Numerical solution to the problem of the complete stabilization of a supersonic boundary layer

A method is suggested for solving numerically the problem of the complete stabilization of a supersonic boundary layer. It is shown that when the surface is significantly cooled, the neutral stability curve splits into two branches. A calculation is given for the temperatures of complete stabilization for both neutral curves. A comparison of the results obtained with those derived from asymptotic calculations shows that above M = 2 (M is the Mach number) the asymptotic method gives incorrect results.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 39–43, March–April, 1972.     

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the “inviscid” and “viscous” parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.     

Original Article Multi-component convection-diffusion in a porous medium

The stability of a triply diffusive fluid-saturated porous layer is investigated. A linear stability analysis similar to that of Pearlstein et al [1] is presented. This allows us to make a thorough investigation of the topology of the neutral curves. For some values of the thermal and solute diffusivities we obtain highly unusual neutral curves, in particular a heart-shaped, disconnected oscillatory curve. The effect of this is that three critical Rayleigh numbers are required to fully specify the linear stability criteria, a novel result in porous convection. The influence of nonlinear terms is likely to have important consequences for the experimental realisation of the linear results and so we investigate the nonlinear stability of the problem by making use of the energy method. This provides an unconditional nonlinear stability boundary and enables us to identify possible regions of subcritical instability. Received: April 4, 1996     

Stability of the center of the inner cylinder in a Couette flow

The plane problem of the stability of the center of a rotating inner cylinder in Couette flow is considered in the linear formulation in the absence of external forces and for a narrow gap. Neutral waves were not detected in [1] and the authors concluded that such motion is always unstable. Some neutral curves for the limiting case of a narrow gap were obtained in [2] by the Runge—Kutta method together with the intersecting line method, and the case of small Reynolds numbers and arbitrary gaps was also considered. In the present paper the asymptotic behavior of a narrow gap is constructed for the corresponding eigenvalue problem. Numerical investigation of the exact solution of first-order equations gives results in the form of neutral curves. For an eigenvalue σ and a parameter γ which characterizes instability, second-order asymptotic terms are obtained by the perturbation method.     

Hydrodynamic stability of boundary layers with surface mass transfer

The linear stability of the flat plate boundary layer with surface blowing and suction is investigated by the application of numerical techniques. Complete neutral stability curves, critical Reynolds numbers and wave numbers, and other stability characteristics are determined for a wide range of surface mass transfer intensities. The critical Reynolds number, based on the displacement thickness, is found to vary from 59 to 32500 between the extreme limits of blowing and suction that are investigated. Comparisons are made between the present results and available linear stability information for boundary layers with surface mass transfer and with free-stream pressure gradients. The universal stability bound of Joseph is evaluated and compared with the corresponding numerically exact neutral stability curve.     

The Linear Vortex Instability of the Near-vertical Line Source Plume in Porous Media

Free convection plumes usually rise vertically, but do not do so when in an asymmetrical environment. In such cases they are susceptible to a thermoconvective instability because warmer fluid lies below cooler fluid in the upper half of the plume. We analyse the behaviour of streamwise vortex disturbances in plumes that are close to being vertical. The linearised equations subject to the boundary layer approximation are parabolic and are solved using a marching method. Our computations indicate that disturbances tend to be centred in the upper half of the plume. A neutral curve is determined and an asymptotic theory is developed to describe the right hand branch of this curve. The left hand branch is not amenable to an asymptotic analysis, and it is found that the onset of convection for small wavenumbers is very sensitively dependent on both the profile of the initiating disturbance and where it is introduced.     

NGP_G-STABILITY OF LINEAR MULTISTEP METHODS FOR SYSTEMS OF GENERALIZED NEUTRAL DELAY DIFFERENTIAL EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｉｎｇｔｈｅｓｔａｂｉｌｉｔｙｂｅｈａｖｉｏｕｒｉｎｔｈｅｎｕｍｅｒｉｃａｌｓｏｌｕｔｉｏｎｏｆｇｅｎｅｒａｌｉｚｅｄｎｅｕｔｒａｌｄｅｌａｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｙ′(ｔ) =Ｌｙ(ｔ) Ｍｙ(ｔτ) Ｎｙ′(ｔτ)　　 (ｔ≥ 0 ) ,( 1 )ｙ(ｔ) =(ｔ)　　 (ｔ≤ 0 ) ,( 2 )ｗｈｅｒｅＬ ,ＭａｎｄＮ∈Ｃｄ×ｄａｒｅｃｏｎｓｔａｎｔｃｏｍｐｌｅｘｍａｔｒｉｃｅｓ,(ｔ) ∈Ｃｄｉｓａｇｉｖｅｎｖｅｃｔｏｒ＿ｖａｌｕｅｄｉｎｉｔｉａｌｆｕｎｃｔｉｏｎ ,ａ…     

Asymptotic behavior of solutions of the Orr-Sommerfeld equations in regions adjacent to two branches of the neutral curve

The asymptotic behavior of the upper and lower branches of the neutral stability curve of a boundary layer found by Lin [1] was determined more accurately by various authors [2–4], who, on the basis of the linearized Navien-Stokes equations, analyzed the higher approximations in the Reynolds number R. In the limit R , neutral perturbations have wavelengths that exceed in order of magnitude the boundary layer thickness. The long-wavelength asymptotic behavior of the Orr-Sommerfeld equation is, in particular, of interest because the characteristic solutions of the linearized equations of free interaction (triple-deck theory) [5–7] are a limiting form of Tollmierr-Schlichting waves in an incompressible fluid with critical layers next to the wall [8–9]. At the same time, the dispersion relation, which is identical to the secular equation of the Orr-Sommerfeld problem, contains an entire spectrum of solutions not considered in the earlier studies [2–4]. The first oscillation mode in the spectrum may be either stable or unstable. In the present paper, solutions are constructed for each of the subregions (including the critical layer) into which the perturbed velocity field in the linear stability problem is divided at large Reynolds numbers. Dispersion relations describing the neighborhood of the upper and lower branches of the neutral curve for the boundary layer are derived. These relations, which contain neutral solutions as a special case, go over asymptotically into each other in the unstable region between the two branches.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–11, July–August, 1984.     

Effect of Inertial Terms on Fluid–Porous Medium Flow Coupling

The study considers an effect of the nonlinear inertial terms in the Brinkman filtration equation on the characteristics of coupled flows in a pure fluid and porous medium in the frameworks of two independent problems. The first problem is the forced boundary-layer flow overlying the Darcy–Brinkman porous medium. The Prandtl theory is used, and the self-similar equations are built to describe it. It is shown that the inertial terms have a valuable effect on the boundary-layer structure because of the large velocity gradient in the transition zone. The boundary-layer thickness in a porous medium rapidly grows at large Reynolds numbers. The velocity magnitude and gradient at the interface also change. The second independent problem is an analysis of the inertial terms effect on the flow stability. The neutral curves of the full and linearized flow models are built using the shooting method. They have different short-wave asymptotic, but there are no significant changes in the critical Reynolds numbers and corresponding wave numbers.     

Stability of fluid conveying nanobeam considering nonlocal elasticity

In this study, the nonlocal Euler–Bernoulli beam theory is employed in the vibration and stability analysis of a nanobeam conveying fluid. The nanobeam is assumed to be traveling with a constant mean velocity along with a small harmonic fluctuation. In the considered analysis, the effects of the small-scale of the nanobeam are incorporated into the equations. By utilizing Hamilton’s principle, the nonlinear equations of motion including stretching of the neutral axis are derived. Damping effect is considered in the analysis. The closed form approximate solution of nonlinear equations is solved by using the multiple scale method, a perturbation technique. The effects of the different value of the nonlocal parameters, mean speed value and ratios of fluid mass to the total mass as well as effects of the simple–simple and clamped–clamped boundary conditions on the linear and nonlinear frequencies, stability, frequency–response curves and bifurcation point are presented numerically and graphically. The solvability conditions are obtained for the three distinct cases of velocity fluctuation frequency. For all cases, the stability areas of system are constructed analytically.     

Asymptotic behavior of the neutral curves of the linear stability problem for a laminar boundary layer

Asymptotic flow schemes corresponding to two branches of the solution for the neutral stability curve of a laminar boundary layer in an incompressible fluid are constructed. Two-term asymptotic solutions are obtained in the limit when the Reynolds number tends to infinity. The linear formulation of the problem is used and the flow is assumed to be two dimensional.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 39–46, September–October, 1981.I should like to thank O. V. Denisenko for making the numerical calculations.     

The Onset of Convection in an Anisotropic Porous Layer Using a Thermal Non-Equilibrium Model

The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases.     

Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness

For a spherical shell of arbitrary thickness which is subjected to an external hydrostatic pressure, symmetrical buckling takes place at a value of μ₁ which depends on and the mode number, where A₁ and A₂ are the undeformed inner and outer radii, and μ₁ is the ratio of the deformed inner radius to the undeformed inner radius. In the large mode number limit, we find that the dependence of μ₁ on has a boundary layer structure: it is a constant over almost the entire region of and decreases sharply from this constant value to unity as tends to unity (the thin-shell limit). Simple asymptotic expressions for the bifurcation condition are obtained. The classical result for thin shells is recovered directly from the equations of finite elasticity, and an asymptotic critical neutral curve (which envelops the neutral curves corresponding to different mode numbers) is obtained.     

On the stability of a thermally stratified conducting fluid under the action of aligned magnetic field

The stability of a thermally stable stratified viscous electrically conducting shear flow is investigated in the presence of an impressed uniform aligned magnetic field. Only two-dimensional disturbances are studied in this paper because Squire's theorem does not apply in general, owing to the presence of the aligned magnetic field. The analysis is partly analytical and partly numerical. The asymptotic solutions for non-viscous fluid are first obtained analytically and they are then improved by introducing viscous and thermal diffusion terms (but only for =1) to get a uniformly valid solution. The neutral stability curves are numerically computed for a range of values of Richardson and Stuart numbers, which show that the flow is completely stabilized when a Stuart number exceeds a certain value for a given R _i>0. It is shown that the combined effects of magnetic field and stratification is to make the system stable to two-dimensional disturbances at lower Stuart number than the one given by Stuart (1954) in the absence of thermal stratification.     

NGPG-Stability of Linear Multistep Methods for Systems of Generalized Neutral Delay Differential Equations

The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was analysed for the solution of the generalized system of linear neutral test equations. After the establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, it is shown that a linear multistep method is NGP _G -stable if and only if it is A-stable.     

Receptivity of Hypersonic Boundary Layer to Wall Disturbances

Theoretical analysis of hypersonic boundary-layer receptivity to wall disturbances is conducted using a combination of asymptotic and numerical methods. Excitation of the second mode by distributed and local forcing on a flat-plate surface is studied under adiabatic and cooled wall conditions. Analysis addresses receptivity to wall vibrations, periodic suction/blowing, and temperature disturbances. A strong excitation occurs in local regions where forcing is in resonance with normal waves. It is shown that the receptivity function tends to infinity as the resonance point tends to the branch point of the discrete spectrum that is typical for boundary layers on cool surfaces. Asymptotic analysis resolves this singularity and provides the receptivity coefficient in the branch-point vicinity. Numerical results indicate extremely high receptivity to vibrations and suction/blowing in the vicinity of the branch point located near the lower neutral branch of the Mack second mode. Received 5 September 2000 and accepted 7 September 2001     

The mode number dependence of neutral stability of cross-waves

The neutral stability of parametrically excited subharmonic cross-waves in a rectangular tank is studied experimentally. The analysis of the experimental data is based on the appropriately scaled dimensionless variables. The variation of the shape and of the location of the neutral stability curves with the mode number is studied. Certain conclusions are drawn on the basis of the present measurements regarding the relative importance of different damping mechanisms.     

Wilton ripples between two uniform streaming magnetic fluids

An analysis is made of the small-amplitude capillary-gravity waves which occur on the interface of two incompressible inviscid magnetic fluids of different densities. The waves arise as a result of second harmonic resonance. The fluids moving with uniform velocities parallel to their interface are stressed by an oblique magnetic field. The linear relations between the oblique magnetic field and the instability criteria of the linear waves are analyzed. At the stability region (away from the neutral curve) of the linear theory, a pair of coupled non-linear partial differential equations are presented. On the neutral curve, a pair of coupled non-linear partial differential equations are introduced. The last pair of equations may be regarded as the counterparts of the single Klein-Gordon equation which occurs in the non-resonant case. In all cases, the wave profile and its stability conditions are obtained. These conditions are discussed analytically and graphically.     

Tailoring of pinched hysteresis for nonlinear vibration absorption via asymptotic analysis

The method of multiple scales is adopted to investigate the dynamic response of a nonlinear Vibration Absorber (VA) whose constitutive behavior is governed by hysteresis with pinching. The asymptotic analysis is first devoted to study the response of the absorber to harmonic excitations and to evaluate its sensitivity to the main constitutive parameters. The frequency response obtained in closed form allows to carry out the stability analysis together with a parametric study leading to behavior charts characterizing multi-valued softening/hardening responses or single-valued, quasi-linear responses. A two-degree-of-freedom model of a primary nonlinear structure endowed with the hysteretic vibration absorber is investigated to explore transfers of energy from the structure to the absorber resulting into optimal vibration amplitude reduction. The asymptotic solution is proved to be in good agreement with the numerical solution obtained via continuation. The asymptotic approach is embedded into a differential evolutionary algorithm to obtain a multi-parameter optimization procedure by which the optimal hysteresis parameters are found.     

Thermocapillary convection in a plane liquid layer with concentration heat sources

Thermocapillary instability of a plane liquid binary-mixture layer with time-dependent surface tension is studied under weightlessness conditions. The liquid is heated (or cooled) due to heat release by an active admixture. The heat release rate is proportional to the active-component concentration. The admixture is transported by convection and diffusion. The active component “burns up” with time. The neutral curves for monotonous and oscillating disturbances are found for different values of the nondimensional parameters. Some nonlinear convection regimes are studied numerically by a finite-difference method. The dependence of the convective flow intensity on the Marangoni number is determined. The phase portraits of unsteady regimes are found.