Showing papers in "Physics of Fluids in 1983"
TL;DR: In this paper, the forces on a small rigid sphere in a nonuniform flow are considered from first prinicples in order to resolve the errors in Tchen's equation and the subsequent modified versions that have since appeared.
Abstract: The forces on a small rigid sphere in a nonuniform flow are considered from first prinicples in order to resolve the errors in Tchen’s equation and the subsequent modified versions that have since appeared. Forces from the undisturbed flow and the disturbance flow created by the presence of the sphere are treated separately. Proper account is taken of the effect of spatial variations of the undisturbed flow on both forces. In particular the appropriate Faxen correction for unsteady Stokes flow is derived and included as part of the consistent approximation for the equation of motion.
3,130 citations
TL;DR: In this article, a definition of coherent structures in turbulent shear flows is proposed and its implications discussed, and the characteristic coherent structure properties are identified and the analytical and experimental constraints in the eduction of coherent structure are examined.
Abstract: The nature and significance of large‐scale coherent structures in turblent shear flows are addressed. A definition for the coherent structure is proposed and its implications discussed. The characteristic coherent structure properties are identified and the analytical and experimental constraints in the eduction of coherent structures are examined. Following a few comments on coherent motions in wall layers, the accumulated knowledge from a number of recent and ongoing coherent structure investigations in excited and unexcited free shear flows in the author’s laboratory is reviewed. Also briefly addressed are effects of initial conditions, the role of coherent structures in jet noise production and broadband noise amplification, the feedback effect of coherent structures, the use of the Taylor hypothesis in coherent structure description, negative production, turbulence suppression via excitation, validity of the Reynolds number similarity hypothesis, etc. From the detailed quantitative results, a picture of the state of the art in coherent structure studies emerges. While coherent structures are highly interesting characteristic features of (perhaps all) turbulent shear flows, it is argued that their dynamical significance has been overemphasized. These are predominant only in their early stages of formation following instability, or in resonant situations and excited flows, or in regions adjacent to a wall of a turbulent boundary layer. The coherent Reynolds stress, vorticity, and production are comparable to (and not an order of magnitude larger than) the time‐average Reynolds stress, vorticity, and production, respectively, in fully developed states of turbulent shear flows, where incoherent turbulence is also important and cannot be ignored. The concept and importance of coherent structures are here to stay; understanding and modeling of turbulent shear flows will be incomplete without them; but they are not all that matter in turbulent shear flows.
772 citations
TL;DR: In this article, an energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, ρ, π, σ, ω, φ, υ, τ, ϵ, ϳ, ς, ψ, ϩ, ϸ, ϴ, Ϡ, ϖ, ϓ, ό, ϐ, Ϻ, ϔ
Abstract: An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest‐descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive‐definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self‐consistent value for λ.
750 citations
TL;DR: In this article, a two-equation turbulence model was developed for predicting two-phase flows, which describes the conservation of turbulence kinetic energy and dissipation rate of that energy for the carrier fluid.
Abstract: A two-equation turbulence model has been developed for predicting two-phase flows. The two equations describe the conservation of turbulence kinetic energy and dissipation rate of that energy for the carrier fluid in a two-phase flow. They have been derived rigorously from the momentum equations of the carrier fluid. Closure of the time-mean equations is achieved by modeling the turbulent correlations up to third order. The new model eliminates the need to simulate in an ad hoc manner the effects of the dispersed phase on turbulence structure. Preliminary testing indicates that the model is successful in predicting the main features of a round gaseous jet laden with uniform-size solid particles.
565 citations
TL;DR: In this paper, a new scheme for particle simulation based on the gyrophase-averaged Vlasov equation has been developed, which is suitable for studying linear and nonlinear low-frequency microinstabilities and the associated anomalous transport in magnetically confined plasmas.
Abstract: A new scheme for particle simulation based on the gyrophase‐averaged Vlasov equation has been developed. It is suitable for studying linear and nonlinear low‐frequency microinstabilities and the associated anomalous transport in magnetically confined plasmas. The scheme retains the gyroradius effects but not the gyromotion; it is, therefore, far more efficient than conventional ones. Furthermore, the reduced Vlasov equation is also amenable to analytical studies.
477 citations
TL;DR: In this article, it is suggested that extremely low-level spatially coherent disturbances in individual facilities change the initial conditions of a laminar shear layer and result in the discrepancies reported in the literature.
Abstract: For about a decade it has been noticed from the measurements of many jets that the value of the preferred mode and the spreading rate vary within a range of about 100%. In the present paper it is suggested that extremely low‐level spatially coherent disturbances in individual facilities change the initial conditions of a laminar shear layer. The various initial conditions are able to cause different downstream developments of the jet and result in the discrepancies reported in the literature.
453 citations
TL;DR: In this paper, the authors derived nonlinear gyrokinetic equations from a systematic Hamiltonian theory, which employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius.
Abstract: Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, only electrostatic fluctuations in slab geometry are considered; however, there is a straightforward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and several limiting forms are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev can ony be derived by an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry–Horton and Hasegawa–Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed. The resulting theory is very similar in content to the recent work of Lee. However, the systematic nature of our derivation provides considerable insight into the structure and interpretation of the equations.
346 citations
TL;DR: In this paper, the structure and dynamics of round turbulent jets were investigated and it was found that the far field region of the jet is dominated by large-scale vortical structures, which appear to be axisymmetric or helical a large part of the time.
Abstract: Laser‐induced fluorescence and particle streak velocity measurements were conducted to investigate the structure and dynamics of round turbulent jets. The results suggest that the far‐field region of the jet is dominated by large‐scale vortical structures, which appear to be axisymmetric or helical a large part of the time. Entrainment and mixing of the reservoir fluid with the jet fluid is found to be intimately connected with the kinematics of these structures. Unmixed reservoir fluid is found to reach and cross the jet axis.
304 citations
TL;DR: In this article, a WKB formalism for constructing normal modes of short-wavelength ideal hydromagnetic, pressure-driven instabilities (ballooning modes) in general toroidal magnetic containment devices with sheared magnetic fields is developed.
Abstract: A WKB formalism for constructing normal modes of short‐wavelength ideal hydromagnetic, pressure‐driven instabilities (ballooning modes) in general toroidal magnetic containment devices with sheared magnetic fields is developed. No incompressibility approximation is made. A dispersion relation is obtained from the eigenvalues of a fourth‐order system of ordinary differential equations to be solved by integrating along a line of force. Higher‐order calculations are performed to find the amplitude equation and the phase change at a caustic. These conform to typical WKB results. In axisymmetric systems, the ray equations are integrable, and semiclassical quantization leads to a growth rate spectrum consisting of an infinity of discrete eigenvalues, bounded above by an accumulation point. However, each eigenvalue is infinitely degenerate. In the nonaxisymmetric case, the rays are unbounded in a four‐dimensional phase space, and semiclassical quantization breaks down, leading to broadening of the discrete eigenvalues and the accumulation point of the axisymmetric unstable spectrum into continuum bands. Analysis of a model problem indicates that the broadening of the discrete eigenvalues is numerically very small, the dominant effect being broadening of the accumulation point.
282 citations
TL;DR: In this paper, the moment equation approach to nonaxisymmetric toroidal systems under the assumption of the existence of magnetic surfaces was generalized to non-axismmetric systems and the parallel plasma flows and bootstrap current were calculated in both the Pfirsch-Schluter and banana regimes.
Abstract: The moment equation approach to neoclassical transport theory has been generalized to nonaxisymmetric toroidal systems under the assumption of the existence of magnetic surfaces. In particular, the parallel plasma flows and bootstrap current are calculated in both the Pfirsch–Schluter and banana regimes. It is found that both parallel plasma flows and the bootstrap current can be reduced as the toroidal bumpiness increases in an otherwise axisymmetric system.
251 citations
TL;DR: In this article, the Schrodinger equation was used to model the two-plasmon instability in warm inhomogeneous plasma for a normally incident pump, and the complex eigenfrequencies of the absolute instability were obtained by reducing the linearized fluid equations to a simple WKB equation in wavenumber space.
Abstract: The two‐plasmon instability in warm inhomogeneous plasma for a normally incident pump is considered. The complex eigenfrequencies of the absolute instability are obtained by reducing the linearized fluid equations to a Schrodinger equation in wavenumber space. These eigenvalues are obtained in several ways. One is by combining a perturbation expansion in powers of the reciprocal scale length with WKB theory. The resulting algebraic equations are solved by three analytical approximations and by direct numerical solution. A second way is by analysis of the Schrodinger equation using an interactive WKB computer code. A third way is by the use of a shooting code. These methods are all used and compared for threshold curves and growth rates above threshold. Some eigenfunction forms are also obtained. The threshold is near (v0/ve)2k0 L =3, and varies weakly with β≂v4e/v20c2, rising from near 2 to about 4 over six decades of variation of β. The corresponding critical value of (ky/k0)2 is near 0.2/β over this ran...
TL;DR: In this paper, a simplified derivation of a second-order partial differential equation which determines axisymmetric equilibrium states is given for general configurations, equations on flux surfaces which determine the Alfven and cusp continuous spectrum are derived and the stability investigated.
Abstract: In a rotating equilibrium state, the velocity and magnetic fields are shown to share the same flux surfaces. A simplified derivation is given of a second‐order (not necessarily elliptic) partial differential equation which determines axisymmetric equilibrium states. For general configurations, equations on flux surfaces which determine the Alfven and cusp continuous spectrum are derived and the stability investigated. These equations are written without the use of any particular coordinate system. Similar equations yield a sufficient condition for global stability of axisymmetric equilibria if the flow is parallel to the magnetic field up to a rigid rotation of the plasma. This condition is also necessary for stability in a mirror configuration with no toroidal field and a pure rigid rotation.
TL;DR: In this paper, a linear secondary instability mechanism is presented that leads to the occurrence of subharmonic three-dimensional disturbances in wall-bounded shear flows, which originates from the periodic redistribution of vorticity in the shear flow by small but finite-amplitude Tollmien-Schlichting waves.
Abstract: A linear secondary instability mechanism is presented that leads to the occurrence of subharmonic three‐dimensional disturbances in wall‐bounded shear flows. The instability originates from the periodic redistribution of vorticity in the shear flow by small but finite‐amplitude Tollmien–Schlichting waves. Low threshold amplitudes and other characteristics of this instability are consistent with experiments and may elucidate various obscure observations.
TL;DR: In this article, it was shown that large classes of plasma equilibria can have identical drift orbits and associated transport and are called isomorphic, and the neoclassical coefficients were given for all equilibrium in which the magnetic field strength depends on one helicity.
Abstract: It is shown that large classes of plasma equilibria can have identical drift orbits and associated transport. Such equilibria are called isomorphic. In particular, the neoclassical coefficients are given for all equilibria in which the magnetic field strength depends on one helicity.
TL;DR: In this paper, large-amplitude rotating magnetohydrodynamic modes were observed to induce significant high-energy beam particle loss during high-power perpendicular netural beam injection on the poloidal divertor experiment (PDX) and a Hamiltonian formalism for drift orbit trajectories was used to study induced particle loss analytically and numerically.
Abstract: Large‐amplitude rotating magnetohydrodynamic modes are observed to induce significant high‐energy beam particle loss during high‐power perpendicular netural beam injection on the poloidal divertor experiment (PDX). A Hamiltonian formalism for drift orbit trajectories in the presence of such modes is used to study induced particle loss analytically and numerically. Results are in good agreement with experiment.
TL;DR: In this article, a method of analyzing ergodic magnetic fields is given, including a generalization of magnetic coordinates to such fields, which can be used in existing Monte Carlo codes to assess the enhanced transport associated with imperfect surfaces.
Abstract: A method of analyzing ergodic magnetic fields is given, including a generalization of magnetic coordinates to such fields. The results of this analysis can be used in existing Monte Carlo codes to assess the enhanced transport associated with imperfect surfaces. The Hamiltonian for a general magnetic field is given as part of this analysis.
Abstract: The Brinkman equation purports to describe low‐Reynolds‐number flow in porous media in situations where velocity gradients are non‐negligible The equation involves modifying the usual Darcy law by the addition of a standard viscosity term whose coefficient is usually identified with the pure‐fluid viscosity It is argued instead that the porous medium induces a renormalization of viscosity, which is calculated in the dilute limit and separately in a self‐consistent approximation The effective Brinkman viscosity is found to decrease from the pore‐fluid value The calculation fails at low porosity but agrees at least in part with experiment In addition, the relationship between the Brinkman equation and the phenomenological boundary condition of Beavers and Joseph is discussed and it is pointed out that their experimental configuration provides a simple means of measuring viscosity renormalization
TL;DR: In this paper, the macrostructure of perpendicular shocks in the supercritical regime is investigated theoretically using a model in which the ions are treated as a multifluid and the electrons as a massless fluid.
Abstract: The macrostructure of perpendicular shocks in the supercritical regime is investigated theoretically. The coupling between the shock precursor (foot region) associated with reflected ions and the magnetic ramp associated with electron Ohmic heating is analyzed using a model in which the ions are treated as a multifluid and the electrons as a massless fluid. The results are found to be in good agreement with a number of features exhibited by recent numerical simulations of perpendicular shocks, including the morphology of the reflected‐gyrating ion stream, the presence of potential and magnetic field overshoots, and the Mach number dependence of the shock structure.
TL;DR: In this paper, the two-component fluid equations describing electron-drift and ion-acoustic waves in a nonuniform magnetized plasma were shown to possess nonlinear two-dimensional solitary wave solutions.
Abstract: The two‐component fluid equations describing electron‐drift and ion‐acoustic waves in a nonuniform magnetized plasma are shown to possess nonlinear two‐dimensional solitary wave solutions. In the presence of magnetic shear, radiative shear damping is exponentially small in Ls/Ln for solitary drift waves, in contrast to linear waves.
TL;DR: In this paper, the variable cross-section hard sphere (VHS) model was used to define a mean free path in a real gas, which properly accounts for the temperature exponent of the coefficient of viscosity of the gas.
Abstract: Attention is drawn to the inconsistency in the conventional procedure for the definition of a mean free path in a real gas through the classical hard sphere result. It is shown that the variable cross‐section hard sphere, or VHS, model can be used to define a mean free path that properly accounts for the temperature exponent of the coefficient of viscosity of the gas. In addition, the VHS model is shown to have advantages over the classical inverse power law models for numerical and analytical studies.
TL;DR: The free energy associated with the anisotropy in the velocity space of a microwave-heated hot electron distribution can drive the mirror mode unstable as mentioned in this paper, and the real frequency of this instability is of the same order as the diamagnetic drift frequency of the hot electrons.
Abstract: The free energy associated with the anisotropy in the velocity space of a microwave‐heated hot electron distribution can drive the mirror mode unstable. The real frequency of this instability is of the same order as the diamagnetic drift frequency of the hot electrons.
TL;DR: In this paper, the effect of the electromagnetic terms and the stability boundaries in both low and high-beta plasmas were examined. And the authors derived an approximate dispersion relation showing the relation of this mode to the whistler and compared the solutions of it with those obtained from the exact dispersion relations.
Abstract: In a high‐beta plasma the so‐called modified‐two‐stream instability, which results from strongly magnetized electrons drifting relative to unmagnetized ions across a homogeneous magnetic field, is misnamed because the mode is highly kinetic, particularly when the relative streaming velocity exceeds the Alfven speed of the plasma. This kinetic cross‐field streaming instability is investigated in detail, examining the effect of the electromagnetic terms and the stability boundaries in both low‐ and high‐beta plasmas. An approximate dispersion relation showing the relation of this mode to the whistler is derived and solutions of it are compared with those obtained from the exact dispersion relation. The kinetic mode, unlike the usual modified‐two‐stream instability, is not stabilized by electromagnetic effects when the relative electron–ion drift speed exceeds the Alfven speed.
TL;DR: In this paper, a method which simplifies the investigation of the form of an obliquely incident wave on a plane parallel layered plasma by reducing it to the case where the wave is normally incident is presented.
Abstract: A method which simplifies the investigation of the form of an obliquely incident wave on a plane parallel layered plasma by reducing it to the case where the wave is normally incident is presented. The resonance at N0=Nc (critical density) is discussed. Another resonance at N0=4Nc is described.
TL;DR: An analytical and numerical study of the stability of tearing modes was carried out using the Braginskii fluid equations in this paper, where an electron temperature gradient coupled with finite (nonzero) parallel thermal conductivity causes large parallel currents to flow in the vicinity of the singular layer (where k⋅B=0).
Abstract: An analytical and numerical study of the stability of tearing modes is carried out using the Braginskii fluid equations An electron temperature gradient coupled with finite (nonzero) parallel thermal conductivity causes large parallel currents to flow in the vicinity of the singular layer (where k⋅B=0) The pressure‐driven currents are stabilizing and in the limit βL2s/L2n>1, where β is the ratio of the thermal to magnetic pressure and Ls and Ln are the magnetic shear and density scale lengths, the linear tearing mode no longer exists In this high‐β limit, the magnetic perturbation of the tearing mode is completely shielded from the singular layer so that no reconnection of the magnetic field can take place The relationship between the tearing mode and previously investigated temperature‐gradient‐driven modes and the implications of the results for resistive modes in present and future tokamak discharges is discussed
TL;DR: McLean and Saffman's model for the fingering in a Hele-Shaw cell is solved numerically as mentioned in this paper, which suggests that a countably infinite number of solutions exist for nonzero surface tension.
Abstract: McLean and Saffman’s model for the fingering in a Hele–Shaw cell is solved numerically. The results suggests that a countably infinite number of solutions exist for nonzero surface tension. This set of solutions contains the solution previously obtained by McLean and Saffman.
TL;DR: In this paper, the small signal theory of a harmonic gyrotron is presented for the case of axis-encircling electron orbits and the beam current required for oscillation is found to be highly dependent on the electron energy.
Abstract: The small‐signal theory of a harmonic gyrotron is presented for the case of axis‐encircling electron orbits. The beam current required for oscillation is found to be highly dependent on the electron energy. Gyrotron cavities operating on this principle at very high harmonic numbers (n≂10) and high frequency in weak magnetic fields are well matched to low‐current, moderate‐energy, rf‐accelerated electron beams (≂50 mA, ≂250 keV), resulting in compact submillimeter wave systems.
TL;DR: In this paper, the authors studied the behavior of unexcited and excited elliptic jets and found that their characteristics are noticeably different from circular jets, suggesting potential applications of excited elliptical jets for enhanced mixing and chemical reaction, and control of aerodynamic noise.
Abstract: Studies of unexcited and excited elliptic jets reveal their characteristics to be noticeably different from circular jets, suggesting applications of excited elliptic jets for enhanced mixing and chemical reaction, and control of aerodynamic noise. The near‐field turbulence characteristics, jet spread, and locations of switching of major and minor axes of the jet cross section can be drastically altered by forcing. The preferred mode and the stable pairing mode of an elliptic jet scale with the exit equivalent diameter.
TL;DR: In this article, a computational and analytical study of the nonlinear evolution of single-helicity, resistive, current-driven modes in the reversed-field pinch utilizing the single-fluid magnetohydrodynamic approximation is presented.
Abstract: A computational and analytical study of the nonlinear evolution of single‐helicity, resistive, current‐driven modes in the reversed‐field pinch utilizing the single‐fluid magnetohydrodynamic approximation is presented. In the start‐up phase it is found that an off‐axis current peak can be anomalously dissipated by globally reconnecting current‐driven modes. The reconnection is seen to be well described by the Kadomtsev reconnection model, which is directly applied to the reversed‐field‐pinch regime. During the sustainment phase of the discharge, stable profiles are driven unstable by transport processes and restabilized by the resulting current‐driven instabilities. The dynamics of the toroidal field is of fundamental importance in the evolution of these modes. The reversed‐field‐pinch profile is restabilized by two successive global reconnections: the first removes the rational surface; the second restores it. It is also found that magnetic reconnection can occur in the reversed‐field pinch without a linear phase. Although the interaction of instabilities with different helicities is not directly investigated, the importance of this phenomenon in the reversed‐field pinch is discussed.
TL;DR: In this paper, a comparison of experimental data and theoretical results is presented that may help to explain the underlying causes of the flux regeneration mechanism in the reversed field pinch (RFP), which is seen on many diagnostic measurements which are coincident with observable increases in toroidal magnetic flux in discharges where the pinch parameter θ (θ≡Bθwall/〈Bφ〉) exceeds the value 1.6.
Abstract: A comparison of experimental data and theoretical results is presented that may help to explain the underlying causes of the flux regeneration mechanism (‘‘dynamo’’) in the reversed‐field pinch (RFP). Discrete events are seen on many diagnostic measurements which are coincident with observable increases in toroidal magnetic flux in discharges where the pinch parameter θ (θ≡Bθwall/〈Bφ〉) exceeds the value 1.6. By observing the relative timing and the θ dependence of these events as the discharge is returned to lower values of θ, a case may be made for associating the m=1 tearing mode with the RFP dynamo.
TL;DR: A survey of the main areas of research to which Kovasznay contributed can be found in this paper, including the use of mass-weighted averages, decrease of turbulence in expansion, increase in compression, strong Reynolds analogy extension, and temperature-velocity correlation conservation.
Abstract: Following a scientific cooperation beginning in 1954 with L. S. G. Kovasznay, and in his memory, in this paper is presented a survey of two of the main areas of research to which he contributed: (1) turbulence properties deduced through statistical conditional sampling and space‐time measurements of correlations and contingencies; these include celerities, optimum correlations, memories, and coherent structures; and (2) the study of turbulence in supersonic flows initiated by Kovasznay; this include the use of mass‐weighted averages, decrease of turbulence in expansion, increase in compression, strong Reynolds analogy extension, and temperature–velocity correlation conservation.