First-mode Waves Research Articles

Effect of pressure gradient on first mode of instability in compressible boundary layers

The effect of a pressure gradient on the first mode of instability of compressible subsonic and supersonic boundary layers is investigated using linear stability theory. A pressure gradient is studied that generates potential-flow Mach number distributions at the edge of the boundary layer of the form Me=cxn, where c is a constant and x is the dimensionless streamwise distance. Variations are calculated for the maximum growth rates of three-dimensional first-mode waves with different edge Mach numbers and different levels of both adverse and favorable pressure gradients. A favorable pressure gradient is shown to have a stabilizing effect on first-mode waves. However, at high edge Mach numbers, a favorable pressure gradient becomes less effective in stabilizing first-mode waves. The frequencies and streamwise and spanwise wave numbers that correspond to the maximum growth rates of first-mode waves decrease as the pressure gradient becomes more favorable at all Mach numbers when the Reynolds number R=1500 and at Me≥2 when R=600. Setting the Prandtl number to unity significantly increases the maximum growth rates of first- and second-mode waves at high Mach numbers compared with setting it to the realistic value for air of 0.72. Predicted transition in flow over a flat plate using the N-factor criterion is found to be due to first-mode waves up to free-stream Mach numbers of 6–6.5.

Oblique-mode breakdown and secondary instability in supersonic boundary layers

Laminar–turbulent transition mechanisms for a supersonic boundary layer are examined by numerically solving the governing partial differential equations. It is shown that the dominant mechanism for transition at low supersonic Mach numbers is associated with the breakdown of oblique first-mode waves. The first stage in this breakdown process involves nonlinear interaction of a pair of oblique waves with equal but opposite angles resulting in the evolution of a streamwise vortex. This stage can be described by a wave–vortex triad consisting of the oblique waves and a streamwise vortex whereby the oblique waves grow linearly while nonlinear forcing results in the rapid growth of the vortex mode. In the second stage, the mutual and self-interaction of the streamwise vortex and the oblique modes results in the rapid growth of other harmonic waves and transition soon follows. Our calculations are carried all the way into the transition region which is characterized by rapid spectrum broadening, localized (unsteady) flow separation and the emergence of small-scale streamwise structures. The r.m.s. amplitude of the streamwise velocity component is found to be on the order of 4–5 % at the transition onset location marked by the rise in mean wall shear. When the boundary-layer flow is initially forced with multiple (frequency) oblique modes, transition occurs earlier than for a single (frequency) pair of oblique modes. Depending upon the disturbance frequencies, the oblique mode breakdown can occur for very low initial disturbance amplitudes (on the order of 0.001% or even lower) near the lower branch. In contrast, the subharmonic secondary instability mechanism for a two-dimensional primary disturbance requires an initial amplitude on the order of about 0.5% for the primary wave. An in-depth discussion of the oblique-mode breakdown as well as the secondary instability mechanism (both subharmonic and fundamental) is given for a Mach 1.6 flat-plate boundary layer.

Internal solitary wave breaking and run-up on a uniform slope

Laboratory experiments have been conducted to study the shoaling of internal solitary waves of depression in a two-layer system on a uniform slope. The shoaling of a single solitary wave results in wave breaking and the production of multiple turbulent surges, or boluses, which propagate up the slope. Significant vertical mixing occurs everywhere inshore of the breaking location. The kinematics of the breaking and bolus runup are described and a breaking criterion is found. The energetics of the breaking are investigated. Over the range of parameters examined, 15 (±5) % of the energy lost from first-mode wave motion inshore of the break point goes into vertical mixing.

Effect of pressure gradient on the stability of compressible boundary layers

An investigation into the influence of pressure gradient (wall shaping) on the stability of compressible boundary layers is presented. The steady, compressible, nonsimilar boundary-layer equations are solved with a potential flow velocity distribution corresponding to a power law edge Mach number distribution. The stability of the mean flow is investigated using the small-disturbance compressible linear stability theory. The results indicate that two-dimensional second-mode (Mack) waves can be stabilised with pressure gradient. However, the effectiveness of the pressure gradient on the natural laminar flow control of two-dimensional second-mode waves decreases at hypersonic speeds. The maximum growth rate varies almost linearly with the pressure gradient level. The effect of pressure gradients on stabilising three-dimensional first-mode waves is much larger than their effect on stabilising two-dimensional first-mode waves.

Effect of heat transfer on the stability of compressible boundary layers

The effect of heat transfer on the spatial stability of compressible boundary layers is investigated using a formulation that accounts for variable-fluid properties of both the mean-flow and stability problems as well as a constant Prandtl number formulation. It is found that heat transfer is more effective in stabilizing or destabilizing flows at low Mach numbers than at high Mach numbers. Cooling always stabilizes first-mode waves and destabilizes second-mode waves. Cooling increases the peak amplifications of second-mode waves and shifts them towards higher frequencies. The least stable second-mode wave remains two-dimensional in the presence of heat transfer. The theoretical results are in good agreement with the experimental data of Lysenko and Maslov [ J. Fluid Mech. 147, 39 (1984)] for the first mode and in reasonable agreement with their experimental data for the second mode. The theoretical results are also compared with the experimental data of Lebiga et al. Archs Mech. 31, 397 (1979)] and Kosinov et al. [ J. Fluid Mech. 219, 621 (1990)].

Effect of Suction on the Stability of Supersonic Boundary Layers. Part II—First-Mode Waves

The effect of suction on the first mode of instability of compressible two-dimensional boundary layers is investigated. Suction is found to be more effective in stabilizing the viscous instability, and hence it is more effective at low Mach numbers. Suction decreases the amplification rates at all frequencies and narrows down the band of unstable frequencies. Moreover, for a given frequency, suction decreases the amplification rates at all streamwise locations. Variations of the growth rates of the most amplified first-mode waves with mass flux are found to be almost linear.

Effect of heat transfer on the subharmonic instability of compressible boundary layers

The effect of heat transfer on the subharmonic instability of a two-dimensional compressible boundary layer over a flat plate is analyzed using the Floquet model. The resulting problem is solved numerically by using both finite differences and the computer code suport. Results are presented for different Mach numbers. For supersonic flows, results for the first and second modes are presented. The results for an adiabatic flat plate are in good agreement with the results of the numerical simulations of Thumm et al. [in Laminar-Turbulent Transition, edited by D. Arnal and R. Michel (Springer-Verlag, Berlin, 1990), p. 271; (private communication)]. When the wall temperature is decreased and the primary wave is a second-mode wave, it is found that the subharmonic instability is shifted to higher frequencies and downstream locations. When the primary wave is a first-mode wave, the effect of cooling might be stabilizing or destabilizing depending on the frequency. When the primary wave is a first mode, cooling stabilizes the subharmonic wave at low spanwise wave numbers and destabilizes it at higher spanwise wave numbers. When the primary wave is either a first- or a second-mode wave, the most amplified subharmonic mode shifts toward a higher spanwise wave number as the cooling level increases.

Scattering of Inertial Waves by Rough Bathymetry

Abstract Interactions between near-inertial waves and rough bathymetry are studied theoretically and numerically. Rough bathymetric features cause scattering, even when their length scales are much smaller than the wavelengths of the incident waves. The scattering efficiency depends on the relative slopes of the incident wave propagation and bottom features. Scattered wavelength are comparable to wavelengths of the bathymetry. In a steady-state situation over an isolated bump, most of the kinetic energy is associated with upward propagating waves, in agreement with observations. A spectral model shows that first-mode incident waves with wavelengths >150 km, and second-mode waves with wavelengths >50 km are completely scattered into smaller wavelengths, primarily into wavelengths smaller than the bathymetry spectrum roll-off, 40 km. This model is applied to a spectrum of incident internal waves. The principal interactions involve the scattering of low-frequency, low-wavenumber incident waves into higher wa...

Large-Scale Wind-Driven Ocean Response in the Australian Coastal Experiment Region

Abstract Theory is used to predict the large-scale wind-driven response on the eastern coast of Australia in the region of the Australian Coastal Experiment. An analytic model for water of constant depth suggested that the response in deep water could be negligible. This was confirmed by a model with more realistic topography, which showed that the ocean response was confined to the continental shelf and upper continental slope. The response was baroclinic except at the middle and outer shelf where it was quasi-barotropic. Because of the dominance of the first coastal-trapped wave mode and the influence of an effective southern boundary, sea level and longshore current propagated northward at a speed comparable to the first-mode coastal-trapped wave speed (4.3 m s−1).

Remarks on disputed numerical results in compressible boundary-layer stability theory

Past work on the influence of Mach number on the viscous and inviscid instability of flat-plate boundary layers is reviewed, and new spatial calculations are presented. These calculations support the previous view that viscosity is only stabilizing for both two- and three-dimensional first-mode waves above M1=3.0, and for second-mode waves at all Mach numbers. It is concluded that the calculations of Wazzan, Taghavi, and Keltner that show viscous instability at M1=6.0 for first-mode 50° waves, and at M1=3.0 for two-dimensional second-mode waves, are not correct.

Internal Tides on the Australian North-West Shelf: A Preliminary Investigation

Abstract An analysis of current meter data and cross-shelf temperature measurements from the Australian North-West Shelf shows the existence of internal waves of semi-diurnal period. Vertical displacements of density interface am seen to reach ∼30 m, a value nearly half the water depth. Baroclinic currents are isolated from the measurements and have amplitudes reaching 0.2 m s−1 in the cross-shelf direction. The baroclinic motion is shown to be consistent with a first-mode internal wave propagating in the onshore direction at phase speed of approximately 0.4 m s−1 with wavelength of 20 km and being rapidly damped in amplitude while propagating across the shelf. The dissipation of the internal tide across the shelf appears to result from the influence of turbulent mixing processes. There is little coherence between the barotropic tide and the internal tide and it is suggested that this results from the internal tide being generated over a large section of the shelf slope down to a depth of approximately 90...

A shear instability induced by a composite internal wave

Experiments have been performed in a thermally stratified two-layer shear flow with a nonzero mean velocity to observe the effects of artifically introduced low frequency internal waves on higher frequency, Kelvin–Helmholtz waves growing spatially on the density interface. Phillips’ model for wave-induced shear across an interface is extended to the two first-mode wave trains which can propagate in such a flow. It is shown that the Richardson number becomes a function of the phase of the internal composite waves and that its minimum value is either at the front or at the back face of the waveform depending on the sign of the background vorticity. Measurements are presented of the phase-averaged temperature and velocity fields of generated internal waves and show the waves breaking down on the face where the Richardson number was found to be at a minimum. The breakdown is characterized by the rapid growth of high frequency waves which appear to be phase locked on the parent internal gravity wave. Possible instances where this type of breakdown could occur in nature are discussed.

Water circulation and shelf waves in the northern Great Barrier Reef lagoon

Currents and sea levels were measured at a number of locations in the Great Barrier Reef (GBR) lagoon from about 10 to 13� S., during the period October-December 1979. A strong non-tidal, low-frequency modulation of all sea-level and current data was found. The currents nearshore were driven northward by the wind, and then at least partially blocked by the dense network of reefs to the north of 10� s. The water then flowed southward in deeper water adjacent to the reef, driven by a longshore pressure gradient. The low- frequency sea-level data, though not the current records, showed northward phase propagation at speeds characteristic of a first-mode shelf wave trapped in the lagoon between the shore and the reef. Data are presented revealing the intrusion of low-salinity water, through Bligh Entrance, in the GBR lagoon, as a result of river discharges in the Gulf of Papua. It is suggested that low-frequency longshore currents may periodically flush these river plumes from the GBR lagoon and enhance interaction between reefs. In the Coral Sea in front of reef passages, the large horizontal velocities may result in forces upwelling by selective withdrawal and jet entrainment.

Excitation of Rayleigh waves

Rayleigh waves recorded at Lubbock, Texas, from natural earthquakes and from nuclear explosions are compared. Natural earthquakes generate both first-mode and second-mode Rayleigh waves, but the explosions generate only first-mode Rayleigh waves. Absence of second-mode Rayleigh waves from the explosions is related to a lack of SV energy in the source mechanism.

A computer program for plotting wavefronts and rays from a point source in dispersive mediums

A computer program to plot wavefronts and rays from a point source in a two-dimensional velocity field is discussed. The program is used to plot ‘fan’ refraction diagrams from London for first-mode Rayleigh waves over a selected area of the North Atlantic, the refraction being assumed to be caused by the variation of ocean depth over an infinite elastic solid. Such diagrams are prepared for frequencies 0.4, 02, 0.133, and 0.1 cps and the theoretical ratio of energy observed on land to the energy actually present, when the sources are situated at different points in the region, is computed for each of the frequencies. It is found that the continental shelf has a ‘low-pass’ filter characteristic over the frequency range. The relevance of this effect in the study of microseismic noise in the ocean bottom is discussed.