we have selected a representation, or a model, of our problem, we need to focus our attention on a particular characteristic. From the point of view of the perturbations’ evolution, the emphasis can be put on their short-time or their long-time behavior. When selecting the features we are more interested in, we must also select the more convenient representation together with a particular numerical approach.
The most natural approach is to observe the system evolving in time, as if it were an experiment. This can be accomplished using a direct numerical simulation (DNS): once an initial condition has been chosen, the governing equations for the perturbations are advanced in time, and the evolution of the flow in physical space can be observed. While this approach can be used for studying both the short-time and the long-time evolution of the flow, it poses two problems: (i) the most obvious one is the dependence of the solution on the initial condition which represents a somewhat arbitrary choice and is of extreme importance for the short-time evolution of the perturbations; (ii) the second one is the difficulty in extracting, from simple observation of the evolution of a particular initial condition, the coherent structures that represent the true interest of our study (but see [60] for a possible approach). The short-time behavior is of particular interest for non-normal system and is usually approached
in physical space by non-modal analysis [61, 59, 12]. The eigenvalue/eigenvector description, while still possible, is not convenient in analyzing the short-time response because the eigenvectors responsible for the non-normal behavior can be difficult to obtain numerically (but see [49, 50], for a counterexample). The main interest lies in the fact that non-normal systems can present transient growth effects which, as shown in Figure 1.2, can lead to transition even in the case of asymptotically stable flows. The main goal of non-modal analysis is then to investigate the possibility of transient growth and identify the effects of the initial conditions on energy amplification. Of particular interest are the initial conditions leading to the maximum possible transient growth, which are called optimal perturbations: they can be identified by solving an optimization problem for the flow energy (or any equivalent scalar quantity we choose to optimize) over all possible initial conditions. The optimization problem can be treated as a constrained optimization, where the constraints are given by the governing equations and implemented by adjoint fields (also called Lagrangian multipliers). The optimal condition satisfying the constraints is found by identifying stationary points of the Lagrangian. Direct and adjoint equations are obtained by this procedure and can be marched forward and backward in time in order to determine the optimal initial condition. In this sense, non-modal analysis removes the arbitrariness of the initial condition by identifying the most interesting one.
In contrast, the eigenvalue/eigenvector description is more often used in the description of the long-time (asymptotic, or modal) behavior of the system. As we will see in detail, this description is used in this work to identify the coherent structures underlying the flow as well as to identify how the behavior of these structures is influenced by external disturbances. A Lagrangian approach similar to the one just described is used to identify receptivity and sensitivity of an eigenvector (and its corresponding eigenvalue) to forcing and structural modifications of the operator. The adjoint equations and their solutions given by the adjoint fields will play a central role in this process.

Stato dell’arte

The swept-wing attachment-line boundary layer has been investigated from multiple points of view in the past. References to most of the relevant work dating before 2003 can be found in the reviews by Reed and Saric [53, 54, 57, 58].
Most of the research has been concentrated on local models: the swept Hiemenz flow, characterizing the flow impinging on a flat plate with a sweep angle, has been extensively studied as a local approximation of the flow close to the attachment line of a swept wing. DNS computations have been performed by Joslin [32, 33] while the short-time optimal growth has been the subject of the work of Guégan et al. [23, 22, 24] and Obrist & Schmid [49]. Comparison of theoretical and experimental results on crossflow instabilities have been provided by Dagenhart & Saric [13].
In this work, the global modal approach for a realistic configuration of the swept wing is attempted. The same approach has been applied to the swept Hiemenz flow starting from the work of Hall, Malik & Poll [25] who studied the stability of the Görtler-Hämmerlin mode (i.e. a mode having the same streamwise structure as the swept Hiemenz flow) and showed that this three-dimensional flow can, in contrast to the two-dimensional (unswept) Hiemenz flow, become unstable above a critical Reynolds number.
Lin & Malik [38] extended the work of Hall, Malik & Poll by computing several modes of the incom-pressible swept Hiemenz flow using a Chebyshev spectral collocation method and regular polynomials P (x) = xn; n = 0; 1; 2; : : : in order to discretize the normal- and chord-wise direction of their domain. They identified a branch of eigenvalues, all moving at approximately the same phase speed of cr = 0:35 in the spanwise direction — i.e. moving at a velocity equal to 0.35 times the spanwise free-stream velocity component. It was shown that the most unstable mode was the symmetric Görtler-Hämmerlin mode already found by Hall et al. [25]. Less unstable modes were shown to alternate between antisymmetric and symmetric as one descends down the branch towards smaller growth rates.
In a subsequent study [39] they addressed the question of the leading-edge curvature by using a second-order boundary-layer approximation in computing the base flow and showed that the flow was stabilized by increasing the leading-edge curvature. For small distances in the chord-wise direction from the attachment line, this effect was mainly related to the curvature terms appearing in the continuity equation, while the centrifugal acceleration terms in the momentum equations have been found of less importance.
With a similar approach, Obrist & Schmid [48] addressed the same problem by replacing the regular polynomials used by Lin & Malik in the chord-wide discretization with Hermite polynomials. They too found the most unstable mode to be the Görtler-Hämmerlin mode, showing an exponential decay outside the boundary layer, and identified a richer spectrum composed of several branches, continuous and discrete. The non-modal analysis previously mentioned [49] was part of the same work.
Almost all the investigations mentioned so far addressed the simplified model of swept Hiemenz flow. The first global analysis of the leading-edge region was performed by Mack, Schmid & Sesterhenn [44] and Mack & Schmid [42, 43]. They addressed the stability of a compressible flow impinging on a parabolic body with a sweep angle. A high-order finite-difference discretization was used in both the normal and chordwise direction. They showed a global spectrum consisting of different branches: boundary layer modes, acoustic modes and wave-packet modes. Of these, the boundary layer and wave-packet branches are of interest for the current, incompressible study. Additionally, they showed, for the first time, evidence of a connection between attachment-line and crossflow instabilities, a feature already suggested but never proven in previous works [26]. This result was made possible by considering a domain extending beyond the attachment-line region.

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Outline

The present work continues in the wake of the global modal approach used by Mack et al. [40], but analyzes an incompressible flow instead of a compressible one and extends Mack’s results by includ-ing a receptivity and sensitivity analysis. This work is organized in six chapters, the first being this introduction.
In chapter 2 we define the swept-wing problem: governing equations for the base flow and the pertur-bations are derived, and the dimensionless parameters governing the base flow and perturbation behavior will be described. The adjoint equations for the perturbations are obtained starting from a Lagrangian description of the flow and, building on that, receptivity to external forcing and sensitivity to struc-tural modifications will be defined. The concept of the wavemaker in the context of global analysis, as introduced by Giannetti & Luchini [19], will be presented.
Once the theoretical framework underlying the swept-wing flow analysis is presented, our numerical approach is specified. Because numerical issues represent a large part of this work, it is split into two parts. The first part, in chapter 3, includes details related to the discrete representation of our problem: the pressure-form of the nonlinear and linear Navier-Stokes equations — where a Poisson equation for the pressure replaces the mass-conservation equation — and their discretization, the boundary conditions, the grid generation process and the validation of the implemented discretization. A final section contains some considerations on the required grid size. The second part, in chapter 4, is dedicated to the solver used to compute the base flow, based on the multigrid framework. An analysis of the multigrid algorithm from the point of view of an iterative solution process is performed, and the most important parts of the algorithm are analyzed using Local Fourier Analysis (LFA), a useful tool providing theoretical estimates on the convergence rates of iterative methods. The two available algorithmic choices are then introduced, starting from the more well-known Correction Scheme and moving on to the less well-known but more powerful Full Approximation Scheme. At the end of the chapter, some simple test cases are used to identify and overcome possible problems leading to a loss of efficiency.
In chapter 5 we return to the theory covered in chapter 2 and present the results obtained for the swept-wing configuration. A description of the main features of the base flow is followed by a presentation of the results obtained by a global analysis of the linearized Navier-Stokes operator, including the computed part of the spectrum and an in-depth analysis of the least stable eigenvector. The corresponding adjoint field is then analyzed, and the significance of its distribution in the domain is clarified. The identification of the wavemaker region concludes this chapter.
Finally, chapter 6 summarizes the main results of this work and describe some paths that can be followed to build upon what has been accomplished here.
A short Matlab code is provided in Appendix A This is a demonstration code used in producing the results for the test cases at the end of the chapter on multigrid (in section 4.6), but it is not the code used in computing the main results of this work presented in chapter 5. It can nonetheless be useful in providing an outline of how a real multigrid code can be structured.
I found Figure 1.4, from a lecture given by Peter Schmid at Institut Henri Poincaré in Paris, very useful in outlining the main steps required in a receptivity and sensitivity analysis. While the chapters in this work do not follow exactly the same outline, this flowchart can be helpful in maintaining a broad view on this work and in avoiding getting lost in details. The unfilled boxes on the right briefly indicate the numerical tools used at each step.
As a final note in this introduction, I would like to mention that in writing these pages I tried as much as possible to follow and apply the suggestions given by McIntyre in his paper Lucidity and science I: Writing skills and the pattern perception hypothesis [46]. Whether or not I accomplished this task is another story, but McIntyre’s paper is nonetheless a very interesting reading that deserves to be mentioned.

The governing equations for the base flow

The flow in the leading-edge region of an infinitely long swept wing is considered. The wing, as represented in Figure 2.1, is subjected to a uniform flow U1 = fU1; 0; W1g, where U1 and W1 are the chordwise and spanwise components of the uniform velocity field. The sweep angle then satisfies the relation tan = W1=U1. Asterisks denote dimensional quantities.
Two reference systems are defined as shown in Figure 2.1, both centered at the leading edge: (i) a Cartesian coordinate system, whose x-axis is parallel to the chord, whose y-axis points upwards and whose z-axis is in the spanwise direction and (ii) a curvilinear coordinate system, with its s-axis running along the profile, its n-axis normal to the profile surface and sharing its z-axis with the Cartesian coordinate system.

Table of contents :

1 Introduction and Motivation 
2 Problem Definition 
2.1 The governing equations for the base flow
2.2 The governing equations for the perturbations
2.3 The adjoint governing equations for the perturbations
2.4 Receptivity and sensitivity
3 Numerical Approach 
3.1 The pressure form of the Navier-Stokes equations
3.2 Computational domain and discretization
3.3 Validation
3.4 Considerations on domain size and grid size
4 Multigrid 
4.1 A generic iterative solver
4.2 Ingredients of a multigrid solver
4.3 Relaxation
4.4 Correction Scheme – the linear equation
4.5 Full Approximation Scheme – the non-linear equation
4.6 Test cases
5 Global analysis 
5.1 Base flow
5.2 Global analysis of the direct operator
5.3 Adjoint field
5.4 The wavemaker
6 Conclusions and Perspectives 
A Matlab multigrid code

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