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Forced and natural isothermal jets.
Isothermal jet ows are typical noise ampli ers, meaning that in an ideal sit-uation the ow would be steady but in practice external perturbations even at low levels are ampli ed by the ow. These disturbances may be hydro-dynamic, e.g. coming from the fan in an experiment, acoustic, or produced by the vibrations of the jet pipe. For a given con guration, if the velocity of a jet U0 is gradually increased, the behavior of the structures excited in the jet varies. The ow is initially laminar, with large-scale oscillations, as schematically represented in gure 1.1(a d). As the Reynolds number based on the initial jet radius R increases, the initially laminar ow experiences a transition to turbulence further downstream, as shown in gure 1.1(e). For even higher jet velocities it becomes di cult to identify coherent ow struc-tures in unforced (or natural) jets where perturbations are driven by the background noise. \Orderly structure in jet turbulence » (Crow and Cham-pagne 1971) can however be observed for Reynolds numbers of the order of 104 to 106 when a low level of controlled forcing is applied to the ow, as also demonstrated by Moore (1977).
Perturbations in jet ows develop on top of a strongly non-parallel mean velocity pro le characterized by several length scales. The jet issues from a cylindrical pipe of radius R, and the ow inside this pipe is characterized by a boundary layer momentum thickness 0. In practice, 0 is tens or hundreds of times smaller than R. The mean ow in the free-jet region, schematically described in gure 1.2, is such that the thickness of the shear layer, which is initially of the order of 0 at the nozzle, increases linearly in the streamwise direction over approximately eight jet radii. This region where the centerline velocity remains approximately equal to the exit velocity U0 is referred to as the potential core. At the end of the potential core the shear layer thickness is of order R, and further downstream velocity pro les are approximately Gaussian and the centerline velocity begins to decrease.
As described by Crighton (1981), two main types of instabilities, char-acterized by di erent time and space scales, are usually found in isothermal jet ows. First is the shear-layer instability that develops on the scale of the shear layer momentum thickness . Michalke (1971) showed that the typical frequency f of this instability (the frequency at which the associated spatial growth rate is maximum) is characterized by a Strouhal number based on the shear layer thickness St 0 f 0=U0 0:017, and that the corresponding wave length is of the order of . In natural jets, the size of these structures in the azimuthal direction may range from the order of 0 to the order of R. Due to the rapid variation of the shear layer thickness, such an instabil-ity wave can only be sustained near the exit of the pipe. This mechanism may however generate large scale oscillations through non-linear processes. Kibens (1980) performed an experiment where a jet is forced at a frequency close to that of the most ampli ed shear-layer instability at the nozzle. The vortex passage frequency at di erent streamwise locations is displayed in g-ure 1.3. It appears that successive vortex pairings cause the frequency to be halved several times until the Strouhal number based on the jet diameter, StD = f D=U0, is of the order 0:4. The frequency decreases through discrete steps roughly as 1=x, i.e. as 1= , so that the local Strouhal number St remains of the same order of magnitude through the potential core.
Figure 1.3: Vortex passage frequency in the shear layer in the forced jet experiment performed by Kibens (1980), as a function of the downstream direction (+ symbols). The dashed line, StD = D=x, corresponds to the vortex pairing model of Laufer and Monkewitz (1980). Data taken from Ho and Huerre (1984).
jet: comparison between an unforced case (thick gray line) and a forced case at StD = 0:3 (thin solid line). For the latter case, the dashed line corresponds to the uctuations at StD = 0:3, the dash-dotted line to the uctuations at StD = 0:6 and the dotted line to the rest. (b): smoke visualization of the ow response to a forcing at StD = 0:3. Data taken from Crow and Champagne (1971).
A di erent behavior has been observed in the experiments of Crow and Champagne (1971). In this case forcing is applied at lower frequencies, for Strouhal numbers based on the jet diameter ranging from 0:15 to 0:6. The amplitude of velocity uctuations measured on the jet axis is displayed in gure 1.4(a) for a forcing at StD = 0:3. Results indicate that the dominant frequency of the ow response, shown in gure 1.4(b), remains equal to the forcing frequency throughout the potential core. The authors report that
Sound radiation from subsonic jets. 5
optimal excitation of these ow structures occurs for StD 0:3. This type of unstable structures is referred to as the preferred mode or jet-column mode. The preferred frequency varies depending on the experimental conditions as well as on the physical quantity measured to evaluate the ampli cation: Moore (1977), who performed experiments at low forcing levels so that non-linear e ects are weak, reports an optimal Strouhal number of 0:5 for velocity uctuations and 0:45 for pressure uctuations. Gutmark and Ho (1983) review results from a dozen of experiments that nd values ranging from 0:3 to 0:6. Figure 1.5 shows that the preferred Strouhal number is also observed in natural jets, i.e. when no controlled forcing is applied and only the incoming noise is ampli ed. The azimuthal distribution of this jet column mode is dominated by azimuthal wavenumbers m = 0; 1 and 2 as the typical size of the vortical structures is of the order of the jet radius (see Parekh et al. (1988) for helical forcing).
Acoustic radiation from subsonic jets has extensively been studied both ex-perimentally and numerically. Figure 1.6 displays instantaneous contours of the acoustic eld radiated by natural subsonic jets obtained in numerical simulations. The acoustic far- eld exhibits a wide variety of wave-lengths as well as a strong dependence on the angle of observation with respect to the jet axis, #, as shown in gure 1.7(b) (# = 0 corresponds to the downstream direc-tion). As reviewed by Karabasov (2010), these features are associated with the presence of two sound generation mechanisms. First, ne scale turbulent uctuations in the near- eld radiate a broad-band sound that dominates for large values of #. Closer to the jet axis, acoustic waves mostly come from large scale structures and are characterized by a more peaky spectrum. A typical frequency spectrum of the sound pressure level is displayed in g-ure 1.7(a): it exhibits a maximum at a Strouhal number StD 0:2, thereby con rming the relevance of the jet-column oscillations in sound generation processes.
Local and global stability
Flow stability has traditionally been studied within the assumption that the wavelength of the instability mechanism is short compared to the typical scale of the streamwise ow development. This allows the stability problem to be Fourier-decomposed in the streamwise and azimuthal (or spanwise) directions, so that only the cross-stream direction remains to be discretized. This assumption results in numerical calculations small enough that they have been performed since the 60’s. Several types of problems may be treated within this framework, and they are referred to in this thesis as local stability analyses:
in temporal stability problems the long time behavior of perturbations with a prescribed real streamwise wavenumber is considered (Michalke 1964).
the signaling problem, i.e. the response of the ow to a time-harmonic localized perturbation, is described in terms of spatial eigenmodes at a given real frequency (Michalke 1965).
the analysis of the dispersion relation between complex frequencies and wavenumbers also allows to study the response of the ow to a spatially and temporally localized impulse, and to distinguish between convec-tive and absolute instabilities (Huerre and Monkewitz 1985).
the short-term temporal ampli cation of spatially distributed pertur-bations is described within the optimal perturbation formalism (Reddy and Henningson 1993) the receptivity to external forcing is analyzed in terms of the resolvent of the ow equations for a given real frequency and real wavenumber (Trefethen et al. 1993).
Non-parallel e ects in the signaling problem may be approximately ac-counted for while still considering the discretization of one-dimensional prob-lems. In the WKB approximation (Crighton and Gaster 1976), the cross-stream distribution of the perturbation is assumed to be that of the k+ spa-tial instability branch (in the sense of e.g. Huerre and Monkewitz (1990)), and its downstream evolution is solved forward in x. The Parabolized Sta-bility Equations (Herbert 1997) also involve the solution of a series of one-dimensional problems by neglecting upstream traveling information, but they consider a general distribution of perturbations in the cross-stream direction.
It is now possible to treat linear stability problems in a framework where no assumption is made regarding the order of magnitude of the perturbation wavelength by discretizing all the non-homogeneous directions. Such ap-proaches are referred to as global, in contrast with the local analyses described above. This distinction does not correspond to a di erence in the methodol-ogy or mathematical concepts involved, but to the investigation of di erent situations. Using a two- or three-dimensional discretization, the temporal eigenmodes, optimal perturbations and optimal forcing (receptivity) can be analyzed by using exactly the same formalism as in the local approach. The modal analysis of the linearized ow equations gives access to the growth or decay rate, to the frequency and to the spatial structures of the eigenmodes. As nite domains are considered in the streamwise direction, the presence of unstable modes characterizes situations where perturbations grow exponen-tially in time at each location, in the same way as the absolute instability introduced in a local framework. On the contrary, in a local approach, tem-porally unstable modes may re ect either convectively or absolutely unstable behavior. Temporal eigenmodes in a global framework allow the characteriza-tion of ow bifurcations (Barkley et al. 2002) and, when the adjoint equations are also considered, of their variation with respect to the base ow (Marquet et al. 2008). The computation of eigenmodes for 2D problems may also be used to study the coupling between instability mechanisms in di erent ow regions (Mack et al. 2008). The meaning of optimal perturbation and optimal forcing results obtained in a global framework (Monokrousos et al. 2010) also di ers from their counterparts in a local approach. In a global framework, the optimal initial condition is streamwise localized, in contrast to the local approach where it is extended in the streamwise direction. In a sense, the global optimal perturbation analysis contains the impulse response problem. The same holds for the global optimal analysis: it contains the signaling problem. The local and global approaches are therefore complementary to describe and understand the dynamics of perturbations in non-parallel ows.
Objectives
The main objective of this thesis is to provide a description of the instability mechanisms that lead to the development of the preferred mode of a jet while still in a linear regime, as experimentally observed by Moore (1977). The local spatial problem, solved by Michalke (1971), exhibits at most one unsta-ble mode associated with the shear layer instability. In order to provide an understanding of the large-scale instability structures, Crighton and Gaster (1976) and Strange and Crighton (1983) treated the signaling problem using a WKB approximation, which amounts to considering that the perturbation continually evolves in the streamwise direction in the form of a local shear layer mode. Such an analysis gives results that are in reasonable agreement with measurements by Crow and Champagne (1971). Piot et al. (2006) and Gudmundsson and Colonius (2009) treated this problem for compressible jets using PSE, and found good agreement with experimental data obtained in natural jets. Global approaches have been used in the context of laminar supersonic isothermal jet ows by Nichols and Lele (2011b). Two families of stable modes were identi ed: downstream traveling Kelvin-Helmholtz waves and upstream traveling disturbances linked to acoustic waves in the outer ow. The optimal perturbations of the jet were also computed, and very high ampli cation levels were reached. In another study (Nichols and Lele 2011a) the authors report the presence of unstable Kelvin-Helmholtz modes at low frequencies in a heated jet con guration. By analogy with the Ginzburg{Landau model, Monkewitz (1989) and Huerre and Monkewitz (1990) described the preferred mode as a \slightly damped global mode » maintained by a low-level of forcing. This hypothesis is investigated in this thesis in terms of the linearized Navier{Stokes equations. Following the recent stability analyses in a global framework by for example Mack et al. (2008), Nichols and Lele (2011b) and Monokrousos et al. (2010), modal and non-modal linear stability concepts are applied to jet ows. In order to consider both near eld uctuations and acoustic radiation, these studies are carried out in both compressible and incompressible settings.
Table of contents :
1 Introduction
1.1 Forced and natural isothermal jets
1.2 Sound radiation from subsonic jets
1.3 Local and global stability
1.4 Objectives
1.5 Outline
2 Numerical methods
2.1 Compressible
2.1.1 Non-dimensional equations
2.1.2 Spatio-temporal discretization
2.1.3 Adjoint equations
2.2 Incompressible
2.2.1 Equations
2.2.2 Spatio-temporal discretization
2.3 External packages
2.3.1 PETSc (Portable, Extensible Toolkit for Scientic Computation)
2.3.2 SLEPc (Scalable Library for Eigenvalue Problem Computations)
3 Eigenvalue solver for compressible ows
3.1 Introduction
3.2 Paper: A relaxation method for large eigenvalue problems
4 Base ows
4.1 Steady jet ows
4.2 Mean turbulent ows
4.3 Model jet ow
4.3.1 Free jet
4.3.2 Pipe
4.3.3 Matching
5 Modal analysis of the jet dynamics
5.1 Introduction
5.1.1 Spatial and temporal instability
5.1.2 Local shear-layer and jet-column modes
5.1.3 Convective instability
5.1.4 Helical perturbations
5.2 Paper: Modal and transient dynamics of jet ows
6 Optimal forcing of incompressible jets
6.1 Introduction
6.2 Paper: The preferred mode of incompressible jets
6.3 Eect of the azimuthal wave number
6.4 Optimal forcing of the laminar base
6.5 A remark on the projection on stable eigenmodes
6.5.1 Method
6.5.2 Ginzburg-Landau problem
6.5.3 Application to the incompressible jet problem
7 Optimal forcing of subsonic jets
7.1 Introduction
7.2 Forcing and measure of the response
7.2.1 Forcing
7.2.2 Measure of the response
7.3 Numerical procedure
7.4 Results
7.4.1 Optimal energy responses: near eld
7.4.2 Acoustic radiation of the responses of maximum energy 128
7.4.3 Optimal acoustic radiation
7.4.4 Transients
7.5 Conclusions and outlook
8 Conclusions and outlook
