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Influence of the temperature ratio S: Baroclinic torque
The effect of a non-uniform temperature profile on the spatio-temporal in-stability of a jet is demonstrated for a temperature ratio S = 0.5, with all other parameters identical to the isothermal case described in the previous section. The axisymmetric linear impulse response of the heated jet is given in figure 1 (thick lines) for comparison with the isothermal case. For the jet column mode (vg < 0.170), the heating is seen to give rise to an overall increase of the growth rate σ, while the real frequency takes on lower values. In agreement with the analysis of Monkewitz & Sohn [85], the S = 0.5 case is found to be absolutely unstable (σ(0) > 0). The complex wavenumbers of the jet column modes are hardly affected by the temperature ratio. The parabola-shaped σ distribution of the shear layer modes is shifted towards lower group velocities as compared to the isothermal case, but the maximum growth rate σmax remains approximately the same. The growth rates of the azimuthal modes of the heated jet, displayed in figure 2.4b, are found to display the same trend. As in isothermal jets, the axisymmetric and first azimuthal modes are in close competition for high group velocities. All azimuthal modes are convectively unstable at S = 0.5.
It has been suggested by Soteriou & Ghoniem [103] that differences in the instability characteristics of homogeneous and non-homogeneous shear layers may be ascribed to the action of the baroclinic torque. According to these authors, the presence of a baroclinic vorticity dipole within a rolled-up eddy may explain the lateral displacement of the eddy core into the low-density stream as well as the bias of its convection speed towards the velocity of the high-density stream. Both of these features are in qualitative agreement with numerical observations [103].
Following this idea, the role of baroclinic effects in the linear impulse response of a heated jet is quantitatively assessed by solving a modified dis-persion relation, in which the baroclinic torque term is counterbalanced by appropriate forcing. Only the axisymmetric case is considered here. In the presence of source terms denoted as Sx and Sr , the linear inviscid momentum equations become ∂u′ ′ ∂ub ∂u′ 1 ∂p′ = −v − ub − + Sx (2.14a)
Influence of the shear layer thickness, Reynolds number and Mach number
The distinction between jet column and shear layer modes implies a sep-aration of scales between the jet radius R and the momentum shear layer thickness θ. For low values of R/θ, towards the end of the potential core in a spatially developing jet, this assumption is no longer valid. The effect of R/θ on the transition from convective to absolute instability in hot jets is ex-plored in figure 2.6. Contours of marginal absolute instability (ω0,i = 0) are displayed in the S–R/θ plane for the axisymmetric and the first azimuthal mode. The absolute/convective boundary of the axisymmetric mode is iden-tical with figure 8 of Ref. [56] and also in excellent agreement with the re-sults given in Ref. [85]. Absolute instability is found to first occur for the axisymmetric jet column mode at a critical temperature ratio S = 0.713 for R/θ = 26. Higher values of R/θ have a slight stabilizing effect. Below R/θ∼ 15 the critical value of S decreases sharply. Monkewitz & Sohn [85] have shown that absolute instability of the m = 1 mode in a top-hat jet profile requires much stronger heating than is necessary for the m = 0 mode. However, in temporal [4, 43] as well as in spatial [78] jet instability stud-ies, the m = 1 mode has been found to display larger growth rates than its axisymmetric counterpart at very low R/θ. The m = 1 absolute instability boundary in the S–R/θ plane has therefore been included in figure 2.6. It is confirmed that absolute instability always occurs first for the axisymmetric mode, even at values of R/θ as low as 6.
Growth rates of the full linear impulse response in a thick shear layer jet with R/θ = 5, S = 1, Ma = 0 and Re = ∞ are displayed in figure 2.7 for azimuthal wave numbers m ≤ 2. Higher-order azimuthal modes are stable everywhere. The σ(vg ) distributions should be compared to the thin shear layer case R/θ = 20 of figure 2.4. Note that the discontinuity that separates the axisymmetric jet column and shear layer modes in the R/θ = 20 jet is not observed in figure 2.7. A detailed inspection of the spatial branches reveals that the axisymmetric absolute instability mode (vg = 0) still arises from the pinching of the k+- and k1−-branches, as defined in section 2.4.1. However, at higher group velocities, both k−-branches first merge with each other, and the pinching at σ(vg ) then takes place between the k+- and a combined k1−/2-branch. This behavior is illustrated in figure 2.8 for a profile with R/θ = 10, S = 1 and a group velocity vg = 0.3. Note that the k1− and k2− branches are no longer distinct for ωi < 0.487, whereas pinching with the k+ branch occurs for ωi = 0.259. The resulting spatio-temporal modes cannot be categorized as being distinctly of the jet column or shear layer type, but rather of mixed character. These mixed axisymmetric modes display lower growth rates than the formerly distinct shear layer modes. In the R/θ = 5 case of figure 2.7, the maximum axisymmetric temporal growth rate has now fallen below the σmax of the first helical mode. The merging of the k1− and k2− branches therefore explains the dominance of the m = 1 over the m = 0 mode observed in temporal stability studies of thick shear layer jets [4, 43].
Computational grids
The grid point distributions used in the simulations presented in the following chapters are briefly documented here. In section 4.2, two types of flow configurations are considered. Both are computed on the same radial grid point distribution (figures 3.6a,c). As only acoustic wave lengths λ ∼ 35 need to be resolved in the far field, Δr can be chosen very large for r 5. However, the stretching rate from one radial point to the next is kept below 4% in the physical region, in order to minimize the generation of spurious waves. The radial grid distribution in figures 3.6a,c within the physical region r < 46 corresponds to the function Δr = 2f (r), with Δr = f (r) being the grid spacing that has been used in the calculation of the baseflow (see figure 3.1). Thus the baseflow quantities are available directly at the grid points used in the DNS, without necessitating interpolation. The axial grid spacing (figure 3.6b) is kept constant throughout the physical region. In the computations of section 4.2.4, Δx = 0.05 was chosen in order to provide a high resolution of baseflow variations. The baseflows considered in section 4.2.5 vary very slowly, and a resolution Δx = 0.1 was found to be sufficient.
Since the computations of section 4.2 were the first published attempt to capture global modes in a direct numerical simulation, the construction of the grid was guided by great prudence. Later tests showed that in particular the radial resolution of the near field (figure 3.6c) is excessive, and that a bicubic interpolation, as implemented in Matlab, of the baseflow onto a well-designed DNS grid yields highly accurate results. The computational grid used in the simulations presented in section 4.3 is conceived to allow more time-efficient simulations for a parametric study. The radial and axial grid point distributions are displayed in figure 3.7. The axial sponge region, not shown entirely in figure 3.7b, extends down to x = 125, where Δx = 3.35. Grid independence of the results has been demonstrated in test calculations with a R/θ = 25, S = 0.5 baseflow profile on a finer grid, where Δr and Δx were decreased by a factor 1.5. Any influence of box effects (see Buell & Huerre [16]) on the simulation results has been excluded in tests on a larger computational domain, with a physical region 0 ≤ x ≤ 50.
Table of contents :
1 Introduction
1.1 Amplifiers and oscillators
1.2 Self-sustained oscillations in hot jets
1.3 Absolute instability in hot jets
1.4 From absolute to global instability
1.5 Acoustic field of instability wave packets
1.6 Objectives
1.7 Outline
2 Linear impulse response in hot round jets
2.1 Introduction
2.2 Problem formulation
2.3 Numerical method
2.4 Results
2.4.1 Incompressible inviscid jet
2.4.2 Influence of temperature ratio: Baroclinic torque
2.4.3 Influence of the shear layer thickness, Reynolds number and Mach number
2.5 Conclusions
2.A Compressible spatial eigenvalue problem
3 Direct numerical simulation method
3.1 Introduction
3.2 Flow model
3.2.1 Flow Variables
3.2.2 Perturbation equations
3.3 Baseflow
3.3.1 Boundary layer equations
3.3.2 Numerical method
3.3.3 Validation
3.4 Numerical solution of the perturbation equations
3.4.1 Time advancement
3.4.2 Spatial derivatives
3.4.3 Spatial filtering
3.4.4 Computational grids
3.5 Boundary Conditions
3.5.1 Symmetry conditions at the jet axis
3.5.2 Sponge zones
3.5.3 Inflow Conditions and Forcing
3.5.4 Tests of the inflow boundary conditions
3.6 Validation: Linear impulse response
4 Nonlinear global instability
4.1 Introduction to front dynamics
4.1.1 Nonlinear versus linear absolute instability
4.1.2 Non-parallel baseflows
4.1.3 Semi-infinite flows with upstream boundary
4.4.2 Nonlinear global modes in hot jets
4.2.1 Introduction
4.2.2 Problem formulation
4.2.3 Numerical methods and validation
4.2.4 Nonlinear global mode in a jet with a pocket of absolute instability
4.2.5 Nonlinear global modes in jets with absolutely unstable inlet
4.2.6 Concluding remarks
4.4.3 Frequency selection in globally unstable round jets
4.3.1 Introduction
4.3.2 Problem formulation
4.3.3 Numerical method
4.3.4 Onset and frequency of self-sustained oscillations
4.3.5 Inner structure of rolled-up stratified vortices
4.3.6 Conclusion
4.4 Further remarks
4.4.1 Influence of inflow boundary conditions
4.4.2 Pseudo-turbulent states in thin shear layer jets
5 Acoustic field of a global mode
5.1 Integration of the Lighthill equation
5.1.1 Axisymmetric source terms
5.2 Test: a linearly diverging forced jet
5.2.1 Baseflow
5.2.2 Forcing
5.2.3 Near field results
5.2.4 Acoustic field in the DNS
5.2.5 Acoustic field according to the Lighthill equation
5.3 Acoustic radiation from a globally unstable jet
5.3.1 Acoustic field in the DNS
5.3.2 Acoustic field according to the Lighthill equation
5.3.3 Decomposition of the entropy-related source term
5.4 Conclusion
6 ´Epilogue
6.1 Conclusions
6.2 Suggestions for future work
Appendix
A.1 Introduction
A.2 Governing equations and numerical method
A.3 Linear stability analysis
A.4 Numerical results
A.5 Comparison with the experiment
A.6 Comparison to theoretical predictions
A.7 Conclusion
