Steady Wave Research Articles

Linear stability of transversely modulated finite‐amplitude capillary waves on deep water

AbstractWe investigate the three‐dimensional linear stability of the periodic motion of pure capillary waves progressing in permanent form on water of infinite depth for the whole range of wave amplitudes. After introducing a coordinate transformation based on a conformal map for two‐dimensional steady capillary waves, we perform linear stability analysis of finite‐amplitude capillary waves in the transformed space. To solve the linearized equations for small amplitude disturbances, it is assumed that the wavelengths of the disturbances in the transverse direction are much longer than those in the propagation direction and, therefore, the disturbances are weakly three‐dimensional. This assumption along with the periodicity of solutions allows us to write the linearized equations as an eigenvalue problem in matrix form. Following a perturbation theory for matrices, we expand the solutions of this eigenvalue problem in terms of a small parameter measuring the weak three‐dimensionality, and numerically obtain approximate eigenvalues. For weakly three‐dimensional superharmonic disturbances, the numerical results demonstrate that the pure capillary waves are two‐dimensionally stable, but three‐dimensionally unstable for almost all wave amplitudes. On the other hand, for subharmonic disturbances that are known to be two‐dimensionally unstable, it is found that the long‐wavelength disturbances in the transverse direction reduce the two‐dimensional growth rate near the critical amplitude beyond which the pure capillary waves are unstable.

Stratified Large-Amplitude Steady Periodic Water Waves with Critical Layers

By means of a conformal mapping and bifurcation theory, we prove the existence of large-amplitude steady stratified periodic water waves, with a density function depending linearly on the streamfunction, which may have critical layers and overhanging profiles. We also provide certain conditions for which these waves cannot overturn.

On the accuracy and applicability of a new implicit Taylor method and the high-order spectral method on steady nonlinear waves.

This paper presents an investigation and discussion of the accuracy and applicability of an implicit Taylor (IT) method versus the classical higher-order spectral (HOS) method when used to simulate two-dimensional regular waves. This comparison is relevant, because the HOS method is in fact an explicit perturbation solution of the IT formulation. First, we consider the Dirichlet-Neumann problem of determining the vertical velocity at the free surface given the surface elevation and the surface potential. For this problem, we conclude that the IT method is significantly more accurate than the HOS method when using the same truncation order, M, and spatial resolution, N, and is capable of dealing with steeper waves than the HOS method. Second, we focus on the problem of integrating the two methods in time. In this connection, it turns out that the IT method is less robust than the HOS method for similar truncation orders. We conclude that the IT method should be restricted to M = 4, while the HOS method can be used with M ≤ 8. We systematically compare these two options and finally establish the best achievable accuracy of the two methods as a function of the wave steepness and the water depth.

Experimental and modeling analysis of detonation in circular arcs of the conventional high explosive PBX 9501

We examine the diffraction dynamics of a two-dimensional (2D) detonation in a circular arc of the conventional HMX-based, high performance, solid explosive PBX 9501, for which the detonation reaction zone length scale is estimated to be of the order of 100–150 µm. In this configuration, a steady propagating detonation will develop, sweeping around the arc with constant angular speed. We report on results from three PBX 9501 arc experiments, exploring the variation in linear speed on the inner and outer arc surfaces for the steady wave along with the structure of the curved detonation front, as a function of varying inner surface radius and arc thickness. Comparisons of the properties of the motion of the steady wave for each arc configuration are then made with a spatially-distributed PBX 9501 reactive burn model, calibrated to detonation performance properties in a 2D planar slab geometry. We show that geometry-induced curvature of the detonation near the inner arc surface has a significant effect on the detonation motion even for conventional high explosives. We also examine the detonation driving zone structure for each arc case, and thus the subsonic regions of the flow that determine the influence of the arc geometry on the detonation propagation. In addition, streamline paths and reaction progress isolines are calculated. We conclude that a common approximation for modeling conventional high explosive detonation, wherein the shock-normal detonation speed is assumed equal to the Chapman–Jouguet speed, can lead to significant errors in describing the speed at which the detonation propagates.

Impact of North Atlantic SST and Tibetan Plateau forcing on seasonal transition of springtime South Asian monsoon circulation

The South Asian circulation and precipitation in spring shows a clear seasonal transition and interannual variation. We investigate how the North Atlantic sea surface temperature (SST) and Tibetan Plateau (TP) forcing affect this seasonal transition over South Asia on interannual timescale. Our results suggest that North Atlantic SST can affect the seasonal transition of South Asian monsoon via TP forcing in spring. The positive tripole pattern of North Atlantic SST anomaly during winter–spring can trigger a steady downstream Rossby wave train with cyclonic circulation over the southwestern TP. This forms a spring dipole mode of surface sensible heating and 10 m winds over the plateau, with a westerly (easterly) flow and positive (negative) surface sensible heating over its southern (northern) regions. A distinct land–air coupling configuration in May is then generated on the southwestern TP via such a positive TP dipole mode, which consists of anomalous positive precipitation, negative surface sensible heating and a baroclinic circulation structure with cyclonic circulation in the mid- to upper troposphere and a shallow anticyclonic circulation in the lower layer. The anticyclonic circulation is opposite to the summertime monsoon circulation. It weakens the cross-equatorial flow and water vapor transport to the South Arabian Sea and Bay of Bengal, resulting in in-situ precipitation reduction. Consequently, the seasonal transition in circulation over South Asia from winter to summer is delayed.

Ludwig Edward Fraenkel. 28 May 1927—27 April 2019

Edward Fraenkel’s professional career began as an experimentalist at the Royal Aircraft Establishment, Farnborough, but his preoccupation with the theoretical and mathematical aspects of aerodynamics led him into academia, working initially in aerodynamics and classical applied mathematics, but later in the modern theory of nonlinear partial differential equations and its applications to fluid mechanics. He made outstanding contributions to the mathematical theories of viscous flow separation, steady vortex rings and surface waves on water.

Editorial: Biographical Memoirs, Volume 69

Editorial: Biographical Memoirs, Volume 69

Solitary waves on constant vorticity flows with an interior stagnation point

Abstract

Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes

This survey presents results on the stability of elevation solitary waves in axisymmetric elastic membrane tubes filled with a fluid. The elastic tube material is characterized by an elastic potential (elastic energy) that depends non-linearly on the principal deformations and describes the compliant elastic media. Our survey uses a simple model of an inviscid incompressible fluid, which nevertheless makes it possible to trace the main regularities of the dynamics of solitary waves. One of these regularities is the spectral stability (linear stability in form) of these waves. The basic equations of the ‘axisymmetric tube – ideal fluid’ system are formulated, and the equations for the fluid are averaged over the cross-section of the tube, that is, a quasi-one-dimensional flow with waves whose length significantly exceeds the radius of the tube is considered. The spectral stability with respect to axisymmetric perturbations is studied by constructing the Evans function for the system of basic equations linearized around a solitary wave type solution. The Evans function depends only on the spectral parameter , is analytic in the right-hand complex half-plane , and its zeros in coincide with unstable eigenvalues. The problems treated include stability of steady solitary waves in the absence of a fluid inside the tube (the case of constant internal pressure), together with the case of local inhomogeneity (thinning) of the tube wall, the presence of a steady fluid filling the tube (the case of zero mean flow) or a moving fluid (the case of non-zero mean flow), and also the problem of stability of travelling solitary waves propagating along the tube with non-zero speed. Bibliography: 83 titles.

A Numerical Solution for the Coexisting Field of Surface and Internal Solitary Waves

The numerical solutions for the coexisting fields of surface and internal solitary waves have been obtained, where the set of nonlinear equations based on the variational principle for steady waves are solved using the Newton- Raphson method. The relative phase velocity of surface-mode solitary waves is smaller in the coexisting fields of surface and internal solitary waves than in the cases without the coexistence of internal waves. The relative phase velocity of internal-mode solitary waves is also smaller in the coexisting fields of surface and internal solitary waves than in the cases without surface waves. The interfacial position of an internal mode internal solitary wave in a coexisting field of surface and internal waves can exceed the critical level determined in the corresponding case without a surface wave. The wave height ratio between internal-mode surface and internal solitary waves is smaller than the corresponding linear shallow water wave solution, and the difference increases, as the relative wave height of internal-mode internal solitary waves is increased.

Rotating Detonation Rocket Engine Operability Under Varied Pressure Drop Injection

Rotating detonation rocket engine (RDRE) experiments have so far been limited to injectors with high pressure drop to limit injector–plenum coupling. However, this corresponds to a reduction in the potential performance benefits of RDREs. To investigate this phenomenon, two injection configurations with different total injection areas are compared with respect to overall engine operability, global performance, and steady-state wave propagation. Notably, the increased area corresponds with a three to four times decrease in pressure drop, with negligible effects on operability and performance. The RDRE is found to operate in a detonative mode for both injectors across equivalence ratios between 0.6 and 2.5, and total mass flows between 0.09 and . The steady-state wave propagation varies significantly between the two injectors, with low pressure drop injection sustaining fewer waves traveling at higher velocities at the maximum performance conditions, but in all other conditions promoted lower wave speeds and counter-propagating behavior, denoting a breakdown in the steady wave propagation. This breakdown is likely due to increased injector–plenum coupling. These results provide insight into the effects of reduced pressure drop on the operation of an RDRE, and indicate challenges that must be overcome in the design of real systems that include RDREs.

Detonation propagation across a stratified layer with a diffuse interface

A study was conducted to examine detonation propagation in a stratified layer of hydrogen-oxygen-nitrogen above an inert gas in a horizontal narrow channel. The stratified layer was produced by a gravity current, generated by retracting a door initially separating a hydrogen-oxygen-nitrogen mixture in the predetonator and a heavier inert gas in the test-section. A steady detonation wave generated in the predetonator was transmitted into the stratified layer. The reactivity of the predetonator mixture was varied via the hydrogen-oxygen equivalence ratio and the amount of nitrogen dilution. Schlieren photography was used to visualize the detonation front in the test-section, and soot foils were used to obtain the cellular structure. Schlieren imaging showed a curved detonation front that decoupled at about mid channel height, into a shock wave and trailing contact surface. Both the hydrogen-oxygen-nitrogen reactivity and the type of inert gas initially in the test-section affected the distance travelled by the detonation wave in the stratified layer. The mixture composition distribution within the test-section before ignition was obtained via a three-dimensional CFD simulation. The lateral extent of the cellular structure captured on the soot foil, coincided with the calculated inert gas mole fraction contour that corresponds to a sharp increase in the ZND induction zone length, e.g., 70% argon dilution for a stoichiometric hydrogen-oxygen predetonator mixture.

Variational formulations of steady rotational equatorial waves

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

Numerical simulations of a ship obliquely advancing in calm water and in regular waves

Numerical simulations of the S-175 containership obliquely advancing at constant forward speed in calm water and in regular waves were conducted using a potential flow method. A double-body potential and trailing vortex sheets, which modeled the flow field around the ship obliquely advancing in calm water, yielded transverse forces and yaw moment due to the lift effect. Perturbation potentials describing the free surface flow yielded first-order wave-induced motions and second-order wave loads. Evaluating and then averaging the associated second-order wave loads obtained the steady wave drift forces and moments. Generally, numerical predictions compared favorably to experimental measurements. A systematic analysis demonstrated that, in short beam waves, the transverse wave drift force attained considerable amplitudes, and these amplitudes increased rapidly at larger drift angles. The transverse force due to wave drift effects in beam waves and due to the lift effects in calm water were nearly of equal magnitude. Therefore, when predicting wave drift loads acting on a maneuvering ship, the effect of drift angle should be accounted for.

Role of instability on the limits of laterally strained detonation waves

The present work examines the role of instability and diffusive phenomena in controlling the limits of detonations subject to lateral strain. Experiments were conducted in mixtures with varying levels of cellular instability, i.e., stoichiometric methane–oxygen (CH4/2O2), ethylene–oxygen (C2H4/3O2), and ethane–oxygen (C2H6/3.5O2). These detonations were propagated in channels with exponentially enlarging cross-sections, following the recent works of Radulescu and Borzou (2018) and Xiao and Radulescu (2020). Steady detonation waves were obtained at the macro-scale, with the near-limit reaction zone structures characterized by significant unreacted gas pockets. The turbulent flame burning velocity of these pockets was evaluated to be 30 m/s to 70 m/s, which is larger than the theoretical laminar value by a factor of 2 to 7 and smaller than the CJ deflagration speed by a factor of 2 to 3. For all the mixtures tested, the characteristic D−K relationships, relating the detonation mean propagation speed with lateral flow divergence, were obtained directly from experiments and as well from the generalized ZND model with lateral strain rates using detailed chemical kinetics. The results showed that the degree of departure between experiments and the theoretical predictions increases significantly with the detonation instability level. As compared to the laminar ZND wave, the more unstable detonations are much more detonable than the more stable detonations, with substantially larger limiting divergence rates and maximum velocity deficits. Such enhanced detonability with detonation instability can be manifested in the significantly enhanced global rates of energy release with the notably suppressed thermal character of ignition for the more unstable detonations. This globally enhanced burning mechanism is found to be realized by the intensified auto-ignition assisted by the turbulent diffusive burning of the unreacted gas pockets, substantially shortening the characteristic reaction zone lengths. Finally, the relatively good universality of the D/DCJ−Keffλ relations implicitly suggests that cell size is also a function of the detonation wave stability, since it is controlled by the reaction zone thickness of unstable detonations.

High- and low-entropy layers in solids behind shock and ramp compression waves

Non-uniform temperature fields are analyzed, which arise in the problems of formation of the steady shock wave at impact and ramp loading of metals, exit of the steady shock wave to the free surface, and the shock wave passing through the interface between two different materials. Theoretical analysis and computations show that high-entropy (with the temperature increase) and low-entropy (with the temperature decrease) layers arise near the interfaces in the above problems of shock and ramp loading. The impact produces the high-entropy layer; while the ramp loading can result in the both high- and low-entropy layers. At the shock wave passing through the interface, the high-entropy layer is formed in the lower-impedance material and the low-entropy—in the higher-impedance one. The formation of high-entropy layer at impact is supported by molecular-dynamics simulations in addition to continuum modeling. The high- and low-entropy layers should be taken into account in simulations of shock-wave processes in thin targets or in other cases where surface effects are important.

Steady shock waves in porous metals: Viscosity and micro-inertia effects

The structure of steady shock waves in porous solids is a complex phenomenon involving in general the interplay of micro-inertia effects with the nonlinear elastic viscoplastic matrix response. Micro-inertia effects are due to the important acceleration of material particles in the vicinity of collapsing voids. By adopting the analytical approach recently developed for porous metals by Czarnota et al. [J. Mech. Phys. Solids 107 (2017)], we analyze the effects of matrix rate sensitivity, shock stress amplitude and micro-inertia on the structure of planar shock waves. We also analyze the relationship that links the strain rate within the shock to the jump of the stress across the shock. The fourth power law experimentally revealed for dense metals, Swegle & Grady [J. Appl. Phys. 58 (1985)] does not hold for heterogeneous materials. By considering the case of porous aluminum, we show that this relationship is characterized by two distinct regimes: (i) the first regime holds for weak shock intensities and is representative of the viscoplastic response of the dense matrix material, (ii) the second regime, that holds for shock of higher amplitude, is dominated by micro-inertia effects and is strongly influenced by the pore size. Micro-inertia effects appear to be quite beneficial since they are conducive to shock mitigation by attenuating the level of strain rate and of acceleration sustained by material particles.

An Existence Theory for Small-Amplitude Doubly Periodic Water Waves with Vorticity

We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the relative velocity field is a Beltrami field, meaning that the vorticity is collinear to the velocity. The existence theory is based on multi-parameter bifurcation theory.

Numerical investigation of a Mach 9 oblique detonation engine with fuel pre-injection

In this study, the performances of a Mach 9 oblique detonation engine fueled by hydrogen are numerically investigated by solving the multi-species reactive Reynolds-averaged Navier-Stokes (RANS) equations with a detailed combustion mechanism. The fuel is perpendicularly pre-injected into the core airflow in the engine inlet by three parallel strut-injectors. It is demonstrated that mixing can be enhanced by the baroclinic effect of oblique shock waves, the incipient expansion of fuel jets and the intensive momentum exchange of vertical jets into the crossflow, resulting in a well-mixed fuel-air core flow before entering the combustor. Analyses of the two most dangerous zones that bear potential of pre-ignition suggest that no pre-combustion occurs in the inlet. Benefiting from the floor bleed structure, the upstream movement of the shock waves stops at the combustor's entrance and they remain stabilized in the combustor thereafter. Finally, the combustor is proved to work under the stable detonation mode of combustion, and the supersonic fuel-air mixture is fast burnt through the steady detonation waves generated in the combustor. The concept of the pre-injection oblique detonation engine has been numerically demonstrated, which provides a significant reference to further experimental studies and future engineering applications.

An investigation on the disturbance evolution and the transition by resonant-triad interactions with a side-frequency disturbance in a boundary layer

In practical engineering problems, there are always side-frequency components whose frequencies are close to those of the dominant-frequency waves. In this paper, the parabolized stability equations are employed to study the influence of a side-frequency component on the development of a dominant-frequency disturbance and on the transition by resonant-triad interactions. The numerical results are qualitatively consistent with the experimental data and the asymptotic analysis results. It is found that the resonant-triad waves and the mean flow distortion cannot trigger transition by themselves. We identify a new mechanism, which we refer to as the Steady-Spanwise-Waves-Working (SSWW) mechanism, which is necessary to cause transition, in that the steady spanwise waves generated by the nonlinear interaction between the pair of three-dimensional waves play an indispensable role. For the transition caused by resonant-triad interactions with a side-frequency component, the side-frequency wave makes transition occur earlier, and the relative amplitude rather than the absolute amplitude of the side-frequency disturbance plays the essential role in the transition advance. If the relative amplitude reaches the threshold level of 40%, the transition location can be affected substantially. In this kind of transition, the SSWW mechanism still works, and the side-frequency perturbation enhances the effects of the SSWW mechanism such that the transition occurs earlier.