First-mode Waves Research Articles

Asymptotic boundary condition for numerical modeling of wave packets in a supersonic boundary layer

Within the framework of linear stability theory, a simple non-stationary boundary condition is developed to simulate the far-field asymptotic behavior of linear wave packets propagating in a compressible boundary layer. This condition allows us to skip the linear stage of instability evolution in direct numerical simulations (DNS) of laminar-turbulent transition. It is also suggested to perform numerical simulations of the linear stage without recalculating the Jacobi matrix in the Newton iteration procedure. Robustness and efficiency of the both approaches are evaluated by computations of the first-mode wave packet propagating in the boundary layer on a flat plate at freestream Mach number 2. DNS of the nonlinear stage show that the wave packet quickly breaks down into a turbulent spot, if its hump amplitude is umax′≥0.1U∞. This confirms an amplitude nature of the transition onset criterion. The spectral analysis indicates that the breakdown mechanism differs from the oblique, fundamental or subharmonic resonance and requires additional studies.

Linear and Nonlinear Disturbance Evolution on the Frustum of Hypersonic Ogive Cylinders

Aspects of transition mechanisms on a 14° sharp-nosed ogive cylinder at Mach 6 are elucidated by considering linear and nonlinear disturbance evolution of freestream stochastic and wave packet forcing at different locations downstream of the ogive-cylinder junction. The spectral response of the stochastic forcing displays favorable agreement with experimental observations from Air Force Research Laboratory on which the configuration is based. Intermittent wave packets are generated from small-amplitude continuous freestream pressure forcing. Linear wave packet evolution reveals that Mack modes are the dominant primary instabilities, followed by relatively weaker first-mode waves. The nonlinear wavepacket displays a three-legged wall pressure perturbation signature, which is traced to spanwise curvature effects, and fundamental resonance is the dominant secondary instability mechanism. At higher amplitudes of excitation, weak signatures of subharmonic and oblique resonance phenomena are identified. Downstream placement of the small-amplitude disturbance diminishes amplification of first-mode waves; however, with high-amplitude forcing, the location of actuation does not significantly vary the disturbance evolution signature or the secondary instability mechanisms.

Estimating propagation speed and direction, and vertical displacement of second-mode nonlinear internal waves from ADCP measurements

Propagation speed and direction, and vertical displacement of the second-mode nonlinear internal waves (NLIWs) over the continental shelf of the East China Sea are estimated by applying the iterative method, previously validated only for the first-mode NLIWs, to the along-beam velocities measured with the standard acoustic Doppler current profiler (ADCP). Two ADCPs mounted in trawl-resistance bottom mounts were deployed in upward looking configurations on the seabed at stations spaced 7.4 km apart, perpendicular to isobaths of the continental slope. During the observation period of two days, two wave trains arrived at an interval of approximately one day, whereas the flow was dominated by the semi-diurnal internal tide with a diurnal modulation. The results of the iteration show that the second-mode NLIWs of the convex type propagate shoreward at a speed of 0.44 (0.36) m s−1 at the offshore (onshore) station. The amplitude of the leading wave is estimated to be about 20 m at the offshore station, which is reduced to about 10 m at the onshore station. All the parameters agree remarkably well with those estimated independently from the backscatter signal of the ADCPs. The propagation time of 6.8 h estimated for the averaged propagation speed and direction is also consistent with a 6.9-h lag of the arrival time of the wave train between the two stations. In addition to the propagating features of the second-mode NLIWs, the iterative method reveals the wave-induced velocity associated with the radiation of short, small-amplitude first-mode waves from the second-mode NLIW, which has been observed only with shipboard echo sounders.

Observations of Internal Structure Changes in Shoaling Internal Solitary Waves Based on Seismic Oceanography Method

High spatial resolution and deep detection depths of seismic reflection surveying are conducive to studying the fine structure of the internal solitary wave. However, seismic images are instantaneous, which are not conducive to observing kinematic processes of the internal solitary waves. We improved the scheme of seismic data processing and used common-offset gathers to continuously image the same location. In this way, we can observe internal fine structure changes during the movement of the internal solitary waves, especially the part in contact with the seafloor. We observed a first-mode depression internal solitary wave on the continental slope near the Dongsha Atoll of the South China Sea and short-term shoaling processes of the internal solitary wave by using our improved method. We found that the change in shape of waveform varies at different depths. We separately analyzed the evolution of the six waveforms at different depths. The results showed that the waveform in deep water deforms before that in shallow water and the waveform in shallow water deforms to a greater degree. We measured four parameters of the six waveforms during the shoaling including phase velocity, amplitude, wavelength, and slopes of leading and trailing edge. The phase velocity and amplitudes of waveforms in shallow water increase, the wavelengths decrease, and the slopes of trailing edge gradually become larger than that of the leading edge, while the amplitudes of the deep water waveforms do not change significantly and the phase velocities decrease. Our results are consistent with previous studies made by numerical simulations, which suggest the effectiveness of the new processing scheme. This improved scheme cannot only study the internal solitary waves shoaling, but also has great potential in the study of other ocean dynamics.

Effect of cavity on hypersonic flat-plate boundary layer instability

The effects of a cavity on hypersonic boundary layer instabilities are investigated experimentally. First- and second-mode wave amplitude evolution is determined using pressure sensors and high-resolution Schlieren technology. The results indicate that when a cavity is located downstream of the fast- and slow-mode synchronization points, it suppresses the second-mode wave and promotes first-mode wave growth. In contrast, when the cavity is located upstream of the synchronization point, the growth of the second-mode wave is promoted while that of the first-mode wave is suppressed. During disturbance evolution, nonlinear interactions are observed due to the phase-locked mechanism that relates the two unstable modes. The transition locations for various cases are confirmed using temperature-sensitive paint technology, and the influences of the cavity on the transition are explained from the perspective of unstable mode amplitude evolution.

Characteristics of dispersion curves for Love channel waves in transversely isotropic media

Coal seams have a pronounced bedding structure with developed cracks and exhibit significant anisotropy. However, few studies have examined the frequency dispersion properties of channel waves in anisotropic coal seams. In this study, numerical solutions are calculated using the generalized reflection-transmission coefficient method for the dispersion curves of Love channel waves in vertical transversely isotropic (VTI) and horizontal transversely isotropic (HTI) medium models. Moreover, the frequency dispersion characteristics of Love channel waves in several typical transversely isotropic models are analyzed. We find that the dispersion curves for isotropic and VTI media differ significantly. In addition, the phase and Airy-phase velocities in VTI media are higher than those in isotropic media. Thus, neglecting this difference in practical channel wave detection will result in large detection errors. The dispersion curves for the isotropic and HTI media do not differ significantly, and the Airy-phase velocities of various modes are similar. The group-velocity curve for a coal seam model containing a dirt band is found to be extremely irregular. The fundamental-mode Airy phase is not pronounced, but the first-mode Airy phase can be clearly observed. Hence, first-mode channel waves are suitable for detecting dirt bands.

Flat-plate hypersonic boundary-layer flow instability and transition prediction considering air dissociation

The effects of air dissociation on flat-plate hypersonic boundary-layer flow instability and transition prediction are studied. The air dissociation reactions are assumed to be in the chemical equilibrium. Based on the flat-plate boundary layer, the flow stability is analyzed for the Mach numbers from 8 to 15. The results reveal that the consideration of air dissociation leads to a decrease in the unstable region of the first-mode wave and an increase in the maximum growth rate of the second mode. High frequencies appear earlier in the third mode than in the perfect gas model, and the unstable region moves to a lower frequency region. When the Mach number increases, the second-mode wave dominates the transition process, and the third-mode wave has little effect on the transition. Moreover, when the Mach number increases from 8 to 12, the N-factor envelope becomes higher, and the transition is promoted. However, when the Mach number exceeds 12, the N-factor envelope becomes lower, and the transition is delayed. The N-factor envelope decreases gradually with the increase in the altitude or Mach number.

Effect of glow discharge on hypersonic flat plate boundary layer

Glow discharge is introduced as an artificial disturbance to investigate the evolution of first- and second-mode instabilities in a hypersonic flat plate boundary layer. Experiments are conducted in a Mach 6.5 quiet wind tunnel using Rayleigh scattering visualization and particle image velocimetry (PIV). Detailed analysis of the experimental observations is provided. It is found that the artificially introduced 17 kHz disturbance, which belongs to the first-mode frequency band, can effectively enhance first-mode waves. Moreover, it can enhance second-mode waves even more intensely. Possible mechanisms to explain this phenomenon are discussed.

Instability wave-streak interactions in a hypersonic boundary layer at flight conditions.

The interaction of stationary streaks undergoing nonmodal growth with modally unstable instability waves in a hypersonic boundary-layer flow is studied using numerical computations. The geometry and flow conditions are selected to match a relevant trajectory location from the ascent phase of the HIFiRE-1 flight experiment; namely, a 7 degree half-angle, circular cone with 2.5 mm nose radius, freestream Mach number equal to 5.30, unit Reynolds number equal to 13.42 m-1, and wall-to-adiabatic temperature ratio of approximately 0.35 over most of the vehicle. This paper investigates the nonlinear evolution of initially linear optimal disturbances that evolve into finite-amplitude streaks, followed by an analysis of the modal instability characteristics of the perturbed, streaky boundary-layer flow. The investigation is performed with stationary direct numerical simulations (DNS) and plane-marching parabolized stability equations (PSE), in conjunction with partial-differential-equation-based planar eigenvalue analysis. The overall effect of streaks is to reduce the peak amplification factors of instability waves, indicating a possible downstream shift in the onset of laminar-turbulent transition. The present study confirms previous findings that the mean flow distorsion of the nonlinear streak perturbation reduces the amplification rates of the Mack-mode instability. More importantly, however, the present results demonstrate that the spanwise varying component of the streak can produce a larger effect on the Mack-mode amplification. The study with selected azimuthal wavenumbers for the stationary streaks reveals that a wavenumber of approximately 1.4 times larger than the optimal wavenumber is more effective in stabilizing the planar Mack-mode instabilities. In the absence of unstable first-mode waves for the present cold-wall condition, transition onset is expected to be delayed until the peak streak amplitude increases to nearly 35 percent of the freestream velocity, when intrinsic instabilities of the boundary-layer streaks begin to dominate the transition process. For streak amplitudes below that limit a significant net stabilization is achieved, yielding a potential transition delay that can exceed 100 percent of the length of the laminar region in the uncontrolled case.

Instability wave-streak interactions in a supersonic boundary layer.

Optimal initial conditions for transient growth in a two-dimensional boundary layer flow correspond to stationary, counter-rotating vortices that subsequently develop into streamwise elongated streaks, which are characterized by an alternating pattern of low and high streamwise velocity. For incompressible flows, previous studies have shown that boundary layer modulation due to streaks below a threshold amplitude level can stabilize the Tollmien-Schlichting instability waves, resulting in a delay in the onset of laminar-turbulent transition. In the supersonic regime, the linearly, most-amplified waves become three-dimensional, corresponding to oblique, first-mode waves. This change in the character of dominant instabilities leads to an important change in the transition process, which is now dominated by oblique breakdown via nonlinear interactions between pairs of first-mode waves that propagate at equal but opposite angles with respect to the free stream. Because the oblique breakdown process is characterized by a rapid amplification of stationary streamwise streaks, artificial excitation of such streaks may be expected to promote transition in a supersonic boundary layer. Indeed, suppression of those streaks has been shown to delay the onset of transition in prior literature. Consistent with those findings, the present study shows that optimally growing stationary streaks indeed destabilize the first-mode waves, but only when the spanwise wavelength of the instability waves is equal to or smaller than twice the streak spacing. Transition in a benign disturbance environment typically involves first-mode waves with significantly longer spanwise wavelengths, and hence, these waves are stabilized by the optimal growth streaks. Thus, as long as the amplification factors for the destabilized, short wavelength instability waves remain below the threshold level for transition, a significant net stabilization is achieved, yielding a transition delay that is comparable to the length of the laminar region in the uncontrolled case.

High-resolution modelling of a large-scale river plume

Evolution of a large-scale river plume is studied numerically using the Massachusetts Institute of Technology general circulation model. The model parameters were set close to those observed in the area of the Columbia River mouth. The fine-resolution grid along with the non-hydrostatic dispersion included in the model allowed for the reproduction of detailed inner plume structure, as well as a system of internal waves radiated from the plume’s boundary. It was found that not only first-mode but also second- and third-mode internal waves are radiated from the plume at the latest stages of its relaxation when the velocity of the front propagation drops below an appropriate wave phase speed of internal baroclinic mode. The model output shows that the amplitude of these high-mode waves is of the same order as the leading first-mode waves, which in combination with the specific vertical structure (location of the maximum structure function beyond the pycnocline layer) creates favourable conditions for the generation of shear instabilities. High-resolution model output also reveals evidence of a fine internal structure of the plume characterized by the presence of secondary fronts inside the plume and secondary internal wave systems propagated radially from the lift-off area to the outer boundary. These structures intensify the mixing processes within the propagating plume with predominance of the entrainment mechanism developing on the lower boundary between the plume’s body and underlying waters. The scheme of horizontal circulation in the plume was reproduced by the methodology of Lagrange drifters released near the mouth at different depths.

Second-mode nonlinear internal waves over a sloping bottom

The characteristic features of second-mode internal wave propagation over a sloping bottom are investigated by numerical simulation based on the Korteweg-de Vries equation. A comparison of the transformations that occur for first- and second-mode internal solitons in the course of their propagation over a sloping bottom under the hydrological conditions of the South China Sea is carried out. Convex and concave second-mode waves are considered, and the possibility of their transition from the first to the second state in the course of their propagation from the deep ocean to the shelf is demonstrated. This is an analog of the effect of a change in the internal wave polarity, which earlier was known to occur for only first-mode internal waves.

RESPONSE ANALYSIS OF CONCRETE PILES SUBJECTED TO LATERAL IMPACT

One-dimensional wave theory is assumed to interpret longitudinal waves impulse response or impact echo testing results in concrete piles. Transient flexural waves tests are alternatively introduced to evaluate piles where the tops are inaccessible for conventional longitudinal wave testing. The resulting interpretation for flexural waves based on one-dimensional wave theory is improper due to various velocities at low frequencies. In this research, treating a pile as a waveguide, three-dimensional wave theory provides the dispersion relation between phase flexural velocity and frequency. Using the resonance solutions and first-mode flexural wave curve is a simple and efficient method to find the resonant frequency and phase velocity by knowing pile dimension and flexural IR results. The phase velocities at resonant frequencies measured in several flexural wave tests have good matches with the first-mode wave theoretical predictions at frequencies below 3,000 Hz. The boundary conditions for practical feasibility are discussed for different in-situ conditions.

Stability of Supersonic Swept-Wing Boundary Layers with or without Crossflow

We examine the stability of three-dimensional supersonic boundary layers over an infinite swept wing made from the semispan model of the NAL supersonic experimental airplane. The original wing was designed so as to suppress both streamwise and crossflow instabilities under cruise conditions. It is confirmed through linear stability analyses of the full boundary layer and the boundary layer without crossflow that the spatial growth of first-mode waves in the full boundary layer is mainly due to streamwise instability and restrained near the design angle of attack.

Propagation of Rossby Waves over Ridges Excited by Interannual Wind Forcing in a Western North Pacific Model

Numerical experiments with a multi-level general circulation model have been performed to investigate basic processes of westward propagation of Rossby waves excited by interannual wind stress forcing in an idealized western North Pacific model with ocean ridges. When the wind forcing with an oscillation period of 3 years is imposed around 180°E and 30°N, far from Japan, barotropic waves excited by the wind can hardly cross the ridges, such as the Izu-Ogasawara Ridge. On the other hand, a large part of the first-mode baroclinic waves are transmitted across the ridges, having net mass transport. The propagation speed of the first-mode baroclinic wave is accelerated (decelerated) when an anticyclonic (cyclonic) circulation is formed at the sea surface, due to a deeper (shallower) upper layer, and to southward (slightly northward) drift of the circulation. Thus, when the anticyclonic circulation is formed on the northern side of the cyclonic one, they propagate almost together. The second-mode baroclinic waves converted from the first-mode ones on the ridges arrive south of Japan, although their effects are small. The resulting volume transport variation of the western boundary current (the Kuroshio) reaches about 60% of the Sverdrup transport variability estimated from the wind stress. These characteristics are common for the interannual forcing case with a longer oscillation period. In the intraseasonal and seasonal forcing cases, on the other hand, the transport variation is much smaller than those in the interannual forcing cases.

The Interaction of a Surface Wave with Waves on a Diffuse Interface

Abstract A resonant interaction analysis is used to study the two-dimensional interaction of a surface wave with two waves on a diffuse interface in a stratified fluid. The fluid system is modeled as a combination of two homogenous finite-depth layers with a free surface, divided by a thin third layer. Two possibilities for the interaction are found: one between the surface wave and two first-mode interfacial waves and a second between the surface wave, a first-mode, and a second-mode interfacial wave. By taking the nondimensional density difference δ as a small perturbation parameter, the evolution equations of the interfacial waves are derived asymptotically. The analysis indicates that growth of the interfacial waves is possible only in the first case. The effects of viscosity on the interaction are discussed, and some preliminary experimental results are presented.

Buoyancy and Mixed Layer Effects on the Sea Surface Height Response in an Isopycnal Model of the North Pacific

Abstract An isopycnal model of the North Pacific is used to demonstrate that the seasonal cycle of heating and cooling and the resulting mixed layer depth entrainment and detrainment cycle play a role in the propagation of wind-driven Rossby waves. The model is forced by realistic winds and seasonal heat flux to examine the interaction of nearly annual wind-driven Rossby waves with the seasonal mixed layer cycle. Comparison among four model runs, one adiabatic (without diapycnal mixing or explicit mixed layer dynamics), one diabatic (with diapycnal mixing and explicit mixed layer dynamics), one with the seasonal cycle of heating only, and one with only variable winds suggests that mixed layer entrainment changes the structure of the response substantially, particularly at midlatitudes. Specifically, the mixed layer seasonal cycle works against Ekman pumping in the forcing of first-mode Rossby waves between 17° and 28°N. South of there the mixed layer seasonal cycle has little influence on the Rossby waves...

Diffraction of internal waves by a circular cylinder near the pycnocline

Propagation of internal waves over a circular cylinder under the conditions of a continuous stratification characterized by the presence of a high-gradient density layer (the pycnocline) of finite thickness is studied. The dependences of the coefficent of wave propagation on the wavelength of the first-mode incident wave for various thicknesses of the pycnocline are obtained. In the diffraction of internal waves, substantial nonlinear effects are shown to occur, which result in the appearance of waves of double oscillation frequency compared to the frequency of the incident waves. The generation coefficient for these waves is found.

Waves along the Equator in the Atlantic*

Abstract Spectra of two hundred days of data from five inverted echo sounders deployed along the equator in 1983–84 and a thousand days of 30 crossings of the equator by the TOPEX/POSEIDON altimeter in 1993–95 are both found to have enhanced variance of sea surface displacement in two frequency bands centered nominally about 50−1 and 25−1 cycles/day. Complex empirical orthogonal function analysis of the two bands finds much of the variance being accounted for by eastward and westward propagating waves, respectively, that are coherent across the basin. The eastward propagating wave is identifiable as a first-mode Kelvin wave with a period of 54 (49) days in the sounder (altimeter) dataset, a basinwide wavelength and a speed of 2.1 (1.8) m s−1. It is suggested that the westward propagating wave is a surface expression of the meridional oscillation of the Equatorial Undercurrent about the equator, presumably induced by a tropical instability wave, with a period of 24 (28) days, a wavelength of 600 (580) km, ...

On transition in supersonic and hypersonic boundary layers

The roles of the first and second modes of instability in high-speed boundary-layer transition are quantified. Linear stability theory and the N-factor criterion are used to predict transition location. The effect of heat transfer is also studied; the results demonstrate that the transition reversal phenomenon can be explained by the opposite effect of cooling on the first and second modes of instability. A heating strip can be used to stabilize first-mode waves at supersonic speeds. Compressibility is destabilizing at free-stream Mach numbers of 2-3.5. The predicted transition location is due to the oblique first modes of instability, up to free-stream Mach numbers between 6 and 6.5. At higher Mach numbers, the predicted transition location is due to a combination of two-dimensional (2-D) first and second modes of instability.