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The simple procedure presented for spatial stability computations can substantially reduce the computational requirements of such analyses, as illustrated for the cases of both internal and external cases of compressible and incompressible flows, and both viscous and inviscid instability modes. Excellent estimates of spatial eigenvalues are obtained.
The eigenvalue sensitivity for hydrodynamic stability operators is investigated. Classical matrix perturbation techniques as well as the concept of epsilon-pseudoeigenvalues are applied to show that parts of the spectrum are highly sensitive to small perturbations. Applications are drawn from incompressible plane Couette, trailing line vortex flow and compressible Blasius boundary layer flow. Parametric studies indicate a monotonically increasing effect of the Reynolds number on the sensitivity. The phenomenon of eigenvalue sensitivity is due to the non-normality of the operators and their discrete matrix analogs and may be associated with large transient growth of the corresponding initial value problem.
Discussion of a problem encountered in the processing of viscoelastic materials that is caused by the presence of a hydrodynamic instability in the extrusion of polymer melts. The importance of the so-called Weissenberg number in determining the onset of the melt fracture is examined using classical linearized hydrodynamic stability analysis. It is shown that the simple shearing flow of a viscoelastic fluid becomes unstable at a critical value of the Weissenberg number. Implications for the extrusion processing of polymers are reviewed.
The spatial instability of a swirling jet is investigated both experimentally and theoretically. A hydrodynamic stability analysis is applied to an inviscid incompressible top-hat jet, with a swirl distribution of solid-body rotation and free vortex in and outside the vortex core, respectively. Both plane and helical instability modes are examined. It is found that the top-hat jet with swirl distribution of Rankine vortex type is unstable in all the modes studied. The higher the positive helicity, the less spatially unstable the jet behavior; the higher the negative helicity, the more spatially unstable this behavior becomes. A comparison is made between theoretical results and experimental data on a low-intensity swirling jet. The trend of the initial growth of the instability waves in the near field is captured.
The influence of the wave amplitude and oil supply pressure on the dynamic behavior of a hydrodynamic three-wave journal bearing is presented. Both, a transient and a small perturbation technique, were used to predict the threshold to fractional frequency whirl (FFW). In addition, the behavior of the rotor after FFW appeared was determined from the transient analysis. The turbulent effects were also included in the computations. Bearings having a diameter of 30 mm, a length of 27.5 mm, and a clearance of 35 microns were analyzed. Numerical results were compared to experimental results obtained at the NASA GRC. Numerical and experimental results showed that the above-mentioned wave bearing with a wave amplitude ratio of 0.305 operates stably at rotational speeds up to 60,000 rpm, regardless of the oil supply pressure. For smaller wave amplitude ratios, a threshold of stability was found. It was observed that the threshold of stability for lower wave amplitude strongly depends on the oil supply pressure and on the wave amplitude. When the FFW occurs, the journal center maintains its trajectory inside the bearing clearance and therefore the rotor can be run safely without damaging the bearing surfaces.
A linear analysis of hydrodynamic stability has been applied to a problem in which a fluid mixture is contained between two horizontal planes. One species diffuses to the lower plane where it is destroyed by a rapid exothermic or endothermic catalytic reaction. Results show that important coupling takes place between thermal and concentration fields. This coupling gives rise to unusual stabilizing or destabilizing effects, depending upon the value of Lewis number. Several examples are discussed. It is also shown how the results can be applied to other problems involving heat and mass transfer.
Accurate, robust, and efficient prediction of transition in viscous flows is a significant challenge in computational fluid dynamics. We present a coupled, high-fidelity, iterative approach that leverages the FUN3D flow solver and the LASTRAC stability code to predict transition in low-disturbance environments, initiated by the linear growth of boundary-layer instability modes. Our method integrates the ability of FUN3D to compute mixed laminar-transitional-turbulent mean flows via transition-sensitized Reynolds-Averaged Navier-Stokes equations with the ability of LASTRAC to perform linear stability analysis, all within an automated framework that requires no intermediate user involvement. Unlike conventional frameworks that rely on classical stability theory or reduced-order metamodels, our approach employs the parabolized stability equations to provide more accurate and reliable estimates of disturbance growth for multiple instability mechanisms, including Tollmien-Schlichting, Kelvin-Helmholtz, and crossflow modes. By accounting for the effects of mean-flow nonparallelism as well as the surface curvature, this approach lays the foundation for improved N-factor correlations for transition onset prediction in a broad class of flows. We apply this method to three distinct flow configurations: 1) flow over a zero-pressure-gradient flat plate, 2) the NLF-0416 airfoil with both natural and separation-induced transitions, and 3) a 6:1 prolate spheroid, where transition is primarily driven by crossflow instability. For the two-dimensional cases, a formulated intermittency distribution is used to model the transition zone in between the laminar and fully turbulent flows. The results include comparisons with experimental measurements, similar numerical approaches, and transport-equations-based models, demonstrating good agreement in surface pressure coefficients, transition onset locations, and skin-friction coefficients for all three configurations. Besides contributing a couple of new insights into boundary-layer transition in these canonical cases, this study provides a powerful tool for transition modeling in both research and design applications in aerodynamics.
The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LBE either analytically by using perturbation technique or numerically. The proposed LBE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation (hyper-viscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters which optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LBE simulations. The stability properties of some 2D LBE models are analyzed and compared with each other in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LBE models. As examples, LBE models with various interpolation schemes are analyzed. Numerical results on shear flow with an initially discontinuous velocity profile (shock) with or without a constant streaming velocity are shown to demonstrate the dispersion effects in the LBE model; the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long wave-length limit (wave vector k = 0), it can also provide results for large values of k. Such results are important for the stability and other hydrodynamic properties of the LBE method and cannot be obtained through Chapman-Enskog analysis.
The growth of three-dimensional perturbations subject to the Crow instability along a vortex dipole resulting from the passage of a shock wave through a heavy gaseous cylinder is examined numerically. A linear stability analysis is performed based on geometric parameters extracted from two-dimensional simulations to determine the range of unstable wavenumbers, which is found to extend from 0.0 to 1.3 when normalized by the core separation distance. The analysis is then verified by comparison to three-dimensional simulations, which clearly show the development of the instability and the pinch-off of the vortex dipole into isolated vortex rings, which manifest as clumps of the original cylinder material. A scaling law is developed to determine the relevant spatiotemporal scales of the instability development, which is then used to assess the feasibility of a high-energy-density experiment visualizing clump formation. Specifically, a shocked cylinder with an initial diameter of 100 μm consisting of a perturbation of approximate wavelength and amplitude of 600 and 10 μm, respectively, is expected to form clumps resulting from the Crow instability approximately 40 ns after it is shocked, with dynamics which can be readily visualized on the Omega EP laser facility.
Laminar-turbulent transition in shear flows is still an enigma in the area of fluid mechanics. The conventional explanation of the phenomenon is based on the instability of the shear flow with respect to infinitesimal disturbances. The conventional hydrodynamic stability theory deals with the analysis of normal modes that might be unstable. The latter circumstance is accompanied by an exponential growth of the disturbances that might lead to laminar-turbulent transition. Nevertheless, in many cases, the transition scenario bypasses the exponential growth stage associated with the normal modes. This type of transition is called bypass transition. An understanding of the phenomenon has eluded us to this day. One possibility is that bypass transition is associated with so-called algebraic (non-modal) growth of disturbances in shear flows. In the present work, an analysis of the optimal disturbances/streamwise vortices associated with the transient growth mechanism is performed for boundary layers in the presence of a streamwise pressure gradient. The theory will provide the optimal spacing of the control elements in the spanwise direction and their placement in the streamwise direction.
The structure and stability of flames in dusty mixtures is investigated. The presence of the dust leads to significant transport of energy by radiation and the fundamental goal of the analysis is to explore to what extent this displaces the classical non-hydrodynamical stability boundaries of the plane degradation. An approximate description of the radiative transport permits analysis for arbitrary values of both the planck length and the Boltzman number. It is shown that the pulsating/traveling-wave instability usually associated with values of Lewis number (L(sub e)) bigger than 1 is stongly enhanced by the presence of radiation and can be present even if L(sub e) is less than 1. On the other hand radiation tends to supress the cellular instability normally associated with values of L(sub e) less than 1. The latter is consisitent with preliminary experimental observations of Abbud-Madrid and Ronney.
A steady incompressible three-dimensional (3-D) viscous flow analysis was conducted for the Space Shuttle Main Propulsion External Tank (ET)/Orbiter (ORB) propellant feed line quick separable 17-inch disconnect flapper valves for liquid oxygen (LO2) and liquid hydrogen (LH2). The main objectives of the analysis were to predict and correlate the hydrodynamic stability of the flappers and pressure drop with available water test data. Computational Fluid Dynamics (CFD) computer codes were procured at no cost from the public domain, and were modified and extended to carry out the disconnect flow analysis. The grid generator codes SVTGD3D and INGRID were obtained. NASA Ames Research Center supplied the flow solution code INS3D, and the color graphics code PLOT3D. A driver routine was developed to automate the grid generation process. Components such as pipes, elbows, and flappers can be generated with simple commands, and flapper angles can be varied easily. The flow solver INS3D code was modified to treat interior flappers, and other interfacing routines were developed, which include a turbulence model, a force/moment routine, a time-step routine, and initial and boundary conditions. In particular, an under-relaxation scheme was implemented to enhance the solution stability. Major physical assumptions and simplifications made in the analysis include the neglect of linkages, slightly reduced flapper diameter, and smooth solid surfaces. A grid size of 54 x 21 x 25 was employed for both the LO2 and LH2 units. Mixing length theory applied to turbulent shear flow in pipes formed the basis for the simple turbulence model. Results of the analysis are presented for LO2 and LH2 disconnects.
Using numerical hydrodynamics techniques, we perform a nonlinear stability analysis of accretion disk systems that contain thick, self-gravitating disks. The systems are initially represented by a point mass M(sub c) at the center and a geometrically thick, axisymmetric disk of mass M(sub d) that supports uniform specific angular momentum and obeys an n = 3/2, polytropic equation of state. The equilibrium disk structure is uniquely defined upon the specification of two key dimensionless system parameters: M(sub d)/M(sub c) and T/absolute value of W (the ratio of rotational kinetic energy if the disk to the gravitational potential energy of the system). The focus of this work is on the identification of systems within this two-dimensional parameter space that are marginally unstable toward the development of nonaxisymmetric distortions. The geometric form and relative pattern speed of the disk's distortion as well as the likelihood of disk fragmentation as a result of such instabilities is examined, particularly in the context of protostellar systems. The value of T/absolute value of W at which thick disks first become dynamically unstable to nonaxisymmetric distortions is found to vary significantly with the mass ratio of the accertion disk system. Nonaxisymmetric eigenmodes with four distinctly different characters are identified in systems with mass ratios in the range 0.2 less than or equal to (M(sub d)/M(sub c)) less than or equal to 5.
This investigation addresses the problem of hydrodynamic stability of a two-component droplet undergoing spherically-symmetrical gasification. The droplet components are assumed to have characteristic liquid species diffusion times that are large relative to characteristic droplet surface regression times. The problem is formulated as a linear stability analysis, with a goal of predicting when spherically-symmetric droplet gasification can be expected to be hydrodynamically unstable from surface-tension gradients acting along the surface of a droplet which result from perturbations. It is found that for the conditions assumed in this paper (quasisteady gas phase, no initial droplet temperature gradients, diffusion-dominated gasification), surface tension gradients do not play a role in the stability characteristics. In addition, all perturbations are predicted to decay such that droplets were hydrodynamically stable. Conditions are identified, however, that deserve more analysis as they may lead to hydrodynamic instabilities driven by capillary effects.
The rotordynamic behavior of the high pressure oxygen turbopump and high pressure fuel pump was analyzed for the Space Shuttle Main Engine. The identification of potential rotordynamic problem areas which might arise during operation of these units prior to their testing was accomplished. Alternative procedures for correcting potential rotordynamic problems should they occur were investigated. An adequate analytic and physical understanding of the turbopump rotordynamics was developed to improve the probability of a correct diagnosis of rotordynamic problems from test data. Transient rotordynamic models were developed for both turbopumps. The transient models model the hydrodynamic forces of the turbopump seals. A linear stability analysis was performed for the turbopump rotordynamics models, which included gyroscopic effects, seal forces, speed-dependent bearing characteristics, and internal rotor damping. Results are presented and discussed.
Understanding the solvation structures of lithium salts in carbonate- and ether-based electrolytes is central to explaining the exceptional stability of high-concentration electrolytes (HCEs) and localized high-concentration electrolytes (LHCEs). Conventional techniques such as vibrational spectroscopy and one-dimensional NMR provide only limited information, typically restricted to coordination ratios and ion-pair distributions, without revealing the actual size and mass of solution complexes. Here, in this study, we introduce an average hydrodynamic radius (AHR) analysis based on internally referenced DOSY NMR, which enables direct estimation of the average volume and molecular weight of lithium–solvent complexes in solution. Using LiFSI–EMC and LiFSI–EMC–TTE electrolytes, we demonstrate that the onset of effective lithium metal stabilization and aluminum corrosion suppression coincides with the formation of very large complexes, whose average volume exceeds 300 times that of free EMC molecules. This finding supports a new “blocking mechanism,” wherein bulky solvated complexes impede direct solvent access to reactive surfaces. The AHR analysis thus not only clarifies the fundamental origin of HCE and LHCE effectiveness, but also provides a broadly applicable experimental framework for probing complex solvation structures and guiding rational electrolyte design.
A linearized stability analysis has been applied to a fluid flowing in a gravity field between horizontal planes in Couette flow under conditions such that the temperature of the bottom plane exceeds that of the top. It is shown that, under conditions likely to be encountered with polymer solutions, oscillatory instabilities will not be controlling. Criteria are offered for ascertaining when an analysis based upon a second-order fluid model may be expected to yield physically meaningful results. It is also shown that, for the fluid model considered, critical conditions for stability are not changed when disturbances which vary in the flow direction are substituted for those which are a function of the coordinate transverse to the flow.
A theoretical formulation and analysis is presented for a study of the stability and interaction of coherent structure in reacting free shear layers. The physical problem under investigation is a premixed hydrogen-oxygen reacting shear layer in the wake of a thin flat plate. The coherent structure is modeled as a periodic disturbance and its stability is determined by the application of linearized hydrodynamic stability theory which results in a generalized eigenvalue problem for reactive flows. Detailed stability analysis of the reactive wake for neutral, symmetrical and antisymmetrical disturbance is presented. Reactive stability criteria is shown to be quite different from classical non-reactive stability. The interaction between the mean flow, coherent structure and fine-scale turbulence is theoretically formulated using the von-Kaman integral technique. Both time-averaging and conditional phase averaging are necessary to separate the three types of motion. The resulting integro-differential equations can then be solved subject to initial conditions with appropriate shape functions. In the laminar flow transition region of interest, the spatial interaction between the mean motion and coherent structure is calculated for both non-reactive and reactive conditions and compared with experimental data wherever available. The fine-scale turbulent motion determined by the application of integral analysis to the fluctuation equations. Since at present this turbulence model is still untested, turbulence is modeled in the interaction problem by a simple algebraic eddy viscosity model. The applicability of the integral turbulence model formulated here is studied parametrically by integrating these equations for the simple case of self-similar mean motion with assumed shape functions. The effect of the motion of the coherent structure is studied and very good agreement is obtained with previous experimental and theoretical works for non-reactive flow. For the reactive case, lack of experimental data made direct comparison difficult. It was determined that the growth rate of the disturbance amplitude is lower for reactive case. The results indicate that the reactive flow stability is in qualitative agreement with experimental observation.