Transonic Point Research Articles

Aerodynamic performance optimization of a UAV's airfoil at low-Reynolds number and transonic flow under the Martian carbon dioxide atmosphere

Unmanned aerial vehicles (UAV) on Mars operate in a thin atmosphere of carbon dioxide, where the flow Reynolds number and local sound velocity are significantly reduced compared to the Earth's atmosphere because of low gas density and temperature. Therefore, a Mars UAV with a rotary-wing design can operate in a state of low-Reynolds number (Re = 1-5 × 104) and transonic flow (Matip = 0.7–1.2), where flow separation and shock waves occur simultaneously. The interaction between flow separation and shock waves not only makes the entire flow field more complex but also seriously affects the aerodynamic performance of the airfoil, thereby reducing the payload of UAVs, which previous aircraft designs based on the thick atmospheric environment of the Earth have not considered. Therefore, in this study, a numerical simulation method is used to conduct airfoil optimization research on a characteristic airfoil under low-Reynolds number transonic flow (Re = 33200, Ma = 0.85) conditions to acquire additional insight, improve aerodynamic performance of the airfoil, and enhance the payload. The results were compared with the optimization results of subsonic flow (Re = 19500, Ma = 0.50). The Hicks-Henne bump functions method was selected for airfoil parameterization, and non-dominated sorting genetic algorithms (NSGA)-Ⅱ were used for optimization. Through an analysis of the pressure coefficient (Cp), surface friction coefficient (Cf), and density gradient, the number and intensity of oblique shock waves on the upper and lower surfaces were observed to significantly affect the aerodynamic performance of airfoils under low-Reynolds number flow. The changes in flow separation and transition positions do not directly affect the aerodynamic performance but weaken the oblique shock wave intensity induced by the separation point or directly transform strong shock waves into weaker compression waves because of the absence of flow separation. The optimization results for both subsonic and transonic flow point to a thinner overall shape with a flatter surface with the drag greatly reduced due to the disappearance of shock waves on the lower surface, which significantly improves aerodynamic performance. Compared with the original airfoil, at Ma = 0.85, this design can increase the lift coefficient (Cl) by at least 124 % and lift drag ratio (Cl/Cd) by 177 %.

Self-similar Solution of Rotating Eruptive Outflows on Its Equatorial Plane

We construct axisymmetric self-similar solutions of transonic outflows emanating from a point source including the effect of the rotation. The solutions are constructed exclusively on the equatorial plane. The features of solutions are determined by three parameters: the adiabatic index γ, the dimensionless coordinate of the transonic point, and the dimensionless azimuthal velocity at the transonic point. We classify the solutions into five groups according to asymptotic behaviors. We find that the behaviors of the self-similar solutions change at γ = 11/9. In addition, some solutions show double-power-law density profiles, which are usually seen in ejecta from a binary merger or nova-like explosion. Thus, our self-similar solutions can be applied not only to the outflow blowing from the central spinning objects but also to the ejecta erupted from the binary merger or nova-like explosion.

Time-dependent evolution of magnetic accretion flow with radial viscosity

In this study, we have considered time dependent evolution of advection dominated accretion flow (ADAF) in the presence of the toroidal magnetic field and radial viscosity. We have used time-dependent self-similar solutions for solving the 1D MHD equations in the spherical coordinates in the equatorial plane () and we have neglected terms with any θ and φ dependence. While the azimuthal viscosity υ as the turbulence factor in transporting the angular momentum and α-prescription for kinematic coefficient of viscosity is used in the most previous studies, recent studies show the disc structure can also be affected by the radial viscosity υr. In this work, we have assumed that the ratio is a dimensionless parameter 𝜉. We use ξ and β variables as free parameters to consider the effects of magnetic field and radial viscosity. The solutions indicate a transonic point in the accretion flow. This point approaches to outward by increasing the magnetic field and radial viscosity. Also, by adding strength of the magnetic field, the radial-velocity of the disc decreases and the disc compresses. Also, the flow is sub-Keplerian at all radii. The 𝜉 parameter has the same behavior in the inner and intermediate regions of the flow but in the outward of the flow, by adding the 𝜉 parameter, accretion rate increases and hence, it is expected that the disc has a shorter lifetime with radial viscosity.

Stationary Accretion Flow with Nuclear Burning

We present a series of numerical solutions of spherically symmetric stationary flows with nuclear burning accreted by a neutron star (or black hole). We consider the accretion of matter composed of carbon and oxygen, which mimics the flow after a neutron star is engulfed by a CO star or the CO core of a massive star. It is found that there are two types of transonic solutions depending on the accretion rate. The flow with a low accretion rate reaches the center (or the surface of the central object) at supersonic speeds. The other type with a high accretion rate has another sonic point inside the transonic point and the flow truncates at the sonic point. The critical accretion rate dividing these two types is derived as a function of the mass of the central object and the specific enthalpy in the ambient matter. We discuss implications from the solutions for a new mechanism of super-Chandrasekhar Type Ia supernovae and Type Icn supernovae.

Adiabatic accretion onto black holes in Einstein-Maxwell-scalar theory

We study the adiabatic accretion process of ordinary baryonic gas onto spherically symmetric black holes in Einstein-Maxwell-scalar theory, with two parameters α and β in the coupling term. Especially, we demonstrate the range of the transonic points in terms of the charge-to-mass ratio squared and the dimensionless coordinate radius, in two important classes of black holes as examples. Further, we find that the two coupling parameters give modifications to the mass accretion rate at different orders of the sound speed at infinity. We also present their different effects on the temperature ratios of the accreted gas.

Surrogate-based aerodynamic shape optimization of hypersonic flows considering transonic performance

Aerodynamic shape optimization of a hypersonic transport aircraft over a wide Mach-number range is challenging. The difficulty is not only associated with the large computational cost of high-fidelity CFD (computational fluid dynamics) simulations but also linked to the reasonable compromise between aerodynamic performances of aircraft at different speed ranges. This article proposes to use efficient global optimization based on surrogate models to address this type of problems. A RANS (Reynolds averaged Navier-Stokes) flow solver is adopted to evaluate objective and constraint functions; Kriging surrogate model combined with a parallel infill-sampling method and a multi-round strategy are employed to find the global optimum. First, a profile optimization with baseline airfoil of NACA64A-204 is conducted to achieve high lift-to-drag ratio (L/D) at both hypersonic (Ma=6.0) and transonic (Ma=0.8) regimes with up to 18 design variables, and significant improvement has been observed. Then, multi-objective wing optimizations of maximizing both hypersonic and transonic L/Ds with up to 54 design variables are performed and multiple optimizations with various sets of weight coefficients are investigated. The optimized wings are further evaluated and compared with the baseline wing at the flow regimes from subsonic to hypersonic speeds. Results show that by using the optimized profile, hypersonic and transonic L/Ds of a typical wing configuration are increased by 13.85% and 7.32%, respectively. After 3-D optimizations, the L/Ds at hypersonic and transonic design points are further improved by 4.47% and 3.19%, respectively, and a better performance over a wide Mach-number range is obtained. It is shown that a multi-round surrogated-based optimization is feasible and effective for aerodynamic shape optimization of hypersonic transport aircrafts over a wide Mach-number range.

Relativistic sonic geometry for isothermal accretion in the Kerr metric

We linearly perturb advective isothermal transonic accretion onto rotating astrophysical black holes to study the emergence of the relativistic acoustic spacetime and to investigate how the salient features of this spacetime is influenced by the spin angular momentum of the black hole. We have perturbed three different quantities—the velocity potential, the mass accretion rate and the relativistic Bernoulli’s constant to show that the acoustic metric obtained for these three cases are the same up to a conformal factor. By constructing the required causal structures, it has been demonstrated that the acoustic black holes are formed at the transonic points of the flow and the acoustic white holes are formed at the shock location. The corresponding acoustic surface gravity has been computed in terms of the relevant accretion variables and the background metric elements. We have performed a linear stability analysis of the background stationary flow.

Polytropic transonic galactic outflows in a dark matter halo with a central black hole

Polytropic transonic solutions of spherically symmetric and steady galactic winds in the gravitational potential of a dark matter halo (DMH) with a supermassive black hole (SMBH) are studied. The solutions are classified in terms of their topological features, and the gravitational potential of the SMBH adds a new branch to the transonic solutions generated by the gravity of the DMH. The topological types of the transonic solutions depend on the mass distribution, the amount of supplied energy, the polytropic index $\gamma$, and the slope $\alpha$ of the DMH mass distribution. When $\alpha$ becomes larger than a critical value $\alpha_\mathrm{c}$, the transonic solution types change dramatically. Further, our model predicts that it is possible for a slowly accelerating outflow to exist, even in quiescent galaxies with small $\gamma$. This slowly accelerating outflow differs from those considered in many of the previous studies focusing on supersonic outflows in active star-forming galaxies. In addition, our model indicates that outflows in active star-forming galaxies have only one transonic point in the inner region ($\sim$ 0.01 kpc). The locus of this transonic point does not strongly depend on $\gamma$. We apply the polytropic model incorporating mass flux supplied by stellar components to the Sombrero Galaxy, and conclude that it can reproduce the observed gas density and the temperature distribution well. This result differs significantly from the isothermal model, which requires an unrealistically large mass flux (Igarashi et al. 2014). Thus, we conclude that the polytropic model is more realistic than the isothermal model, and that the Sombrero Galaxy can have a slowly accelerating outflow.

A new concept of transonic galactic outflows and its application to the Sombrero galaxy

AbstractWe study fundamental properties of transonic galactic outflows in the gravitational potential of a cold dark matter halo (DMH) with a central super-massive black hole (SMBH) assuming a polytropic, steady and spherically symmetric state. We have classified the transonic solutions with respect to their topology in the phase space. As a result, we have found two types of transonic solutions characterized by a magnitude relationship between the gravity of DMH and that of SMBH. These two types of solutions have different loci of the transonic points; one transonic point is formed at a central region (< 0.01kpc) and another is at a distant region (> 100kpc). Also, mass fluxes and outflow velocities are different between the two solutions. These two transonic solutions may play different roles on the star formation history of galaxies and the metal contamination of intergalactic space. Furthermore, we have applied our model to the Sombrero galaxy. In this galaxy, the wide-spread hot gas is detected as an apparent trace of galactic outflows while the star-formation rate is disproportionately low, and the observed gas density distribution is quite similar to the hydrostatic state (Li et al. 2011). To solve this discrepancy, we propose a slowly accelerating outflow in which the transonic point forms in a distant region (~ 120 kpc) and the subsonic region spreads across the stellar distribution. In the subsonic region, the gas density distribution is similar to that of the hydrostatic state. Our model predicts the possibility of the slowly accelerating outflow in the Sombrero galaxy. Igarashi et al. 2014 used the isothermal model and well reproduced the observed gas density distribution, but the estimated mass flux (1.8M⊙/yr) is lager than the mass of the gas supplied by stars (0.3-0.4M⊙/yr). Then, we expect that the polytropic model may reproduce the observational mass of the supplied gas (Igarashi et al. 2015). Such slowly accelerating outflows should be distinguished from the conventional supersonic outflows frequently argued in star-forming galaxies.

Transonic galactic outflows in a dark matter halo with a central black hole

AbstractWe study fundamental properties of transonic galactic outflows in the gravitational potential of a cold dark matter halo (DMH) with a central super-massive black hole (SMBH) assuming an isothermal, steady and spherically symmetric state. Transonic solutions of galactic outflows are classified according to their topological features. As result, we find two types of transonic solutions distinguished by a magnitude relationship between the gravity of DMH and that of SMBH. The loci of transonic points for two types are different; one transonic point is formed at a central region (< 0.01kpc) and another is at a very distant region (> 100kpc). Also, mass fluxes and outflow velocities are different for two solutions. Thus, these solutions may differently influence the evolution of galaxies and the release of metals into the intergalactic space.Furthermore, we apply our model to the Sombrero galaxy. In this galaxy, the wide-spread hot gas is detected as the trace of galactic outflows while the star-formation rate is low, and the observed gas density distribution is similar to the hydrostatic state (Li et al. 2011). To solve this discrepancy, we propose a solution that this galaxy has a slowly accelerating outflow; the transonic point forms in a very distant region (~ 120 kpc) and the wide subsonic region spreads across the stellar distribution. Thus, the gas density distribution in the observed region is similar to the hydrostatic state. Such slowly accelerating outflows are different from high-velocity outflows conventionally studied (Igarashi et al. 2014).However, this isothermal model requires an unrealistically large mass flux. Then, we apply the polytropic model to this galaxy incorporating mass flux supplied by stellar components. We find that it can reproduce the observed gas density and the temperature distributions with the realistic mass flux. Thus, our polytropic model successfully demonstrates the existence of the slowly accelerating outflow in the Sombrero galaxy (Igarashi et al. 2015).

Transonic galactic outflows in a dark matter halo with a central black hole and its application to the Sombrero galaxy

We have classified possible transonic solutions of galactic outflows in the gravitational potential of the dark matter halo (DMH) and super massive black hole (SMBH) under the assumptions of isothermal, spherically symmetric and steady state. It is clarified that the gravity of SMBH adds a new branch of transonic solutions with the transonic point in very close proximity to the centre in addition to the outer transonic point generated by the gravity of DMH. Because these two transonic solutions have substantially different mass fluxes and starting points, these solutions may have different influences on the evolution of galaxies and the release of metals into intergalactic space. We have applied our model to the Sombrero galaxy and obtained a new type of galactic outflow: a slowly accelerated transonic outflow through the transonic point at very distant region ($\simeq 126$\ kpc). In this galaxy, previous works reported that although the trace of the galactic outflow is observed by X-ray, the gas density distribution is consistent with the hydrostatic state. We have clarified that the slowly accelerating outflow has a gas density profile quite similar to that of the hydrostatic solution in the widely spread subsonic region. Thus, the slowly accelerating transonic solution cannot be distinguished from the hydrostatic solution in the observed region ($\leq 25$\ kpc) even if slow transonic flow exists. Our model provides a new perspective of galactic outflows and is applicable even to quiescent galaxies with inactive star formation.

The density and thermal structure of Pluto’s atmosphere and associated escape processes and rates

The original Strobel et al. (Strobel, D.F., Zhu, X., Summers, M.E., Stevens, M.E. [1996]. Icarus 120, 266–289) model for Pluto’s stratospheric density and thermal structure is augmented to include a radial momentum equation with radial velocity associated with atmospheric escape of N2 and in the energy equation to also include the solar far ultraviolet and extreme ultraviolet (FUV–EUV) heating in the upper atmosphere and adiabatic cooling due to hydrodynamic expansion. The inclusion of radial velocity introduces important negative feedback processes such as increased solar heating leading to enhanced escape rate and higher radial velocity with stronger adiabatic cooling in the upper atmosphere accompanied by reduced temperature. The coupled set of equations for mass, momentum, and energy are solved subject to two types of upper boundary conditions that represent two different descriptions of atmospheric escape: Jeans escape and hydrodynamic escape. For the former which is physically correct, an enhanced Jeans escape rate is prescribed at the exobase and parameterized according to the direct simulation Monte Carlo kinetic model results. For the latter, the atmosphere is assumed to remain a fluid to infinity with the escape rate determined by the temperature and density at the transonic point subject to vanishing temperature and pressure at infinity. For Pluto, the two escape descriptions approach the same limit when the exobase coincides with the transonic level and merge to a common escape rate ∼1028N2s−1 under elevated energy input. For Pluto’s current atmosphere, the hydrodynamic approach underestimates the escape rate by about 13%. In all cases, the escape rate is limited by the solar FUV–EUV power input.Specific results for the New Horizons Pluto flyby July 2015 are escape rate ∼3.5×1027N2s−1, exobase at 8r0∼9600km, with Jeans λ∼5 for a reference Pluto atmosphere model. With Pluto’s highly elliptic orbit and variable solar activity affecting its atmosphere, Pluto’s escape rates’ range is (1–10)×1027N2s−1, exobase radius is bounded by ∼(5–13)r0, and at the exobase Pluto is locked in the enhanced Jeans regime with λ∼(6–4).Finally, a systematic review of previous approximate hydrodynamic escape models is presented to compare the constraints which determine the escape rate in each model.

Dynamics of hot accretion flow with thermal conduction

The purpose of this paper is to explore the dynamical behaviour of hot accretion flow with thermal conduction. The importance of thermal conduction on hot accretion flow is confirmed by observations of the hot gas that surrounds Sgr A$^*$ and a few other nearby galactic nuclei. In this research, the effect of thermal conduction is studied by a saturated form of it, as is appropriate for weakly collisional systems. The angular momentum transport is assumed to be a result of viscous turbulence and the $\alpha$-prescription is used for the kinematic coefficient of viscosity. The equations of accretion flow are solved in a simplified one-dimensional model that neglects the latitudinal dependence of the flow. To solve the integrated equations that govern the dynamical behaviour of the accretion flow, we have used an unsteady self-similar solution. The solution provides some insights into the dynamics of quasi-spherical accretion flow and avoids from limits of the steady self-similar solution. In comparison to accretion flows without thermal conduction, the disc generally becomes cooler and denser. These properties are qualitatively consistent with performed simulations in hot accretion flows. Moreover, the angular velocity increases with the magnitude of conduction, while the radial infall velocity decreases. The mass accretion rate onto the central object is reduced in the presence of thermal conduction. We found that the viscosity and thermal conduction have the opposite effects on the physical variables. Furthermore, the flow represents a transonic point that moves inward with the magnitude of conduction or viscosity.

Time dependence of advection-dominated accretion flow with a toroidal magnetic field

The present study examines self-similarity evolution of advection dominated accretion flow (ADAF) in the presence of a toroidal magnetic field. In this research, it was assumed that the angular momentum transport is due to viscous turbulence and $\alpha$-prescription was used for kinematics coefficient of viscosity. In order to solve the integrated equations which govern the dynamical behavior of the accretion flow, self-similar solution was used. The solutions show that the behavior of physical quantities in a dynamical ADAF are different from steady accretion flow and a disk with polytropic approach. The solution indicates a transonic point in the accretion flow for all selected amounts of advection parameter. Also, by adding strength of the magnetic field and the degree of advection, the radial-thickness of the disk decreased and the disk compressed. The model implies that the flow has differential rotation and is sub-Keplerian at small radii and is super-Keplerian in large radii.

Time dependence of accretion flow with a toroidal magnetic field

In the present study time evolution of quasi-spherical polytropic accretion flow with toroidal magnetic field is investigated. The study especially focused the astrophysically important case in which the adiabatic exponent $\gamma=5/3$. In this scenario, it was assumed that the angular momentum transport is due to viscous turbulence and used $\alpha$-prescription for kinematic coefficient of viscosity. The equations of accretion flow are solved in a simplified one-dimensional model that neglects the latitudinal dependence of the flow. In order to solve the integrated equations which govern the dynamical behavior of the accretion flow, self-similar solution was used. The solution provides some insight into the dynamics of quasi-spherical accretion flow and avoids many of the strictures of the steady self-similar solution. The effect of the toroidal magnetic field is considered with additional variable $\beta[=p_{mag}/p_{gas}]$, where $p_{mag}$ and $p_{gas}$ are the magnetic and gas pressure, respectively. The solution indicates a transonic point in the accretion flow, that this point approaches to central object by adding strength of the magnetic field. Also, by adding strength of the magnetic field, the radial-thickness of the disk decreases and the disk compresses. It was analytically indicated that the radial velocity is only a function of Alfv'en velocity. The model implies that the flow has differential rotation and is sub-Keplerian at all radii.

Mass loss from “Hot Jupiters”—Implications for CoRoT discoveries, Part II: Long time thermal atmospheric evaporation modeling

We investigate the efficiency of the atmospheric mass loss due to hydrodynamic blow-off over the lifetime of the exoplanet HD209458b by studying numerically its hydrogen wind for host star X-ray and EUV (XUV) fluxes between 1 and 100 times that of the present Sun. We apply a time-dependent numerical algorithm which is able to solve the system of hydrodynamic equations straight through the transonic point of the flow including Roche lobe effects. The mass loss rates are calculated as functions of the absorbed energy in the thermosphere. Depending on the heating efficiency for a hydrogen-rich thermosphere the maximum temperature obtained in our study at 1.5 R pl by neglecting IR cooling is about 5000–10,000 K for heating efficiencies of 10% and 60%, respectively. We find that the upper atmosphere of HD209458b experiences hydrodynamic blow-off even at such low temperatures if one does not neglect gravitational effects caused by the proximity of the planet to its Roche lobe boundary. Depending on the heating efficiency, we find from the solution of the hydrodynamic equations of mass, momentum, and energy balance that energy-limited mass loss rate estimations overestimate the realistic mass loss rate at present time for HD209458b by several times. Using the maximum heating efficiency for hydrogen–rich atmospheres of 60% we find that HD209458b may experience an atmospheric mass loss rate at present time of about 3.5 × 10 10 g s - 1 . The mass loss rate evolves to higher values for higher XUV fluxes expected during the early period of the planet's host star evolution, reaching values of several times 10 12 g s - 1 . The integrated mass loss is found to be between 1.8% and 4.4% of the present mass of HD209458b. We found that the influence of the stellar tidal forces on atmospheric loss (the Roche lobe effect) is not significant at 0.045 AU. For a similar exoplanet, but at closer orbital distances ⩽ 0.02 AU , the combined effect of the Roche lobe and the high XUV radiation result in much higher thermal loss rates of about 2.6 × 10 11 g s - 1 and even more for early stages. This leads to a total loss over 4 Gyr of 27.5% of the planetary mass.

Application of inertia-induced excitation theory for nonlinear acoustic modes in colloidal plasma equilibrium flow

Application of inertia-induced acoustic excitation theory offers a new resonant excitation source channel of acoustic turbulence in the transonic domain of plasma flow. In bi-ion plasmas like colloidal plasma, two well-defined transonic points exist corresponding to the parent ion and the dust grain-associated acoustic modes. As usual, the modified ion acoustic mode (also known as dust ion-acoustic (DIA) wave) dynamics associated with parent ion inertia is excitable for both nanoscale-and micronscale-sized dust grains. It is found that the so-called (ion) acoustic mode (also known as dust-acoustic (DA) wave) associated with nanoscale dust grain inertia is indeed resonantly excitable through the active role of weak but finite parent ion inertia. It is interestingly conjectured that the same excitation physics, as in the case of normal plasma sound mode, operates through the active inertial role of plasma thermal species. Details of the nonlinear acoustic mode analyses of current interest in transonic domains of such impure plasmas in hydrodynamic flow are presented.

Propagating wave pattern on a falling liquid curtain

A regular pattern of surface waves is observed on a liquid curtain falling from a horizontal, wetted tube, maintained between two vertical wires. Since the upper boundary is not constrained in the transverse direction, the top of the curtain enters a pendulum-like motion, when the flow rate is progressively reduced, coupled to the propagation of curtain undulations, structured as a checkerboard. This structure is formed by two patterns of propagating waves. In some sense, these propagating patterns replace the stationary pattern of liquid columns observed at a lower flow rate. Measurements of phase velocity, frequency, and wavelength are reported. The data are in agreement with a simple dimensional argument suggesting that the wave velocity is proportional to the surface tension divided by the mass flux of liquid per unit length. This scaling is also that followed by the fluid velocity at the so-called transonic point, i.e., the point where the fluid velocity equals that of sinuous waves. We finally discuss the implications of these results for the global stability of liquid curtains.

Transonic liquid bells

The shape of a liquid bell resulting from the overflow of a viscous liquid out of a circular dish is investigated experimentally and theoretically. The main property of this bell is its ability to sustain the presence of a “transonic point,” where the liquid velocity equals the speed of antisymmetric—or sinuous—surface waves. Their shape and properties are thus rather different from usual “hypersonic” water bells. We first show that the bell shape can be calculated very accurately, starting from the sonic point. We then demonstrate the extreme sensitivity of the shape of these bells to the difference of pressure across the interface, making them a perfect barometer. Finally, we discuss the oscillations of the bell which occur close to the bursting limit.

Stability criterion for self-similar solutions with perfect fluids in general relativity

A stability criterion is derived for self-similar solutions with perfect fluids which obey the equation of state P = kρ in general relativity. A wide class of self-similar solutions turn out to be unstable against the so-called kink mode. The criterion is directly related to the classification of sonic points. The criterion gives a sufficient condition for instability of the solution. For a transonic point in collapse, all primary-direction nodal-point solutions are unstable, while all secondary-direction nodal-point solutions and saddle-point ones are stable against the kink mode. The situation is reversed in expansion. The applications are the following: the expanding flat Friedmann solution for 1/3 ≤ k < 1 and the collapsing one for 0 < k ≤ 1/3 are unstable; the static self-similar solution is unstable; nonanalytic self-similar collapse solutions are unstable; the Larson–Penston (attractor) solution is stable for this mode for 0 < k ≲ 0.036, while it is unstable for 0.036 ≲ k; the Evans–Coleman (critical) solution is stable for this mode for 0 < k ≲ 0.89, while it is unstable for 0.89 ≲ k. The last application suggests that the Evans–Coleman solution for 0.89 ≲ k is not critical because it has at least two unstable modes.