Sensitivity Analysis Using Adjoint Parabolized Stability Equations for Compressible Flows

An input/output framework is used to analyze the sensitivity of two- and three-dimensional disturbances in a compressible boundary layer for changes in wall and momentum forcing. The sensitivity is defined as the gradient of the kinetic disturbance energy at a given downstream position with respect to the forcing. The gradients are derived using the parabolized stability equations (PSE) and their adjoint (APSE). The adjoint equations are derived in a consistent way for a quasi-two-dimensional compressible flow in an orthogonal curvilinear coordinate system. The input/output framework provides a basis for optimal control studies. Analysis of two-dimensional boundary layers for Mach numbers between 0 and 1.2 show that wall and momentum forcing close to branch I of the neutral stability curve give the maximum magnitude of the gradient. Forcing at the wall gives the largest magnitude using the wall normal velocity component. In case of incompressible flow, the two-dimensional disturbances are the most sensitive ones to wall inhomogeneity. For compressible flow, the three-dimensional disturbances are the most sensitive ones. Further, it is shown that momentum forcing is most effectively done in the vicinity of the critical layer. This revised version was published online in July 2006 with corrections to the Cover Date.     

Three-dimensional disturbances of a plane-parallel two-layer flow of a viscous, heat-conducting fluid

The stability of the Marangoni-Poiseuille flow in an inclined channel against three-dimensional disturbances is studied. It is shown that the nature of instability is determined by wall heating conditions and the system orientation with respect to its horizontal position. With variation in the angle of inclination of the system spiral disturbances leading to the system crisis can develop.     

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger , two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.     

Numerical simulation of the horizontal Bridgman growth. Part II: Three-dimensional flow

We study the steady-state three-dimensional flow which occurs in a horizontal crucible of molten metal under the action of a horizontal temperature gradient. The geometry and the boundary conditions are similar to those encountered in the Bridgman growth process of semiconductor crystals. We find that three-dimensional effects can have a dramatic influence upon the flow, which, before the onset of periodic disturbances, differs appreciably from its two-dimensional counterpart. We also investigate the sensitivity of the flow to non-symmetric disturbances.     

Development of three-dimensional disturbances in a boundary layer

In the region of transition from a two-dimensional laminar boundary layer to a turbulent one, three-dimensional flow occurs [1–3]. It has been proposed that this flow is formed as the result of nonlinear interaction of two-dimensional and three-dimensional disturbances predicted by linear hydrodynamic stability theory. Using many simplifications, [4, 5] performed a calculation of this interaction for a free boundary layer and a boundary layer on a wall with a very coarse approximation of the velocity profile. The results showed some argreement with experiment. On the other hand, it is known that disturbances of the Tollmin—Schlichting wave type can be observed at sufficiently high amplitude. This present study will use the method of successive linearization to calculate the primary two- and three-dimensional disturbances, and also the average secondary flow occurring because of nonlinear interaction of the primary disturbances. The method of calculation used is close to that of [4, 5], the disturbance parameters being calculated on the basis of a Blazius velocity profile. A detailed comparison of results with experimental data [1] is made. It developed that at large disturbance amplitude the amplitude growth rate differs from that of linear theory, while the spatial distribution of disturbances agree s well with the distribution given by the natural functions and their nonlinear interaction. In calculating the secondary flow an experimental correction was made to the amplitude growth rate.     

Linear stability analysis in compressible, flat-plate boundary-layers

The stability problem of two-dimensional compressible flat-plate boundary layers is handled using the linear stability theory. The stability equations obtained from three-dimensional compressible Navier–Stokes equations are solved simultaneously with two-dimensional mean flow equations, using an efficient shoot-search technique for adiabatic wall condition. In the analysis, a wide range of Mach numbers extending well into the hypersonic range are considered for the mean flow, whereas both two- and three-dimensional disturbances are taken into account for the perturbation flow. All fluid properties, including the Prandtl number, are taken as temperature-dependent. The results of the analysis ascertain the presence of the second mode of instability (Mack mode), in addition to the first mode related to the Tollmien–Schlichting mode present in incompressible flows. The effect of reference temperature on stability characteristics is also studied. The results of the analysis reveal that the stability characteristics remain almost unchanged for the most unstable wave direction for Mach numbers above 4.0. The obtained results are compared with existing numerical and experimental data in the literature, yielding encouraging agreement both qualitatively and quantitatively.      

Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 101–106, January–February, 1976.     

Applications of parabolized stability equation for predicting transition position in boundary layers

The phenomenon of laminar-turbulent transition exists universally in nature and various engineering practice.The prediction of transition position is one of crucial theories and practical problems in fluid mechanics due to the different characteristics of laminar flow and turbulent flow.Two types of disturbances are imposed at the entrance,i.e.,identical amplitude and wavepacket disturbances,along the spanwise direction in the incompressible boundary layers.The disturbances of identical amplitude are consisted of one two-dimensional(2D) wave and two three-dimensional(3D) waves.The parabolized stability equation(PSE) is used to research the evolution of disturbances and to predict the transition position.The results are compared with those obtained by the numerical simulation.The results show that the PSE method can investigate the evolution of disturbances and predict the transition position.At the same time,the calculation speed is much faster than that of the numerical simulation.     

Rayleigh稳定性准则的一种推广

将不可压无黏旋流在轴对称扰动情况下名的Rayleigh稳定性准则推广到可压缩情况,由物理机理的分析出发导出可压缩无流在轴对称扰下旋转流稳定性的准同,并将此准则与其它稳定性条件进行了比较。     

Stability of Hartmann flows

The stability of Hartmann flows for arbitrary magnetic Reynolds numbers is investigated in the framework of linear theory. The initial three-dimensional problem reduces to the equivalent two-dimensional problem. Perturbation theory is used to find asymptotic expressions for the eigenvalues. Distinguishing two types of disturbances — magnetic and hydrodynamic — is shown to be advantageous in a number of cases. Simple features of the stability are considered for particular cases. The well-know Lundquist result is generalized. An energy approach is applied to the problem of stability. The results of simulations involving the solution of the linear stability problem are described. A distinctive picture of stability is developed. There are several types of instability and they can develop simultaneously. The hydrodynamic and magnetic phenomena interact with each other in a very complex fashion. The magnetic field can either enhance flow stability or reduce it.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–31, November–December, 1972.     

Necessary and sufficient conditions for the stability of flows of incompressible viscous fluids

Summary We perturb a steady flow of an incompressible viscous fluid and derive a necessary and sufficient condition for the marginal cases for monotonie energy stability and stability against small (infinitesimal) disturbances to coincide. Evaluation of this condition in two examples singles out, in terms of the parameters of the problem, the cases where necessary and sufficient conditions for stability coincide and thus the steady flow first becomes unstable, together with the class of perturbations responsible for the instability. The analysis is done within the range of strict solutions of each underlying problem; the precise regularity and existence classes are given in Sec. 0. The examples we treat are plane parallel shear flow with a non-symmetric profile in an infinite rotating layer and the effect of rotation on convection.     

Effect of a Standing Acoustic Wave on the Development of Large-Scale Convection in a Horizontal Layer

The effect of a standing acoustic wave on the development of long-wave convective perturbations in a horizontal layer with thermally insulated boundaries is investigated. The main two-dimensional flow is determined. A nonlinear amplitude equation with spatially-periodic coefficients is derived for investigating the stability of the main flow and secondary convection flows in the neighborhood of the stability threshold. The intensity of the acoustic field is assumed to be low. It is shown that the acoustic action leads to destabilization of the layer. Plane and three-dimensional perturbations are critical at large and small Prandtl numbers, respectively. Nonlinear one-dimensional steady-state solutions of the amplitude equation are obtained and their stability is investigated.     

Instability due to three-dimensional disturbances

In the linear theory of the stability of parallel flows of a viscous fluid, most attention is usually given to plane-wave disturbances. The reason is the validity in many cases of the Squire theorem, which states that the critical Reynolds number R is determined by two-dimensional disturbances [1]. It is shown in the present paper that for large R the region generating the turbulence in the initial stage of its development is formed by three-dimensional disturbances. This feature applies both to the generating range of wave numbers and the dimension of the wall layer, where the fluctuating energy is produced. The consequences of the Squire transformations for parallel flows are analyzed. The contribution of resonant nonlinear triad coupling to the rapid growth of fluctuating energy is studied for the case of an explosive instability in an extended laminar mode. It is shown that the rate of turbulent energy production is not governed by the small derivatives of linear theory, but by nonlinear triad coupling of neutral and growing disturbances, with their three-dimensional nature playing an important role.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 29–34, September–October, 1976.The author thanks M. A. Gol'dshtik for his interest in the work and for discussion of the results.     

Evolution Equations for the Surface Concentration of an Insoluble Surfactant; Applications to the Stability of an Elongating Thread and a Stretched Interface

The general form of the convection–diffusion equation governing the evolution of the surface concentration of an insoluble surfactant over an evolving interface is reviewed and discussed for three-dimensional, axisymmetric, and two-dimensional configurations. The linearized form of the evolution equation is then derived around cylindrical and planar shapes in a framework that is suitable for carrying out a linear stability analysis for axisymmetric or two-dimensional perturbations. Particular attention is paid to the cases of quiescent unperturbed fluids, unidirectional shear flow, and elongational flow. By way of application, the linearized transport equations are combined with Stokes-flow hydrodynamics to investigate the stability of an elongating cylindrical viscous thread suspended in an ambient viscous fluid or in a vacuum, and the stability of a two-dimensional interface separating two semi-infinite fluids and stretched under the action of an orthogonal stagnation-point flow. The results illustrate the possibly important role of the surfactant on the linear growth of periodic waves on the cylindrical interface, and reveal that the surfactant has no effect on the stability of the planar interface.     

THREE-DIMENSIONAL VISCOUS FLOW THROUGH A ROTATING CHANNEL: A PSEUDOSPECTRAL MATRIX METHOD APPROACH

A Fourier–Chebyshev pseudospectral method is used for the numerical simulation of incompressible flows in a three-dimensio nal channel of square cross-section with rotation. Realistic, non-periodic boundary conditions that impose no-slip conditions in two directions (spanwis e and vertical directions) are used. The Navier–Stokes equations are integrated in time using a fractional step method. The Poisson equations for pressure and the Helmholtz equation for velocity are solved using a matrix diagonalization (eigenfunction decomposition) method, through which we are able to reduce a three-dimensional matrix problem to a simple algebraic vector equation. This results in signficant savings in computer storage requirement, particularly for large-scale computations. Verification of the numerical algorithm and code is carried out by comparing with a limiting case of an exact steady state solution for a one-dimensional channel flow and also with a two-dimensional rotating channel case. Two-cell and four-cell two-dimensional flow patterns are observed in the numerical experiment. It is found that the four-cell flow pattern is stable to symmetri cal disturbances but unstable to asymmetrical disturbances.     

Optimal disturbances of a dusty-gas plane-channel flow with a nonuniform distribution of particles

The classical stability theory for multiphase flows, based on an analysis of one (most unstable) mode, is generalized. A method for studying an algebraic (non-modal) instability of a disperse medium, which consists in examining the energy of linear combinations of three-dimensional modes with given wave vectors, is proposed. An algebraic instability of a dusty-gas flow in a plane channel with a nonuniform particle distribution in the form of two layers arranged symmetrically with respect to the flow axis is investigated. For all possible values of governing parameters, the optimal disturbances of the disperse flow have zero wavenumber in the flow direction, which indicates their banded structure (“streaks”). The presence of dispersed particles in the flow increases the algebraic instability, since the energy of optimal disturbances in the disperse medium exceeds that for the pure-fluid flow. It is found that for a homogeneous particle distribution the increase in the energy of optimal perturbations is proportional to the square of the sum of unity and the particle mass concentration and is almost independent of particle inertia. For a non-uniform distribution of the dispersed phase, the largest increase in the initial energy of disturbances is achieved in the case when the dust layers are located in the middle between the center line of the flow and the walls.     

The stability of vortex arrays to two- and three-dimensional disturbances

Results are presented on the structure of arrays of two-dimensional uniform vortex patches and their stability to two- and three-dimensional infinitesimal disturbances. Single rotating vortices and vortices in a uniform strain, vortex pairs, and single and double rows are considered.     

Stability of detonation combustion with respect to the variation in the hydrogen concentration at the supersonic nozzle inlet

Detonation combustion of hydrogen-air mixtures entering an axisymmetric convergent-divergent nozzle at a supersonic velocity is considered. The nozzle geometry does not ensure gas self-ignition; for this reason, forced ignition is used, which, under certain conditions, leads to the formation of stationary detonation combustion in the case of both uniform and nonuniform hydrogen distribution at the channel entry. The nonlinear problem of the stability of these combustion regimes against periodic disturbances of the hydrogen concentration in the oncoming flow is numerically solved. The study is performed on the basis of the two-dimensional gasdynamic Euler equations for a multicomponent reacting gas. A detailed model of chemical reactions is used.     

Estimation of three-dimensional flame surface densities from planar images in turbulent premixed combustion

Turbulence motions are, by nature, three-dimensional while planar imaging techniques, widely used in turbulent combustion, give only access to two-dimensional information. For example, to extract flame surface densities, a key ingredient of some turbulent combustion models, from planar images implicitly assumes an instantaneously two-dimensional flow, neglecting the unresolved flame front wrinkling. The objective here is to estimate flame surface densities from two-dimensional measurements assuming that (1) the flow is statistically two dimensional; (2) the measuring plane is a plane of symmetry of the mean flow, either by translation (homogeneous third direction as in slot burners for example) or by rotation (axi-symmetrical flows such as jets) and (3) flame movements in transverse directions are similar. The unknown flame front wrinkling is then modelled from known quantities. An excellent agreement is achieved against direct numerical simulation (DNS) data where all three-dimensional quantities are known, but validations in other conditions (larger DNS, experiments) are required.     

On the cyclone-anticyclone asymmetry in the stability of rotating shear flows

The manifestations of the cyclone-anticyclone asymmetry on the stability of rotating shear flows are investigated both theoretically and experimentally. The stability of certain classes of shear flows, namely, rotating tangential discontinuities and flows with a constant shear, is analyzed. The dependence of the disturbance growth rate on the sign and absolute value of the shear is determined. The three-dimensional disturbances leading to longitudinal flow modulations are shown to be most dangerous. The results of the observations of the cyclone-anticyclone asymmetry effect in the laboratory conditions are presented.