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We study a singular initial value problem for a nonlinear non-autonomous ordinary differential equation of the second order, defined on a semi-infinite interval and degenerating in the initial data for the phase variable. The problem arises in the dynamics of a viscous incompressible fluid as an auxiliary problem in the study of self-similar solutions of the boundary layer equations for a stream function with a zero pressure gradient (plane-parallel laminar flow in a mixing layer). It is also of independent mathematical interest. Using the previously obtained results on singular nonlinear Cauchy problems and parametric exponential Lyapunov series, a correct formulation and a complete mathematical analysis of this singular initial value problem are given. Restrictions on the “self-similarity parameter” for the global existence of solutions are formulated, two-sided estimates of solutions, and results of calculations of the phase trajectories of solutions for different values of this parameter are given.

In this note we derive MHD boundary layer equations according to viscosity and resistivity coefficients. Especially, when these viscosity and resistivity coefficients are of different orders, it leads to degenerate MHD boundary layer equations. We prove these degenerate boundary layers are stable around a steady solution.

Solving the three-dimensional boundary layer equations carries theoretical significance and practical applications, which also poses substantial challenges due to its inherent complexity. In this paper, the laminar boundary layer equations for the symmetry plane of three-dimensional bodies are derived in an orthogonal curvilinear coordinate system associated with the principal curvatures. The derivation of the boundary layer equations is based not only on the common symmetric properties of the flow, as given by Hirschel et al. (Three-Dimensional Attached Viscous Flow, 2014, Academic Press, pp. 183–187), but also incorporates the geometric symmetry properties of the body. The derived equations are more representative and simplified. Notably, these equations can degenerate to a form consistent with or equivalent to the commonly used boundary layer equations for special bodies such as flat plates, cones and spheres. Furthermore, for hypersonic flows, the crossflow velocity gradient at the boundary layer edge on the symmetry plane is derived based on Newtonian theory. Subsequently, this parameter can provide the necessary boundary condition needed for solving the boundary layer equations using existing methods. Finally, as examples, the equations developed in this paper are solved using the difference-differential method for several typical three-dimensional blunt shapes that appeared on hypersonic vehicles. They prove to be useful in the analysis and interpretation of boundary layer flow characteristics in the symmetry plane of blunt bodies.

Unsteadiness and horizontal heterogeneities frequently characterize atmospheric motions, especially within convective storms, which are frequently studied using large-eddy simulations (LES). The models of near-surface turbulence employed by atmospheric LES, however, predominantly assume statistically steady and horizontally homogeneous conditions (known as the equilibrium approach). The primary objective of this work is to investigate the potential consequences of such unrealistic assumptions in simulations of tornadoes. Cloud Model 1 (CM1) LES runs are performed using three approaches to model near-surface turbulence: the “semi-slip” boundary condition (which is the most commonly used equilibrium approach), a recently proposed nonequilibrium approach that accounts for some of the effects of turbulence memory, and a nonequilibrium approach based on thin boundary layer equations (TBLE) originally proposed by the engineering community for smooth-wall boundary layer applications. To be adopted for atmospheric applications, the TBLE approach is modified to account for the surface roughness. The implementation of TBLE into CM1 is evaluated using LES results of an idealized, neutral atmospheric boundary layer. LES runs are then performed for an idealized tornado characterized by rapid evolution, strongly curved air parcel trajectories, and substantial horizontal heterogeneities. The semi-slip boundary condition, by design, always yields a surface shear stress opposite the horizontal wind at the lowest LES grid level. The nonequilibrium approaches of modeling near-surface turbulence allow for a range of surface-shear-stress directions and enhance the resolved turbulence and wind gusts. The TBLE approach even occasionally permits kinetic energy backscatter from unresolved to resolved scales.

This work deals with the flow and heat transfer in upper-convected Maxwell fluid above an exponentially stretching surface. Cattaneo-Christov heat flux model is employed for the formulation of the energy equation. This model can predict the effects of thermal relaxation time on the boundary layer. Similarity approach is utilized to normalize the governing boundary layer equations. Local similarity solutions are achieved by shooting approach together with fourth-fifth-order Runge-Kutta integration technique and Newton's method. Our computations reveal that fluid temperature has inverse relationship with the thermal relaxation time. Further the fluid velocity is a decreasing function of the fluid relaxation time. A comparison of Fourier's law and the Cattaneo-Christov's law is also presented. Present attempt even in the case of Newtonian fluid is not yet available in the literature.

In this paper, we conduct a systematic study of the instability of a boundary layer over a rotating cone that is inserting into a supersonic stream with zero angle of attack. The base flow is obtained by solving the compressible boundary-layer equations using a marching scheme, whose accuracy is confirmed by comparing with the full Navier–Stokes solution. Setting the oncoming Mach number and the semi-apex angle to be 3 and 7$^\\circ$, respectively, the instability characteristics for different rotating rates ($\\bar \\varOmega$, defined as the ratio of the rotating speed of the cone to the axial velocity) and Reynolds numbers ($R$) are revealed. For a rather weak rotation, $\\bar \\varOmega \\ll 1$, only the modified Mack mode (MMM) exists, which is an extension of the supersonic Mack mode in a quasi-two-dimensional boundary layer to a rotation configuration. Further increase of $\\bar \\varOmega$ leads to the appearance of a cross-flow mode (CFM), coexisting with the MMM but in the quasi-zero frequency band. The unstable zones of the MMM and CFM merge together, and so they are referred to as the type-I instability. When $\\bar \\varOmega$ is increased to an $O(1)$ level, an additional unstable zone emerges, which is referred to as the type-II instability to be distinguished from the aforementioned type-I instability. The type-II instability appears as a centrifugal mode (CM) when $R$ is less than a certain value, but appears as a new CFM for higher Reynolds numbers. The unstable zone of the type-II CM enlarges as $\\bar \\varOmega$ increases. The vortex structures of these types of instability modes are compared, and their large-$R$ behaviours are also discussed.

This work investigates the three-dimensional, spatio-temporal flow development in the aft portion of a laminar separation bubble. The bubble is forming on a flat plate geometry, subjected to an adverse pressure gradient, featuring maximum reverse flow of approximately 2 % of the local free-stream velocity. Time-resolved velocity measurements are performed by means of planar and tomographic particle image velocimetry, in the vicinity of the reattachment region. The measurements are complemented with a numerical solution of the boundary layer equations in the upstream field. The combined numerical and measured boundary layer is used as a baseline flow for linear stability theory analysis. The results provide insight into the dynamics of dominant coherent structures that form in the separated shear layer and deform along the span. Stability analysis shows that the flow becomes unstable upstream of separation, where both normal and oblique modes undergo amplification. While the shear layer roll up is linked to the amplification of the fundamental normal mode, the oblique modes at angles lower than approximately $30^{\\circ }$ are also amplified substantially at the fundamental frequency. A model based on the stability analysis and experimental measurements is employed to demonstrate that the spanwise deformations of rollers are produced due to a superposition of normal and oblique instability modes initiating upstream of separation. The degree of the initial spanwise deformations is shown to depend on the relative amplitude of the dominant normal and oblique waves. This is confirmed by forcing the normal mode through a controlled impulsive perturbation introduced by a spanwise invariant dielectric-barrier-discharge plasma actuator, resulting in the formation of spanwise coherent vortices. The findings elucidate the link between important features in the bubble shedding dynamics and stability characteristics and provide further clarification on the differences in the development of coherent structures seen in recent experiments. Moreover, the results present a handle on the development of effective control strategies that can be used to either promote or suppress shedding in separation bubbles, which is of interest for system performance improvement and control of aeroacoustic emissions in relevant applications.

The development of a bubble plume from a vertical gas-evolving electrode is driven by buoyancy and hydrodynamic bubble dispersion. This canonical fluid mechanics problem is relevant for both thermal and electrochemical processes. We adopt a mixture model formulation for the two-phase flow, considering variable density (beyond Boussinesq), viscosity and hydrodynamic bubble dispersion. Introducing a new change of coordinates, inspired by the Lees–Dorodnitsyn transformation, we obtain a new self-similar solution for the laminar boundary layer equations. The results predict a wall gas fraction and gas plume thickness that increase with height to the power of 1/5 before asymptotically reaching unity and scaling with height to the power 2/5, respectively. The vertical velocity scales with height to the power of 3/5. Our analysis shows that self-similarity is only possible if gas conservation is entirely formulated in terms of the gas specific volume instead of the gas fraction.

This article addresses the boundary layer flow and heat transfer in third grade fluid over an unsteady permeable stretching sheet. The transverse magnetic and electric fields in the momentum equations are considered. Thermal boundary layer equation includes both viscous and Ohmic dissipations. The related nonlinear partial differential system is reduced first into ordinary differential system and then solved for the series solutions. The dependence of velocity and temperature profiles on the various parameters are shown and discussed by sketching graphs. Expressions of skin friction coefficient and local Nusselt number are calculated and analyzed. Numerical values of skin friction coefficient and Nusselt number are tabulated and examined. It is observed that both velocity and temperature increases in presence of electric field. Further the temperature is increased due to the radiation parameter. Thermal boundary layer thickness increases by increasing Eckert number.

The stability of an incompressible boundary layer flow over a wing in the presence of free stream turbulence (FST) has been investigated by means of direct numerical simulations and compared with the linearised boundary layer equations. Four different FST conditions have been considered, which are characterised by their turbulence intensity levels and length scales. In all cases the perturbed flow develops into elongated disturbances of high and low streamwise velocity inside the boundary layer, where their spacing has been found to be strongly dependent on the scales of the incoming free stream vorticity. The breakdown of these streaks into turbulent spots from local secondary instabilities is also observed, presenting the same development as the ones reported in flat plate experiments. The disturbance growth, characterised by its root mean squares value, is found to depend not only on the turbulence level, but also on the FST length scales. Particularly, higher disturbance growth is observed for our cases with larger length scales. This behaviour is attributed to the preferred wavenumbers that can exhibit maximum transient growth. We study this boundary layer preference by projection of the flow fields at the leading edge onto optimal disturbances. Our results demonstrate that optimal disturbance growth is the main cause of growth of disturbances on the wing boundary layer.