Physical Modeling of Vanishing Bores in Open-Channel Flows

When an undular bore is propagating upstream against a subcritical flow in a flat channel, its leading front may weaken and vanish. The present study physically investigated the whole decaying process of vanishing bores in a relatively large facility. The invert was made of mobile or fixed natural river gravels. Detailed sampling of water surface elevations and velocities was performed in combination with high-resolution photo and video recordings. Despite some increased turbulence characteristics linked to the bore front arrival, no sediment motion was observed underneath the vanishing bores. The quantitative results provide a unique benchmark data set for further study of the later-stage bore propagations.

Removal of a dense bottom layer by a gravity current

<p>We investigate the removal of a dense bottom layer by a gravity current, via Navier-Stokes Boussinesq simulations. The problem is governed by a dimensionless thickness parameter for the bottom layer, and by the ratio of two density differences. A quasisteady gravity current propagates along the interface and displaces some of the dense bottom fluid, which accumulates ahead of the gravity current and forms an undular bore or a series of internal gravity waves. Depending on the ratio of the gravity current front velocity to the linear shallow-water wave velocity, we observe small-amplitude waves or a train of steep, nonlinear internal waves. We develop a self-contained model based on the conservation principles for mass and vorticity that does not require empirical closure assumptions. This model is able to predict the gravity current height and the internal wave or bore velocity, generally to within about 10% accuracy. An energy budget analysis provides information on the rates at which potential energy is converted into kinetic energy and then dissipated, and on the processes by which energy is transferred from the gravity current fluid to the dense and ambient fluids. We observe that the energy content of thicker and denser bottom layers grows more rapidly.</p>

Wave Breaking in Undular Bores with Shear Flows

It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

Surface Evolution and Wave Loads on a Horizontal Plate in Different Tsunami-like Waves

Physical experiments are conducted to study the interaction between the tsunami-like waves and the horizontal plate. The surface evolution and wave-induced loads are measured to explore the hydrodynamic characteristics when different waves are employed to simulate the tsunami. The solitary wave, surge wave, and undular bore are generated in laboratory as the simplification of the offshore tsunami wave. The bottom-fixed plate places near the free surface. It is found that the elevated plate attenuates the solitary waves locally, while the submerged plate leads to the wave focus phenomenon. The plate has less influence on the surface variation of the surge wave propagating. Results of loads show the different loading process of each tsunami-like wave. The inertial wave force and the local pressure from the rising surface dominate the inline force and vertical force, respectively. The value of loads induced by the surge wave is less than that of the solitary wave. The undular bore is generated by the superimposition of the solitary wave on the surge wave. The part of the solitary wave plays a local role in the wave force, while the surge part dominates the surface evolution.

Μη γραμμικά κύματα στη ροή κοκκώδους ύλης

Η κοκκώδης ύλη όταν ρέει μέσα σε έναν ανοικτό κεκλιμένο αγωγό μπορεί να θεωρηθεί ως συνεχές μέσο και κατά συνέπεια η κίνησή της επιδέχεται μια περιγραφή υδροδυναμικού τύπου. Η ιδιαίτερη φύση του κοκκώδους ρευστού λαμβάνεται υπόψη μέσω κατάλληλων καταστατικών σχέσεων για την τριβή με τον πυθμένα του αγωγού (αποτέλεσμα των διατμητικών τάσεων) και τις ιξώδεις ορθές τάσεις που αναπτύσσονται στο κοκκώδες στρώμα. Στα δύο πρώτα κεφάλαια της παρούσας διατριβής περιγράφεται αυτή η υδροδυναμική προσέγγιση του κοκκώδους στρώματος, στο πλαίσιο της οποίας η δυναμική του διέπεται από δύο συζευγμένες μη γραμμικές μερικές διαφορικές εξισώσεις (ΜΔΕ). Στις εν λόγω ΜΔΕ, που αποτελούν το κοκκώδες ανάλογο των εξισώσεων Saint-Venant για τη ροή του αβαθούς νερού, κατοπτρίζεται η διατήρηση της μάζας και η μεταβολή της ορμής του κοκκώδους στρώματος αντιστοίχως. Το κοκκώδες στρώμα μπορεί να φιλοξενήσει μια μεγάλη ποικιλία κυματομορφών ανάλογα με την τιμή του αριθμού Froude (Fr) της εισερχόμενης ροής. Στο Κεφάλαιο 3 μελετάμε την περίπτωση Fr < 2/3, όπου η ομοιόμορφη ροή είναι ευσταθής. Για αυτές τις τιμές του Fr, στο κοκκώδες στρώμα μπορεί να αναπτυχθεί ένα μονοκλινές κύμα πλημμύρας όταν η εισροή υλικού από την κορυφή του αγωγού αυξηθεί. Πρόκειται για την κρουστική οδεύουσα δομή που συνδέει δύο περιοχές ομοιόμορφης ροής (πλατώ) με διαφορετικό βάθος. Στο Κεφάλαιο 4, ξεκινώντας από το μονοκλινές κύμα, παρακολουθούμε την αλληλουχία των κυματομορφών που εμφανίζονται στο κοκκώδες στρώμα όταν ο αριθμός Fr αυξάνεται σταδιακά. Προς τούτο εστιάζουμε στις λύσεις οδεύοντος κύματος των προαναφερθέντων ΜΔΕ και διατυπώνουμε το αντίστοιχο δυναμικό σύστημα που αποτελείται από δύο συνήθεις διαφορικές εξισώσεις (ΣΔΕ) πρώτης τάξης. Η ανάλυση ευστάθειας του εν λόγω δυναμικού συστήματος, επικουρούμενη από τα πορτραίτα του χώρου φάσεων για αυξανόμενο Fr, αποκαλύπτει όλες τις διαδοχικές κυματομορφές που δύνανται να αναπτυχθούν στο σύστημα. Λίγο πριν την κρίσιμη τιμή Fr = 2/3 βρίσκουμε μια καινοφανή κυματομορφή, το κοκκώδες undular bore, που αποτελεί μια διαταραγμένη μορφή του μονοκλινούς κύματος, η οποία παρουσιάζει κυματισμούς στο υψηλό πλατώ. Όταν ο αριθμός Fr της εισερχόμενης ροής ξεπεράσει την τιμή 2/3, το υψηλό πλατώ του undular bore αποσταθεροποιείται. Τότε οι κυματισμοί του λειτουργούν ως εστίες περαιτέρω ανάπτυξης των διακυμάνσεων, οδηγώντας στην οργάνωση μιας περιοδικής αλληλουχίας από roll waves. Στο Κεφάλαιο 5 θεωρούμε την αλληλεπίδραση ανάμεσα σε roll waves διαφορετικού πλάτους. Ένα μεγάλο roll wave οδεύει ταχύτερα από ένα μικρό και τα δύο τους συνενώνονται όταν συναντηθούν. Αυτό συνιστά τη βάση μιας διαδικασίας εκτράχυνσης που παρατηρείται στις κοκκώδεις ροές για Fr > 2/3. Ταυτοχρόνως, τα roll waves τείνουν να εξισώσουν το πλάτος τους (και επομένως την ταχύτητά τους) με μια συγκεκριμένη τιμή που αντιστοιχεί στην ισοκατανομή του κοκκώδους υλικού μεταξύ τους. Προφανώς, η τιμή αυτή αυξάνεται κάθε φορά που συμβαίνει μια συνένωση κυμάτων. Όταν πλέον οι διαδοχικές συνενώσεις έχουν μειώσει το πλήθος των roll waves σε τέτοιο βαθμό ώστε τα κύματα να προλαβαίνουν να φτάσουν στο προαναφερθέν επιθυμητό πλάτος πριν συμβεί μια νέα συνένωση, τότε το σύστημα παραμένει ες αεί στην τρέχουσα κατάσταση, στην οποία όλα τα εναπομείναντα roll waves έχουν ίδιο πλάτος και ταχύτητα. Το φαινόμενο αυτό καλείται "πάγωμα" της διαδικασίας εκτράχυνσης της κοκκώδους ροής. Εν κατακλείδι, στο Κεφάλαιο 6 συνοψίζουμε τα πιο σημαντικά μας ευρήματα και παρουσιάζουμε προτάσεις για μελλοντική έρευνα.

The Evolution of Undular Bore in Coastal Zone: Effect of Bottom Slope, Friction and Special Topography

When tsunami waves propagate into shallow and gentle continental shelves, the effects of submarine topography and wave dispersion lead to the appearance of undular bore. This kind of wave is quite different from the initial tsunami wave in the waveform, and its amplitude is enhanced significantly. The undular bore will cause great and repeated runups on the beach, which brings larger destruction to the coastal region. In order to investigate the characteristics of the propagation of undular bore, this paper simulates the evolution from the sinusoidal long wave to the undular bore and analyzes the influencing factors. The beach slope has an important effect on the development of undular bores in terms of the undulation form, amplitude, wave length and occurrence time. The influence of bottom friction and submarine topography on the undulations and soliton fissions are discussed. These bottom conditions will lead to great change of the waveform and amplitude of undular bores.

The Lifecycle of Nonlinear Internal Waves in the Northwestern South China Sea

The life cycle of nonlinear internal waves (NIWs) to the southeast of Hainan Island in the northwestern South China Sea is investigated using synergistic satellite observations, in situ measurements, and numerical simulations. A three-dimensional, fully nonlinear and nonhydrostatic model with ultrafine resolution shows that a diurnal internal tide emanates from a sill in the Xisha Islands at approximately 215 km away from the local shelf break. The internal tide transits the deep basin toward the shelf break and reflects at the sea bottom and seasonal thermocline in the form of a wave beam. Arriving at the shelf break, the internal tide undergoes nonlinear transformation and produces an undular bore. Analyses of in situ measurements reveal that the undular bore appears as sharp depressions of the strong near-surface seasonal thermocline. The undular bore gradually evolves into an internal solitary wave train on the midshelf, which was detected by the spaceborne synthetic aperture radar. This finding has great implications for investigating NIWs in other coastal oceans where waves are controlled by remotely generated internal tides.

Hydrodynamic Features of an Undular Bore Traveling on a 1:20 Sloping Beach

The hydrodynamic characteristics, including local and convective accelerations as well as pressure gradient in the horizontal direction, of the external stream of an undular bore propagating on a 1:20 sloping beach are experimentally studied. A bore with the water depth ratio of 1.70 was generated downstream of a suddenly lifted gate. A high-speed particle image velocimetry was employed to measure the velocity fields during the run-up and run-down motions. The time series of free surface elevation and velocity field/profile of the generated bore, comprising a pure bore accompanied by a series of dispersive leading waves, are first demonstrated. Based on the fast Fourier transform (FFT) and inverse FFT (IFFT) techniques, the free surface elevation of leading waves and the counterpart of pure bore are acquired separately at a specified measuring section (SMS), together with the uniform horizontal velocity of the pure bore. The effect of leading-wave-induced velocity shift on the velocity profiles of the generated bore are then evaluated at the SMS. To understand the calculation procedure of accelerations and pressure gradient, three tabulated forms are provided as illustrative examples. Accordingly, the relationships among the partially depth-averaged values of the non-dimensional local acceleration, convective acceleration, total acceleration and pressure gradient of the generated/pure bore acquired at the SMS versus the non-dimensional time are elucidated. The trends in the non-dimensional accelerations and pressure gradient of the external stream of generated bore are compared with those of the pure bore. During the run-up motion from the instant of arrival of the bore front to the moment of the peak level at the SMS, continuous decrease in the onshore uniform horizontal velocity, and successive deceleration of the pure bore in the onshore direction are evidenced, exhibiting the pure bore under the adverse pressure gradient with decreasing magnitude. However, the pure bore once ridden by the leading waves is decelerated/accelerated spatially and accelerated/decelerated temporally in the onshore direction during the rising/descending free surface of each leading wave. This fact highlights the effect of pre-passing/post-passing of the leading wave crest on the velocity distribution of generated bore. It is also found that, although the leading waves have minor contribution on the power spectrum of the free surface elevation as compared with that of the pure bore, the leading waves do play an important role on the magnitudes of both accelerations and pressure gradient. The largest magnitude of the acceleration contributed by the leading waves is around 26 times the counterpart contributed by the pure bore. Further, during the run-down motion right after the moment for the peak level of the bore, a linear increase in the magnitude of the offshore uniform horizontal velocity and a constant local acceleration with increasing time are both identified. The partially depth-averaged value of the non-dimensional pressure gradient is equal to a small negative constant (−0.0115) in the offshore direction, indicating that the bore is subject to a constant favorable pressure gradient.

Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves

A new type of wave–mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket–mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wavepacket by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit so-called hydrodynamic reciprocity recently described in Maiden et al. (Phys. Rev. Lett., vol. 120, 2018, 144101) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models.