Instability Of Boundary Layer Research Articles

Near-wall vortical structures indomains with and without curved surfaces.

Taylor-Couette flow is a canonical flow to study Taylor-Görtler (TG) instability or centrifugal instability and the associated vortices. TG instability has been traditionally associated with flow over curved surfaces or geometries. In the computational study, we confirm the presence of TG-like near-wall vortical structures in two lid-driven flow systems, the Vogel-Escudier (VE) and the lid-driven cavity (LDC) flows. The VE flow is generated inside a circular cylinder by a rotating lid (top lid in the present study), while the LDC flow is generated inside a square or rectangular cavity by the linear movement of the lid. We look at the emergence of these vortical structures through reconstructed phase space diagrams and find that the TG-like vortices are seen in the chaotic regimes in both flows. In the VE flow, these vortices are seen when the side-wall boundary layer instability sets in at large [Formula: see text]. The VE flow is observed to go to a chaotic state in a sequence of events from a steady state at low [Formula: see text]. In contrast to VE flows, in the LDC flow with no curved boundaries, TG-like vortices are seen at the emergence of unsteadiness when the flow exhibits a limit cycle. The LDC flow is observed to have transitioned to chaos from the steady state through a periodic oscillatory state. Various aspect ratio cavities are examined in both flows for the presence of TG-like vortices. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical transactions paper (Part 2)'.

Thermodynamic instability of the atmospheric boundary layer stimulated by tectonic and seismic activity

The number of researches during the recent two decades clearly demonstrated that a few weeks/days before moderate and strong earthquakes the specific thermal anomalies emerge within the earthquake preparation zones. In these papers some physical mechanisms were proposed, but they could not explain the avalanche character of the observed anomalies. In our works we studied a rich collection of pre-earthquake thermal/meteorological cases and proposed an explanation of the physical mechanism. We consider the Boundary Layer (BL) within the earthquake preparation zone as an open non-equilibrium system with energy dissipation. We show, that the external impact of ionization and following plasma chemical reactions between new-formed ions provide a specific autocatalytic exothermic reaction with a sharp increase of the thermal energy release. We use atmosphere chemical potential (ACP) as a parameter to estimate energy for processes connected with condensation. Monitoring of the latent heat release and ACP in the boundary layer can be used for the detection of tectonic and seismic activity and artificial phenomena, connected with air ionization. The statistical studies of strong M ≥ 6.5 global earthquakes demonstrated the high statistical confidence of ACP as the short-term earthquake precursor.

Three-dimensional wall heating elements effect on the instability in zero pressure gradient supersonic boundary layers

The streaky boundary layers have very important roles in laminar-turbulent transition. Streaks of appropriate size can influence stabilities in boundary layers. In this paper, the effect of steady streamwise elongated, spanwise periodic wall heating elements on the first mode instability in supersonic flat plate boundary layers was investigated. For the balance of the efficient and accuracy, the linearized Navier–Stokes equations are used to obtain the base flow and compared with compressible Navier–Stokes equations. A bi-global analysis tool is used for the instability analysis because the heating source has a much larger length-scale in the streamwise direction than that in the spanwise direction, and the streamwise velocity is much larger than the normal velocity and the spanwise one. Results indicated that the distortion caused by the three-dimensional surface heating elements could modify the first mode, resulting in a lower frequency but with an uncertain effect on the higher frequency modes. Additionally, the streaks make the lower spanwise wave number components of the even first mode disturbance in a three-dimensional supersonic boundary layer in the freestream. As a result, the spontaneous radiation of an acoustic wave to the far field was found for the even mode. These findings suggest that laminar-turbulence transition can be suppressed or enhanced by the three-dimensional wall heating.

Heat transfer modulation in Rayleigh–Bénard convection by an oscillatory bottom plate

In this paper, we consider a heat transfer modulation in Rayleigh–Bénard convection by imposing a periodic sinusoidal oscillation to the bottom hot plate parallel to itself. Two-dimensional numerical simulations are carried out under lateral periodic conditions, over a Rayleigh number range of 106≤Ra≤109 and for a fixed Prandtl number of Pr = 7.1. For a given Rayleigh number, it is found that the Nusselt number, characterizing the global heat transfer efficiency of the system, shows a counter-intuitive initial drop and subsequent rise behavior, as the characteristic oscillatory velocity Vosc increases. Accordingly, taking the classical Rayleigh–Bénard convection as a reference, a heat transfer reduction regime for low Vosc and a heat transfer enhancement regime for high Vosc are recognized. The reduction regime is resulted from the thickening of the thermal boundary layer due to the amplified viscous effect by the oscillation, which increases the thermal resistance of the system. In addition to thickening the thermal boundary layer, a stronger oscillation could also trigger a thermal boundary layer instability, inducing massive emission of the thermal plumes and eventually giving rise to a significant global heat transfer enhancement. Moreover, the combined effect of thickening and destabilizing of the thermal boundary layer leads to a temporal periodic evolution of the Nusselt number at the bottom plate in the enhancement regime. A critical oscillatory velocity Vc is selected at the crossover between two regimes, and it is found decreasing with an increasing Ra as Vc∼Ra−0.2. Through dimensional analysis, we provide a physical explanation for this dependence.

Attachment-line, crossflow and Tollmien-Schlichting instabilities on swept ONERA-D and Joukowski airfoils.

Linear stability analyses are performed to investigate the boundary layer instabilities developing in an incompressible flow around the whole leading-edge of swept ONERA-D and Joukowski airfoils of infinite span. The stability analyses conducted in our study are global in the chordwise direction and local in the spanwise direction. A neutral curve is drawn at a given leading-edge Reynolds number ReR and several overlapping regions, called "lobes", are identified on a physical basis. A detailed study of the marginal modes reveals the presence of attachment-line and crossflow instabilities, as well as modes whose features do not fall within the standards of a specific type. Connected crossflow/Tollmien-Schlichting modes, that show a dominant spatial structure reminiscent of Tollmien-Schlichting waves but whose destabilization is linked to a crossflow mechanism, have been identified. The comparison of several neutral curves at different ReR values reveals the greater stabilizing effect of the increase of ReR on the crossflow instability compared to the attachment-line instability. The influence of the airfoil shape is also studied by comparing the neutral curves of the ONERA-D with the neutral curves of the Joukowski airfoil. These curves reveal similar characteristics with the presence of distinct lobes and their comparison at constant sweep angle shows that, under the conditions studied, the ONERA-D airfoil is more stable than the Joukowski airfoil, even for crossflow instabilities. The absolutely or convectively unstable nature of the flow in the spanwise direction is also tackled and our results suggest that the flow is only convectively unstable.

Investigation of hypersonic cone boundary layer stability regulation with plasma actuation

Boundary layer transition has always been a frontier in the field of fluid mechanics, and hypersonic boundary layer stability experiments can help reveal the physical mechanisms behind such transitions. In particular, the regulation of unstable waves in boundary layers is critical for transition prediction and control. Plasma actuation is a popular flow control method that has made progress in moderating the stability of supersonic boundary layer. However, there have been few studies on regulating the stability of hypersonic boundary layers with plasma actuation. In this paper, wind tunnel experiments are carried out under Mach 6 flow to study the stability regulation of a hypersonic sharp cone boundary layer with nanosecond pulsed plasma actuation. First, the typical characteristic structure of the rope-like structure is captured by the high-speed schlieren method. Then, combining the sensor results and theoretical analysis, the rope-like structure and the dominant instability wave resolved by the schlieren power spectrum density method are determined to correspond to the second-mode wave. The characteristic unstable quasi-ordered structure of the boundary layer under actuation is then extracted, and the impact effect and modulation effect of this structure on the second-mode wave are analyzed. Finally, the mechanism by which actuation influences the boundary layer instability is studied using proper orthogonal decomposition. The results show that actuation can enhance boundary layer pulsation, and the coupling effect between the actuation and boundary layer can produce regular unstable quasi-ordered structures. The intrinsic mechanism works by impinging on and modulating the second-mode waves, and there are characteristic modes of the rope-like structure and the unstable structures distributed over the whole flow-direction range. This verifies the ability of plasma actuation to stimulate the instability of hypersonic cone boundary layers and provides technical support for the further development of transition control methods.

Linear instability of a supersonic boundary layer over a rotating cone

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.

On the identification of cross-flow mode in three-dimensional boundary layers

Parametric zones are obtained where the cross-flow instability can be identified as a mode in the three-dimensional boundary layers with the Mach number ranging from 0 to 10. Although the term cross-flow mode is widely used in the investigations on boundary-layer instability, the previous work [Z. Liu, Phys. Fluids 34, 094110 (2022)] has demonstrated that a cross-flow mode cannot be designated in certain circumstances. The identification of the cross-flow mode is significant not only in the justifiable use of the term but also in judging whether a disturbance is more inclined to a cross-flow type or not. In this work, a criterion is built to identify the cross-flow mode based on the growth-rate peak and disturbance shape. By means of extensive calculations and identifications, parametric zones are presented for the unstable cross-flow mode. It is found that the cross-flow mode cannot be identified at large local sweep or at Mach numbers larger than 1.6. In parametric zones, the cross-flow mode can be distinguished from the Tollmien–Schlichting mode, or identified as the solely dominant mode. Based on the identifications, the maximum growth rates of the cross-flow mode, the Tollmien–Schlichting mode, and the Mack modes in the three-dimensional boundary layers are provided. The dominant mode can be determined at different Mach numbers. The cross-flow mode is revealed to dominate the boundary-layer instability at low Mach numbers. Under large cross-flow strengths, the second mode could not be the dominant mode in the hypersonic boundary layers, which contrasts sharply with the two-dimensional case.

Neural prediction model for transition onset of a boundary layer in presence of two-dimensional surface defects

Predicting the laminar to turbulent transition is an important aspect of computational fluid dynamics because of its impact on skin friction. Traditional transition prediction methods such as local stability theory or the parabolized stability equation method do not allow for the consideration of strongly non-parallel boundary layer flows, as in the presence of surface defects (bumps, steps, gaps, etc.). A neural network approach, based on an extensive database of two-dimensional incompressible boundary layer stability studies in the presence of gap-like surface defects, is used. These studies consist of linearized Navier–Stokes calculations and provide information on the effect of surface irregularity geometry and aerodynamic conditions on the transition to turbulence. The physical and geometrical parameters characterizing the defect and the flow are then provided to a neural network whose outputs inform about the effect of a given gap on the transition through the ${\rm \Delta} N$ method (where N represents the amplification of the boundary layer instabilities).

PLUME STRUCTURES IN NATURAL CONVECTION WITH TRANSPIRATION

We present plume structures at high Rayleigh numbers (Ra<sub>H</sub> ~ 10<sup>11</sup>) and Schmidt numbers (Sc ~ 675), in convection driven by density differences across a horizontal membrane, when vertical transpiration velocities in the range of 0.002 cm/s ≤ V<sub>o</sub> ≤ 0.065 cm/s are imposed. The density differences across the membrane are created by having brine with higher concentrations above the membrane and ammonium chloride with lower concentrations below it, with the bottom fluid being forced upward across the membrane by gravitational heads to create transpiration. Planar laser induced fluorescence (PLIF) visualizations in a horizontal plane grazing the top of the membrane, with refractive index matching, brings out the plume structures and their evolution with Ra<sub>H</sub> and V<sub>o</sub>. The plume structures show complex dendritic patterns of line plumes with plume-free patches; the plume-free patches decrease in area and number, with decrease in Ra<sub>H</sub> and V<sub>o</sub>. At low V<sub>o</sub>, these plume-free patches have aligned plumes around them, while at larger V<sub>o</sub>, such aligned plumes are not seen. We expect these patches to be due to the impingement of large-scale flow at low V<sub>o</sub>, while they could be due to the modification of boundary layer instability at large V<sub>o</sub>. These plumes become closer with increase in Ra<sub>H</sub>, while they become more separated with increase in V<sub>o</sub>. We quantify this variation by measuring the total plume length (L<sub>p</sub>) by skeletonizing the plume structures and calculate the mean plume spacing (λ) from L<sub>p</sub>; λ show nontrivial, non-power law dependence on Ra<sub>H</sub> and V<sub>o</sub>, different from those observed for the case of no throughflow and low throughflows.

Internal Hydraulic Transition and Turbulent Mixing Observed in the Kuroshio over the I-Lan Ridge off Northeastern Taiwan

Abstract Complex small-scale processes and energetic turbulence are observed at a sill located on the I-Lan Ridge that spans across the strong Kuroshio off Taiwan. The current speed above the sill is strong (1.5 m s−1) and unsteady (±0.5 m s−1) due to the Kuroshio being modulated by the semidiurnal tide. Above the sill crest, isothermal domes, with vertical scales of ∼20 and ∼50 m during the low and high tides, respectively, are generated by turbulent mixing as a result of shear instability in the bottom boundary layer. Tidally modulated hydraulic character modifies the small-scale processes occurring on the leeward side of the sill. Criticality analysis, performed by solving the Taylor–Goldstein equation, suggests that the observed lee waves and intermediate layer sandwiched by two free shear layers are related to the mode-1 and mode-2 critical control between the sill crest and immediate lee, respectively. Around high tide, lee waves are advected further downstream, and only mode-1 critical control can occur, leading to a warm water depression. The shear instabilities ensuing from the hydraulic transition processes continuously mediate flow kinetic energy to turbulence such that the status of marginal instability where the Richardson number converges at approximately 0.25 is reached. The resultant eddy diffusivity Kρ is concentrated from O(10−4) to O(10−3) m2 s−1 and has a maximum value of 10 m2 s−1. The sill on the western flank of the Kuroshio is a hotspot for energetic mixing of Kuroshio waters and South China Sea waters.

Effect of the leading-edge protuberances on the aeroacoustic and aerodynamic performances of the wind turbine airfoil

Based on the restraining effect of the leading-edge protuberances (LEPs) on flow separation, this work focuses analyzing the aeroacoustic and aerodynamic performances for the wind turbine airfoil with LEPs near the outboard region of the blades, by using the experimental testing and numerical calculation methods. The associated flow patterns for the aeroacoustic and aerodynamic performances are revealed within the pre-stall and post-stall regions. The shedding of the separation vortices enhances the instability of the laminar boundary layer, generating the aerodynamic noise from the wind turbine airfoil. The LEPs suppress the generation of the tonal noise in pre-stall region by changing the distribution and reducing the intensity of the trailing edge shedding vortices. Moreover, LEPs can improve the aerodynamic performance of the wind turbine airfoil greatly by inhibiting the flow separation at a high inflow angle of attack. As a flow control method for the wind turbine blades, LEPs will help to improve the aeroacoustic performance of the blades under normal inflow wind conditions, and the aerodynamic performance under extreme wind situations where the inflow wind changes on a large scale frequently.

Correlated Pressure–Velocity Instability in a Transonic High-Pressure Turbine Blade

Abstract The present work describes the coupled pressure–velocity instability in the suction side boundary layer of a transonic high-pressure turbine blade. The application case is the VKI LS89 turbine blade MUR237. Modal decomposition techniques such as the proper orthogonal decomposition (POD) and the dynamic mode decomposition (DMD) are applied to data obtained by large eddy simulations (LES) to inspect the existing correlation between pressure fluctuations induced by traveling pressure waves and the velocity disturbances growing into the boundary layer. To this end, POD and DMD were applied to both velocity and pressure fields from the leading edge up to the end of the blade surface. DMD was computed on a 2D slice oriented parallel to the blade surface inside the boundary layer. Data were ordered along the curvilinear abscissas, so that the most spatially amplified waves were obtained. Interestingly, unstable velocity modes were detected in the front blade suction side where pressure waves dominate. The unstable velocity modes computed upstream of the peak suction highlighted the occurrence of elongated streaky structures showing spanwise fluctuations which are typical of their instability. The comparison of pressure and velocity modes highlighted the occurrence of pressure-related fluctuations in the velocity field at the same time instants at which streak instability is observed. The analysis carried out in this work suggests that pressure waves can provoke localized instability of boundary layer streaks, which then break up further downstream causing transition.

Instabilities in three-dimensional boundary-layer flows with a highly non-ideal fluid

The present work investigates the linear instability of three-dimensional boundary layers in thermodynamically non-ideal regimes. As a representative fluid, we consider carbon dioxide at supercritical pressure (80 bar). The flow set-up is matched to the redesigned DLR (German Aerospace Center) experiment on cross-flow instability, with identical pressure-coefficient distribution (accelerating the flow), sweep angle and Reynolds number, at a low Mach number. The flow temperature relative to the Widom line – also known as the pseudocritical line – thus characterises the non-ideality of the flow. We consider supercritical (gas-like), subcritical (liquid-like) and transcritical (pseudoboiling) regimes, where the flow temperature remains above, below or crosses the Widom line. The stability analyses of the parabolised Navier–Stokes baseflows indicate that wall heating destabilises the flow in the supercritical regime while wall cooling stabilises both effects similar to the ideal-fluid situation but being stronger. On the contrary, wall heating/cooling exhibits reversed effects in the subcritical regime, like for an ideal liquid. In the transcritical regime, with its sharp gradients of the thermodynamic and transport properties, wall heating stabilises the flow. Most substantially, however, wall cooling provokes a changeover of the leading instability mechanism: the accelerated streamwise flow attains inflectional wall-normal profiles, and the invoked inviscid Tollmien–Schlichting instability prevails with growth rates up to one order of magnitude larger than those of the cross-flow mode. We establish a two-fold mathematical relation from the momentum equation that explains the consequence of non-ideality and wall heating/cooling. The streamwise perturbation patterns of the flows in their linear instability regime are shown by mimicking wave trains emanating from virtual point-disturbance sources. From the viewpoint of keeping laminar flows, the transcritical thermodynamic state with a cooling wall must be avoided.

The Microstructural Record of Convection in the Little Minch Sill Complex, Scotland

Abstract Detailed microstructural analysis of three basaltic sills of the Little Minch Sill Complex demonstrates that convection leaves a detectable signature in fully solidified bodies. The presence of dense clusters of equant grains of olivine and clinopyroxene in the central parts of sills can only be accounted for if they formed and were enlarged while suspended in convecting magma, with delayed settling to the sill floor. An associated stratigraphic invariance of plagioclase grain shape is consistent with growth while suspended in convecting magma. These microstructural indicators demonstrate that convection during solidification was vigorous and long-lived in the 135-m-thick picrodolerite-crinanite unit (PCU) of the composite Shiant Isles Main sill and vigorous and likely short-lived in the PCU of the composite Creagan Iar sill. In contrast, convection in the Meall Tuath sill was weak and short-lived: plagioclase grain shape in this sill varies with stratigraphic height, indicative of primarily in situ nucleation and growth at the magma-mush interface, while olivine and clinopyroxene were kept suspended in the overlying convecting magma. The magma in all three sills fractionated during solidification, permitting convection driven by the instability of an upper thermal boundary layer. The comparative vigour and longevity of convection in the Shiant Isles Main sill and the Creagan Iar sill was due to their emplacement above an earlier, still-hot, intrusion, resulting in highly asymmetric cooling.

Instability and Transition of Hypersonic Boundary Layer on a Blunted Delta Wing

The instability and transition of hypersonic boundary layer on a blunted delta wing is performed by numerical simulation of experiments conducted in the quiet wind tunnel. Disturbances are calculated using linear stability theory to obtain the integrated disturbance growth rates and amplification factors from laminar steady base flows. Comparisons are made between the results of the stability analysis and the experimentally measured transition characteristics. The results show that the generalized inflections existing in velocity profiles reveal both the streamwise instability and crossflow instability of the boundary layer on the delta wing. In the parameter spaces of frequency and wave number, there is only one unstable region in which the first-mode and second-mode waves are integrated with each other at angle of attack, whereas the unstable regions of the first-mode and second-mode waves are separately distributed at angle of attack of . The numerically predicted factor distribution is in good agreement with the measured transition front, and the factor is about 3 when boundary-layer transition occurs. The low-frequency unstable waves with about 25–30 kHz play a key role in dominating the boundary-layer transition of the delta wing.

Stability analysis of roughness-disturbed boundary layer controlled by wall-blowing

Roughness-induced transition control is of considerable importance for high-speed vehicles. In this paper, the instability of a roughness-disturbed boundary layer controlled by spanwise-uniform wall-blowing is investigated through BiGlobal and three-dimensional parabolized stability equation (PSE-3D) analysis. Without wall-blowing, symmetric and antisymmetric unstable modes are observed when using BiGlobal analysis, with PSE-3D analysis suggesting that the symmetric mode is the dominant instability. Both modes are associated with the instability of the entire separated shear layer behind the roughness region rather than the components in certain directions, as both the wall-normal shear and the spanwise component resulting from the bending shear layer contribute to the growth of the disturbance. Upstream wall-blowing delays the roughness-induced transition by modifying the wake instability. The antisymmetric mode is the first to disappear as the blowing intensity increases while the symmetric instability is also suppressed. Upstream wall-blowing also reduces both the strength and bending of the shear layer by affecting the inflow boundary layer. This leads to a decrease in the wall-normal and spanwise contributions to the disturbance energy. Downstream wall-blowing achieves a control effect by decelerating the development of the dominant symmetric mode through the direct interaction between wall-blowing and the wake. Although the reduction in shear strength is not as strong as with upstream wall-blowing, downstream wall-blowing still relaxes the bending of the shear layer, which is related to the production of disturbance energy. In conclusion, two-dimensional wall-blowing can delay the roughness-induced transition by modifying the wake structure and instability.

Wave Structure of Heterogeneous Detonations at High Operating Pressures

ABSTRACT An analytical model has been constructed to evaluate the behavior of a single droplet in a heterogeneous detonation flowfield. In this study, mass shed off the droplet is estimated based on viscous boundary layer instabilities at the free surface of the gas/liquid interface at high operating pressures. The trajectory of the droplet is computed by assuming that it travels through a gas medium that matches a Zel’dovich-Von Neumann-Döring (ZND) profile, i.e., gas/liquid momentum exchange is ignored. With application to rocket combustors, three different mixtures are considered: liquid oxygen droplets with both methane and hydrogen gas, as well as kerosene droplets in gaseous oxygen. It is identified that the initial droplet diameter and the dynamic pressure behind the leading shock front are the two most important factors influencing the droplet breakup process. It is found that smaller droplet diameters, higher base pressures, and reactants with lower sound speeds are desired for close coupling of the leading shock front to the heat-release zone. With the droplet atomization model employed, it is concluded that increasing the operating pressure may increase the minimum droplet size that will permit coupling of the heat release with the detonation front. This is due to the increased dynamic pressure behind the wavefront when the operating pressure is increased. Finally, it is noted that the sound speed in the compressed reactant mixture is also of prime importance as induction zone lengths scale with this parameter. As a result the liquid kerosene/gaseous oxygen system provided the smallest induction zone length of the three systems studied.

Cross-flow linear instability in compressible boundary layers over a flat plate

The linear instability of compressible boundary layers over a flat plate in the presence of parameterized crossflow has been investigated by means of linear stability theory. A family of boundary layer with crossflow is obtained as the base flow from the compressible Falkner–Skan–Cooke (FSC) flow model. Two factors, that is, the local swept angle and the pressure-gradient parameter, are designed to create the crossflow with different directions and magnitudes, which further results in the variation of the cross-flow instability. Modal properties related to the cross-flow instability are emphasized. The association between the cross-flow mode and the Mack's mode is clarified by extending the base flow from two dimension to three dimension. The cross-flow instability is discovered to be integrated with the slow-mode instability, that is, the instability related to the Tollmien–Schlichting (T–S) mode or the first mode, and it can hardly be distinguished as an individual mode in most cases. The effects of Mach number, pressure gradient, local sweep, and wall temperature are studied. The behaviors of the cross-flow instability under such effects resemble those of the slow-mode instability in the two-dimensional boundary layers. It is found that the unstable modes in the three-dimensional boundary layers are mainly affected by the streamwise pressure gradient and the crossflow per se. Specifically, the first mode is much more sensitive to the crossflow than the second mode. As a result, more marked variations are commonly observed for the first mode in the boundary layers with crossflow.

Solution approaches for evaporation-driven density instabilities in a slab of saturated porous media

This work considers the gravitational instability of a saline boundary layer formed by an evaporation-induced flow through a fully saturated porous slab. Evaporation of saline waters can result in the formation of salt lakes as salt accumulates near the top surface. As gravitational instabilities can impede the accumulation of salt, establishing a relation between their occurrence and the value of physical parameters such as evaporation rate is crucial. One step toward determining when gravitational instabilities can arise is to compute the ground-state salinity. In this work, this ground state is analytically derived within the framework of Sturm–Liouville theory. Then, a linear stability analysis in conjunction with the quasi-steady state approach is employed to investigate the occurrence of instabilities. These instabilities can develop and grow over time depending on the Rayleigh number and the dimensionless height of the porous medium. To calculate the critical Rayleigh number, which is used to determine the stability of a particular system, the eigenvalues of the linear perturbation equations have to be computed. Here, a novel fundamental matrix method is proposed to solve this perturbation eigenvalue problem and shown to coincide with an established Chebyshev–Galerkin method in their shared range of applicability. Finally, a two-dimensional direct numerical simulation of the full equation system via the finite volume method is employed to validate the time of onset of convective instabilities predicted by the linear theory. Moreover, the fully nonlinear convection patterns are analyzed.