Chester E. Grosch

The Continuous Spectrum of the Orr-Sommerfeld Equation. Part 1. The Spectrum and the Eigenfunctions

It is shown that the Orr-Sommerfeld equation, which governs the stability of any mean shear flow in an unbounded domain which approaches a constant velocity in the far field, has a continuous spectrum. This result applies to both the temporal and the spatial stability problem. Formulae for the location of this continuum in the complex wave-speed plane are given. The temporal continuum eigenfunctions are calculated for two sample problems: the Blasius boundary layer and the two-dimensional laminar jet. The nature of the eigenfunctions, which are very different from the Tollmien-Schlichting waves, is discussed. Three mechanisms are proposed by which these continuum modes could cause transition in a shear flow while bypassing the usual linear Tollmien-Schlichting stage.

Inviscid spatial stability of a compressible mixing layer

Presented are the results of a study of the inviscid spatial stability of a parallel compressible mixing layer. The parameters of this study are the Mach number of the moving stream, the ratio of the temperature of the stationary stream to that of the moving stream, the frequency and the direction of propagation of the disturbance wave. Stability characteristics of the flow as a function of these parameters are given. It is shown that if the Mach number exceeds a critical value there are always two groups of unstable waves. One of these groups is fast with phase speeds greater than 1/2, and the other is slow with speeds less than 1/2. Phase speeds for the neutral and unstable modes are given, as well as growth rates for the unstable modes. It is shown that three-dimensional modes have the same general behavior as the two-dimensional modes but with higher growth rates over some range of propagation direction. Finally, a group of very low frequency unstable modes was found for sufficiently large Mach numbers. These modes have very low phase speeds but large growth rates.

A numerical study of the two-dimensional Navier-Stokes equations in vorticity-velocity variables

Abstract Solution methods for compact finite-difference schemes are applied to the vorticity-velocity form of the two-dimensional unsteady Navier-Stokes equations. Numerical experiments for stagnation point and driven cavity flows are described.

The stability of steady and time-dependent plane Poiseuille flow

The linear stability of plane Poiseuille flow has been studied both for the steady flow and also for the case of a pressure gradient that is periodic in time. The disturbance streamfunction is expanded in a complete set of functions that satisfy the boundary conditions. The expansion is truncated after N terms, yielding a set of N linear first-order differential equations for the time dependence of the expansion coefficients. For the steady flow, calculations have been carried out for both symmetric and antisymmetric disturbances over a wide range of Reynolds numbers and disturbance wave-numbers. The neutral stability curve, curves of constant amplification and decay rate, and the eigenfunctions for a number of cases have been calculated. The eigenvalue spectrum has also been examined in some detail. The first N eigenvalues are obtained from the numerical calculations, and an asymptotic formula for the higher eigenvalues has been derived. For those values of the wave-number and Reynolds number for which calculations were carried out by L. H. Thomas, there is excellent agreement in both the eigenvalues and the eigenfunctions with the results of Thomas. For the time-dependent flow, it was found, for small amplitudes of oscillation, that the modulation tended to stabilize the flow. If the flow was not completely stabilized then the growth rate of the disturbance was decreased. For a particular wave-number and Reynolds number there is an optimum amplitude and frequency of oscillation for which the degree of stabilization is a maximum. For a fixed amplitude and frequency of oscillation the wave-number of the disturbance and the Reynolds number has been varied and a neutral stability curve has been calculated. The neutral stability curve for the modulated flow shows a higher critical Reynolds number and a narrower band of unstable wave-numbers than that of the steady flow. The physical mechanism responsible for this stabiIization appears to be an interference between the shear wave generated by the modulation and the disturbance. For large amplitudes, the modulation destabilizes the flow. Growth rates of the modulated flow as much as an order of magnitude greater than that of the steady unmodulated flow have been found.

Visualizing 3D flow

Line integral convolution (LIC) is an elegant and versatile technique for representing directional information via patterns of correlation in a texture. In this article we discuss some of the facto...We discuss volume line integral convolution (LIC) techniques for effectively visualizing 3D flow, including using visibility-impeding halos and efficient asymmetric filter kernels. Specifically, we suggest techniques for selectively emphasizing critical regions of interest in a flow; facilitating the accurate perception of the 3D depth and orientation of overlapping streamlines; efficiently incorporating an indication of orientation into a flow representation; and conveying additional information about related scalar quantities such as temperature or vorticity over a flow via subtle, continuous line width and color variations.

A criterion for vortex breakdown

A criterion for the onset of vortex breakdown over a wide range of the Reynolds number is proposed. Based upon previous experimental, theoretical, and numerical studies, as well as a new numerical study, an appropriately defined local Rossby number is used to delineate the region where breakdown occurs. Comparisons are made with previously suggested criticality parameters and the unique features of the proposed Rossby number parameter are shown. A number of previous theoretical studies concentrating on inviscid standing‐wave analyses for trailing wing‐tip vortices are reviewed and reinterpreted, along with the previous numerical and experimental studies, in terms of the Rossby number parameter. For the case of trailing wing‐tip‐type vortices, it is shown that previous numerical studies were performed at lower Reynolds numbers than the corresponding laboratory experiments. Utilizing a consistently defined Reynolds number, Rossby number–Reynolds number plots of these previous vortex breakdown studies, for b...

Langmuir turbulence in shallow water. Part 2. Large-eddy simulation

Results of large-eddy simulation (LES) of Langmuir circulations (LC) in a wind-driven shear current in shallow water are reported. The LC are generated via the well-known Craik–Leibovich vortex force modelling the interaction between the Stokes drift, induced by surface gravity waves, and the shear current. LC in shallow water is defined as a flow in sufficiently shallow water that the interaction between the LC and the bottom boundary layer cannot be ignored, thus requiring resolution of the bottom boundary layer. After the introduction and a description of the governing equations, major differences in the statistical equilibrium dynamics of wind-driven shear flow and the same flow with LC (both with a bottom boundary layer) are highlighted. Three flows with LC will be discussed. In the first flow, the LC were generated by intermediate-depth waves (relative to the wavelength of the waves and the water depth). The amplitude and wavelength of these waves are representative of the conditions reported in the observations of A. E. Gargett & J. R. Wells in Part 1 (J. Fluid Mech. vol .000, 2007, p. 00). In the second flow, the LC were generated by shorter waves. In the third flow, the LC were generated by intermediate waves of greater amplitude than those in the first flow. The comparison between the different flows relies on visualizations and diagnostics including (i) profiles of mean velocity, (ii) profiles of resolved Reynolds stress components, (iii) autocorrelations, (iv) invariants of the resolved Reynolds stress anisotropy tensor and (v) balances of the transport equations for mean resolved turbulent kinetic energy and resolved Reynolds stresses. Additionally, dependencies of LES results on Reynolds number, subgrid-scale closure, size of the domain and grid resolution are addressed. In the shear flow without LC, downwind (streamwise) velocity fluctuations are characterized by streaks highly elongated in the downwind direction and alternating in sign in the crosswind (spanwise) direction. Forcing this flow with the Craik– Leibovich force generating LC leads to streaks with larger characteristic crosswind length scales consistent with those recorded by observations. In the flows with LC, in the mean, positive streaks exhibit strong intensification near the bottom and near the surface leading to intensified downwind velocity ‘jets’ in these regions. In the flow without LC, such intensification is noticeably absent. A revealing diagnostic of the structure of the turbulence is the depth trajectory of the invariants of the resolved Reynolds stress anisotropy tensor, which for a realizable flow must lie within the Lumley triangle. The trajectory for the flow without LC reveals the typical structure of shear-dominated turbulence in which the downwind component of the resolved normal Reynolds stresses is greater than the crosswind and vertical components. In the near bottom and surface regions, the trajectory for the flow with LC driven by

Linear stability of Poiseuille flow in a circular pipe

Correction of an error in the matrix elements used by Salwen & Grosch (1972) has brought the results of the matrix-eigenvalue calculation of the linear stability of Hagen–Poiseuille flow into complete agreement with the numerical integration results of Lessen, Sadler & Liu (1968) for azimuthal index n = 1. The n = 0 results were unaffected by the error and the effect of the error for n > 1 is smaller than for n = 1. The new calculations confirm the conclusion that the flow is stable to infinitesimal disturbances. Further calculations have led to the discovery of a degeneracy at Reynolds number R = 61·452 ± 0·003 and wavenumber α = 0·9874 ± 0·0001, where the second and third eigenmodes have equal complex wave speeds. The variation of wave speed for these two modes has been studied in the vicinity of the degeneracy and shows similarities to the behaviour near the degeneracies found by Cotton and Salwen (see Cotton 1977) for rotating Hagen-Poiseuille flow. Finally, new results are given for n = 10 and 30; the n = 1 results are extended to R = 10 6 ; and new results are presented for the variation of the wave speed with α R at high Reynolds number. The high- R results confirm both Burridge & Drazins (1969) slow-mode approximation and more recent fast-mode results of Burridge.

The stability of Poiseuille flow in a pipe of circular cross-section

The stability of Poiseuille flow in a pipe of circular cross-section to azimuthally varying as well as axisymmetric disturbances has been studied. The perturbation velocity and pressure were expanded in a complete set of orthonormal functions which satisfy the boundary conditions. Truncating the expansion yielded a matrix differential equation for the time dependence of the expansion coefficients. The stability characteristics were determined from the eigenvalues of the matrix, which were calculated numerically. Calculations were carried out for the azimuthal wavenumbers n = 0,…, 5, axial wavenumbers α between 0·1 and 10·0 and α R [les ] 50000, R being the Reynolds number. Our results show that pipe flow is stable to infinitesimal disturbances for all values of α, R and n in these ranges.

Blending HF radar and model velocities in Monterey Bay through normal mode analysis

Nowcasts of the surface velocity field in Monterey Bay are made for the period August 1–9, 1994, using HF radar observations blended with results from a primitive equation model. A spectral method called normal mode analysis was used. Objective spatial and temporal filtering were performed, and stream function, velocity potential, relative vorticity, and horizontal divergence were calculated over the domain. This type of nowcasting permits global spectral analysis of mode amplitudes, calculation of enstrophy, and additional analyses using tools like empirical orthogonal functions. The nowcasts reported here include open boundary flow information from the numerical model. Nowcasts using no open boundary flow information, however, still provide excellent results for locations within the observation footprint. This method, then, is useful for filtering high-resolution data like HF radar observations, even when open boundary flow information is unavailable. Also, since the nowcast velocity gradient fields were much less noisy than the observations, this may be an effective method for preconditioning high-resolution observation sets for assimilation into a numerical model.