Rotational Mach Number Research Articles

Closed-Loop Control of Dynamic Oblique Shock with Arc Plasma

The fluctuations caused by the dynamic shock wave and its interaction with the boundary layer bring formidable challenges for aerodynamic design and stable control of supersonic aircraft. The mechanism of closed-loop control for dynamic oblique shock stabilizing with arc plasma is numerically studied. Two types of dynamic oblique shocks are formed in supersonic airflow by a compression ramp rotation and incoming Mach number variation, respectively. A closed-loop controller is established to connect the arc plasma deposited power with the pressure ratio across the dynamic shock by using the proportional-integral method. It realizes the shock pressure ratio stabilized in the desired target value with a local overshoot appearing at the initial excitation and a more stable flow downstream of the oblique shock during excitation state for both types of dynamic shock control. The effects of arc plasma length and ramp rotating speed on the closed-loop control for the dynamic shocks induced by ramp rotation are further discussed. Results show that the arc plasma with longer length is more favorable to the dynamic oblique shock control with lower energy consumption and total pressure loss. The pressure ratio fluctuation is increased at higher ramp rotation speed, but its standard deviation is still limited to 0.3% of the desired target value during excitation state.

Nonlinear excitation of geodesic acoustic mode by drift waves in anisotropic tokamak plasmas with toroidal rotation

Nonlinear excitation of geodesic acoustic mode (GAM) by ions drift waves is investigated in anisotropic toroidally rotating tokamak plasmas by using an ideal magnetohydrodynamic model and standard double adiabatic equations. It is found that the radial wave number kx of the effectively excited GAM and the GAM frequency increase with the ratio of the perpendicular pressure to the parallel pressure χ as well as the toroidal rotation Mach number. The equilibrium anisotropy and the ratio of the radial scale length of the equilibrium density to the tokamak major radius are shown to play a significant role in the growth rate and frequency shift of GAM. The growth rate increases with the normalized GAM frequency first and then decreases to zero as the GAM frequency increases. When the GAM frequency increases again, the growth rate continues to increase but with a relatively smaller magnitude.

Sonic Eddy Model of the Turbulent Boundary Layer

The effects of Mach number on the skin friction and velocity fluctuations of the turbulent boundary layer are considered through a sonic eddy model. Originally proposed for free shear flows, the model assumes that the eddies responsible for momentum transfer have a rotation Mach number of unity, with the entrainment rate limited by acoustic signaling. Under this assumption, the model predicts that the skin friction coefficient should go as the inverse Mach number in a regime where the Mach number is larger than unity but smaller than the square root of the Reynolds number. The velocity fluctuations normalized by the friction velocity should be the inverse square root of the Mach number in the same regime. Turbulent transport is controlled by acoustic signaling. The density field adjusts itself such that the Reynolds stresses correspond to the momentum transport. In contrast, the conventional van Driest–Morkovin view is that the Mach number effects are due to density variations directly. A new experiment or simulation is proposed to test this model using different gases in an incompressible boundary layer, following the example of Brown and Roshko in the free shear layer.

Acoustic modes of rapidly rotating ellipsoids subject to centrifugal gravity.

The acoustic modes of a rotating fluid-filled cavity can be used to determine the effective rotation rate of a fluid (since the resonant frequencies are modified by the flows). To be accurate, this method requires a prior knowledge of the acoustic modes in rotating fluids. Contrary to the Coriolis force, centrifugal gravity has received much less attention in the experimental context. Motivated by on-going experiments in rotating ellipsoids, we study how global rotation and buoyancy modify the acoustic modes of fluid-filled ellipsoids in isothermal (or isentropic) hydrostatic equilibrium. We go beyond the standard acoustic equation, which neglects solid-body rotation and gravity, by deriving an exact wave equation for the acoustic velocity. We then solve the wave problem using a polynomial spectral method in ellipsoids, which is compared with finite-element solutions of the primitive fluid-dynamic equations. We show that the centrifugal acceleration has measurable effects on the acoustic frequencies when MΩ≳0.3, where MΩ is the rotational Mach number defined as the ratio of the sonic and rotational time scales. Such a regime can be reached with experiments rotating at a few tens of Hz by replacing air with a highly compressible gas (e.g., SF6 or C4F8).

Towards the definition of a benchmark for low Reynolds number propeller aeroacoustics

Experimental and numerical results of a propeller of 0.3 m diameter operated in quiescent standard ambient conditions at 5000 RPM and axial velocity ranging from 0 to 20 m/s and advance ratio ranging from 0 to 0.8 are presented as a preliminary step towards the definition of a benchmark configuration for low Reynolds number propeller aeroacoustics. The corresponding rotational tip Mach number is 0.231 and the Reynolds number based on the blade sectional chord and flow velocity in the whole radial and operational domain ranges from about 54000 to 106000. Force and noise measurements carried out in a low-speed semi-anechoic wind-tunnel are compared with scale-resolved CFD and low-fidelity numerical results. Results identify the experimental and numerical challenges of the benchmark and the relevance of fundamental research questions related to transition and other low Reynolds number effects.

Definition of a benchmark for low Reynolds number propeller aeroacoustics

Experimental and numerical results of a propeller of 0.3 m diameter operated at 5000 RPM and axial velocity ranging from 0 to 20 m/s and advance ratio ranging from 0 to 0.8 are presented as a preliminary step towards the definition of a benchmark configuration for low Reynolds number propeller aeroacoustics. The corresponding rotational tip Mach number is 0.23 and the Reynolds number based on the blade sectional chord and flow velocity varies from about 46000 to 106000 in the operational domain and in the 30% to 100% blade radial range. Force and noise measurements carried out in a low-speed semi-anechoic wind-tunnel are compared to scale-resolved CFD and low-fidelity numerical predictions. Results identify the experimental and numerical challenges of the benchmark and the relevance of fundamental research questions related to transition and other low Reynolds number effects.

Numerical study of equilibrium solutions for axisymmetric plasmas with toroidal flow obtained using Solovev approach

Solovev approach of finding equilibrium solutions, which was extended to include the vacuum solutions provided by Zheng, Wooton, and Solano, was found extremely useful for the purpose of shaping studies. Its extension to toroidal equilibria with a general plasma flow was examined theoretically in a companion paper by Chu, Hu and Guo. The only meaningful extension was found for plasmas with a pure toroidal rotation and with a constant Mach number. A set of functions {SOLOVEV_ZWSm} was obtained which fixed location of the magnetic axis for equilibria with quasi-constant current density profile, with toroidal flow at constant Mach number and with specific heat capacity 1. The set {Solovev_ZWSm} should have complete shaping capability for plasma shapes with positive curvature at the boundary; but not for plasmas with negative curvature boundary points, i.e. the doublets or bean shaped tokamaks. We report here extensive numerical studies showing the shaping capability of {Solovev_ZWSm} for plasmas with pure toroidal rotations, including the change in topology of the solution when the rotation mach number changes. Included plasma topology are the sphere (spheromaks); and the tokamaks (including the doublets).

Scaling Sealing Effectiveness in a Stator–Rotor Cavity for Differing Blade Spans

As engine development continues to advance toward increased efficiency and reduced fuel consumption, efficient use of compressor bypass cooling flow becomes increasingly important. In particular, optimal use of compressor bypass flow yields an overall reduction of harmful emissions. Cooling flows used for cavity sealing between stages are critical to the engine and must be maintained to prevent damaging ingestion from the hot gas path. To assess cavity seals, the present study utilizes a one-stage turbine with true-scale engine hardware operated at engine-representative rotational Reynolds number and Mach number. Past experiments have made use of part-span (PS) rather than full-span (FS) blades to reduce flow rate requirements for the test rig; however, such decisions raise questions about potential influences of the blade span on sealing effectiveness measurements in the rim cavity. For this study, a tracer gas facilitates sealing effectiveness measurements in the rim cavity to compare data collected with FS engine airfoils and simplified, PS airfoils. The results from this study show sealing effectiveness does not scale as a function of relative purge flow with respect to main gas path flow rate when airfoil span is changed. However, scaling the sealing effectiveness for differing spans can be achieved if the fully purged flow rate is known. Results also suggest reductions of purge flow may have a relatively small loss of seal performance if the design is already near a fully purged condition. Rotor tip clearance is shown to have no effect on measured sealing effectiveness.

Direct numerical simulation of flow past a transversely rotating sphere up to a Reynolds number of 300 in compressible flow

In this study, direct numerical simulation of the flow around a rotating sphere at high Mach and low Reynolds numbers is conducted to investigate the effects of rotation rate and Mach number upon aerodynamic force coefficients and wake structures. The simulation is carried out by solving the three-dimensional compressible Navier–Stokes equations. A free-stream Reynolds number (based on the free-stream velocity, density and viscosity coefficient and the diameter of the sphere) is set to be between 100 and 300, the free-stream Mach number is set to be between 0.2 and 2.0, and the dimensionless rotation rate defined by the ratio of the free-stream and surface velocities above the equator is set between 0.0 and 1.0. Thus, we have clarified the following points: (1) as free-stream Mach number increased, the increment of the lift coefficient due to rotation was reduced; (2) under subsonic conditions, the drag coefficient increased with increase of the rotation rate, whereas under supersonic conditions, the increment of the drag coefficient was reduced with increasing Mach number; and (3) the mode of the wake structure becomes low-Reynolds-number-like as the Mach number is increased.

Validity of sound-proof approaches in rapidly-rotating compressible convection: marginal stability versus turbulence

The validity of the anelastic approximation has recently been questioned in the regime of rapidly-rotating compressible convection in low Prandtl number fluids (Calkins, Julien and Marti, Proc. R. Soc. A, 2015, vol. 471, 20140689). Given the broad usage and the high computational efficiency of sound-proof approaches in this astrophysically relevant regime, this paper clarifies the conditions for a safe application. The potential of the alternative pseudo-incompressible approximation is investigated, which in contrast to the anelastic approximation is shown to never break down for predicting the point of marginal stability. Its accuracy, however, decreases close to the parameters corresponding to the failure of the anelastic approach, which is shown to occur when the sound-crossing time of the domain exceeds a rotation time scale, i.e. for rotational Mach numbers greater than one. Concerning the supercritical case, which is naturally characterised by smaller rotational Mach numbers, we find that the anelastic approximation does not show unphysical behaviour. Growth rates computed with the linearised anelastic equations converge toward the corresponding fully compressible values as the Rayleigh number increases. Likewise, our fully nonlinear turbulent simulations, produced with our fully compressible and anelastic models and carried out in a highly supercritical, rotating, compressible, low Prandtl number regime show good agreement. However, this nonlinear test example is for only a moderately low convective Rossby number of 0.14.

Rayleigh-Taylor instability in non-uniform magnetized rotating strongly coupled viscoelastic fluid

The Rayleigh-Taylor instability (RTI) in an incompressible strongly coupled viscoelastic fluid is investigated considering the effects of inhomogeneous magnetic field, density gradient, and uniform rotation. The generalized hydrodynamic equations have been formulated, and linear dispersion relation is derived taking appropriate density and magnetic field profiles for the considered system. The gravity induced stable and unstable configurations of RTI are analyzed in hydrodynamic and kinetic limits. In the kinetic limit, shear wave modified dispersion relation and the condition of RTI are derived in terms of magnetic-viscoelastic Mach number and viscoelastic Froude number. The criteria of RTI and critical wavenumber for the growth of RTI to be unstable are estimated numerically for white dwarf and inertial confinement fusion target. It is observed that magnetic field, rotation, and viscoelastic effects play a significant role in the suppression of RTI in these systems. The stabilizing influence of magnetic field, rotation, and magnetic-viscoelastic Mach number while the destabilizing influence of viscoelastic Froude on the growth rate of RTI number is observed graphically. The growth rate of RTI decreases faster in kinetic limit as compared to the hydrodynamic limit.

Analytical Acoustic Power Spectrum Formulations for Rotating Monopole and Dipole Point Sources

Analytical acoustic power spectrum formulations for the rotating monopole and dipole point sources are proposed by employing the spherical harmonic series expansion method. Both the analytical acoustic power spectra and the overall acoustic power show a good agreement with the results obtained from other methods. A nondimensional acoustic power ratio (APR) is employed to investigate the effects of the rotational Mach number, the direction of the dipole source, and the number of sources on the acoustic power output.

Simulation of Broadband Noise Sources of an Axial Fan under Rotating Stall Conditions

Study on the influence of rotating stall on the aerodynamic noise of axial fan has important value to warn of the occurrence of stall through monitoring the noise variations. The present work is to analyze the aerodynamic noise before and after the phenomenon of rotating stall by solving Navier-Stokes equations, coupled with the throttle condition and the broadband noise sources model. The impeller exit rotational Mach number and rotational Reynolds number are separately 0.407 and 8.332 × 106. The results show that the aerodynamic noise source of the fan is mainly the rotation noise under the design condition. The vortex noise accounts for the major part of fan noise after the occurrence of stall, and the maximum acoustic power level of the fan appears in the rotor domains. In the evolution process from the stall inception to the stall cell, the high noise regions of the rotor develop along the radial, circumferential, and axial directions, and the area occupied by high noise regions increases from 33% to 46% impeller channels area. On rotating stall condition, the high noise regions occupying about 46% impeller channels area propagate with the stall cell along the circumferential direction at a half of rotor speed.

Numerical simulations of Z-pinch experiments to create supersonic differentially-rotating plasma flows

The aim of this work is to produce and study a high energy density laboratory plasma relevant to astrophysical accretion disks. To this end, an experimental setup based on a modified cylindrical wire array was devised, which employs a cusp magnetic field to introduce angular momentum into the system. The setup was studied numerically with the three-dimensional, resistive magneto-hydrodynamic code GORGON. Simulations show that a differentially-rotating flow is formed, with typical rotation velocity and Mach number values of 60 km/s and Mφ ∼ 5 respectively. The plasma is radiatively cooled and presents a Reynolds number higher than 107. In addition, the magnetic Reynolds number and the plasma β are >1. Such a plasma is of interest for the study of hydrodynamic and magneto-hydrodynamic instabilities, and turbulence generation in differentially-rotating plasma flows.

Plasma rotation in a quasi-symmetric stellarator

The equilibrium plasma rotation in a general toroidal magnetic field is nearly always subsonic and is determined by the requirement that the collisional particle transport should be ambipolar in lowest order in the small-ion-gyroradius expansion. Only in quasi-symmetric fields, where collisional particle transport is intrinsically ambipolar, can the plasma rotate freely and then only in the quasi-symmetry direction. Sonic rotation velocities are allowed in this case. In this paper the effect of rotation in a quasi-axisymmetric field is investigated, and it is found that the symmetry is broken when the rotation speed exceeds the diamagnetic speed appreciably, leading to reappearance of the non-intrinsically ambipolar 1/ν-transport regime. Fortunately, this transport scales with the fourth power of the rotation Mach number and is expected to be modest in most plasmas of interest.

Instability of coupled gravity-inertial-Rossby waves on a β-plane in solar system atmospheres

Abstract. This paper provides an analysis of the combined theory of gravity-inertial-Rossby waves on a β-plane in the Boussinesq approximation. The wave equation for the system is fifth order in space and time and demonstrates how gravity-inertial waves on the one hand are coupled to Rossby waves on the other through the combined effects of β, the stratification characterized by the Väisälä-Brunt frequency N, the Coriolis frequency f at a given latitude, and vertical propagation which permits buoyancy modes to interact with westward propagating Rossby waves. The corresponding dispersion equation shows that the frequency of a westward propagating gravity-inertial wave is reduced by the coupling, whereas the frequency of a Rossby wave is increased. If the coupling is sufficiently strong these two modes coalesce giving rise to an instability. The instability condition translates into a curve of critical latitude Θc versus effective equatorial rotational Mach number M, with the region below this curve exhibiting instability. "Supersonic" fast rotators are unstable in a narrow band of latitudes around the equator. For example Θc~12° for Jupiter. On the other hand slow "subsonic" rotators (e.g. Mercury, Venus and the Sun's Corona) are unstable at all latitudes except very close to the poles where the β effect vanishes. "Transonic" rotators, such as the Earth and Mars, exhibit instability within latitudes of 34° and 39°, respectively, around the Equator. Similar results pertain to Oceans. In the case of an Earth's Ocean of depth 4km say, purely westward propagating waves are unstable up to 26° about the Equator. The nonlinear evolution of this instability which feeds off rotational energy and gravitational buoyancy may play an important role in atmospheric dynamics.

Non-uniform mass transfer or wall enthalpy into a compressible flow over a rotating sphere

A study for the effect of non-uniform slot injection/suction and non-uniform total enthalpy at the wall for a steady non-similar laminar compressible boundary layer flow over a rotating sphere is presented. The effects of rotation and free stream Mach number have also been considered. Non-similar solutions have been obtained from the starting point of the stream wise coordinate to the point of separation using an implicit finite difference scheme in combination with the quasi-linearization technique. The results indicate that the separation can be delayed by non-uniform slot suction and also by moving the slot downstream but the non-uniform slot injection has just the opposite effect. An increase of rotation and total enthalpy at the wall causes separation to occur earlier while cooling delays it. An increase of the Mach number shifts the point of separation upstream due to the adverse pressure gradient. The non-uniform total enthalpy at the wall has a very little effect on the skin friction and hence on the point of separation.

Axisymmetric stability criteria for a composite system of stellar and magnetized gaseous singular isothermal discs

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Lou & Zou recently constructed exact global stationary configurations for both axisymmetric and nonaxisymmetric coplanar MHD perturbations in such a composite MSID system and proposed the MHD $D_s-$criteria for axisymmetric stability by the hydrodynamic analogy. In a different perspective, we derive and analyze here the time-dependent WKBJ dispersion relation in the low-frequency and tight-winding regime to examine axisymmetric stability properties. By introducing a rotational Mach number $D_s$ for the ratio of the stellar rotation speed $V_s$ to the stellar velocity dispersion $a_s$, one readily determines the stable range of $D_s^2$ numerically to establish the $D_s-$criteria for axisymmetric MSID stability. Those MSID systems rotating either too fast (ring ragmentation) or too slow (Jeans collapse) are unstable. Our WKBJ results of (M)SID instability provide physically compelling explanations for the stationary analysis of Lou & Zou. We further introduce an effective MHD $Q$ parameter for a composite MSID system and compare with the earlier work of Elmegreen, Jog and Shen & Lou. As expected, an axisymmetric dark matter halo enhances the stability against axisymmetric disturbances in a composite partial MSID system.

Gravitationally coupled scale-free discs

In a composite fluid system of two gravitationally coupled barotropic scale-free discs bearing a rotation curve v ∝ r −β and a power-law surface mass density � 0 ∝ r −α with α = 1 + 2β ,w econstruct classes of coplanar stationary aligned and logarithmic spiral perturbation configurations in the two discs. Owing to the mutual gravitational interaction, there are two independent classes of perturbation modes, with surface mass density disturbances in the two coupled discs being either in phase or out of phase. We derive analytical criteria for such perturbation modes to exist and show numerical examples. We compute the aligned and spiral perturbation modes systematically to explore the entire parameter regime. For the axisymmetric m = 0 case with radial oscillations, there are two unstable regimes of ring fragmentation and Jeans collapse corresponding to short and long radial wavelengths, respectively. Only within a certain range of the rotation parameter D 2 (square of the effective rotational Mach number for the stellar disc) can a composite disc system be stable against all axisymmetric perturbations. Compared with a single-disc system, the coupled two-disc system becomes less stable against such axisymmetric instabilities. Our investigation generalizes the previous work of Syer & Tremaine on the single-disc case and of Lou & Shen on two coupled singular isothermal discs. Non-axisymmetric instabilities are briefly discussed. These stationary models for various largescale patterns and morphologies may be useful in the context of disc galaxies with bars and spirals or barred spirals. Ke yw ords: wave s‐ stars: formation ‐ ISM: general ‐ galaxies: kinematics and dynamics ‐ galaxies: spiral ‐ galaxies: structure.

Establishment of strong velocity shear and plasma density profile modification with associated low frequency fluctuations

Strong shear of azimuthal plasma rotation velocity and a large density profile modification from a hollow to peaked profiles were successfully obtained in a cylindrical magnetized plasma, using voltage biased electrodes. The shear region and density profiles could be finely controlled by changing the voltage biased position as well as by the magnetic field configurations. A low frequency (<4 kHz) density oscillation was identified as a drift wave type: propagation in the electron diamagnetic direction with a rigid rotation (Mach number M∼−1 at the edge), opposite to the direction of the edge plasma rotation with M∼1.