Transonic Solutions Research Articles

Transonic Shock Problem for the Euler System in a Nozzle

In this paper, we study the well-posedness problem on transonic shocks for steady ideal compressible flows through a two-dimensional slowly varying nozzle with an appropriately given pressure at the exit of the nozzle. This is motivated by the following transonic phenomena in a de Laval nozzle. Given an appropriately large receiver pressure P r , if the upstream flow remains supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the flow is compressed and slowed down to subsonic speed, and the position and the strength of the shock front are automatically adjusted so that the end pressure at exit becomes P r , as clearly stated by Courant and Friedrichs [Supersonic flow and shock waves, Interscience Publishers, New York, 1948 (see section 143 and 147)]. The transonic shock front is a free boundary dividing two regions of C2,α flow in the nozzle. The full Euler system is hyperbolic upstream where the flow is supersonic, and coupled hyperbolic-elliptic in the downstream region Ω+ of the nozzle where the flow is subsonic. Based on Bernoulli’s law, we can reformulate the problem by decomposing the 3 × 3 Euler system into a weakly coupled second order elliptic equation for the density ρ with mixed boundary conditions, a 2 × 2 first order system on u2 with a value given at a point, and an algebraic equation on (ρ, u1, u2) along a streamline. In terms of this reformulation, we can show the uniqueness of such a transonic shock solution if it exists and the shock front goes through a fixed point. Furthermore, we prove that there is no such transonic shock solution for a class of nozzles with some large pressure given at the exit.

Uniqueness of transonic shock solutions in a duct for steady potential flow

We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.

On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges

In this paper, under certain downstream pressure condition at infinity, we study the globally stable transonic shock problem for the perturbed steady supersonic Euler flow past an infinitely long 2-D wedge with a sharp angle. As described in the book of Courant and Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948] (pages 317–318): when a supersonic flow hits a sharp wedge, it follows from the Rankine–Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge in terms of the different pressure states in the downstream region, which correspond to the supersonic shock and the transonic shock respectively. It has frequently been stated that the strong shock is unstable and that, therefore, only the weak shock could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this open problem. More concretely, we will establish the global existence and stability of a transonic shock solution for 2-D full Euler system when the downstream pressure at infinity is suitably given. Meanwhile, the asymptotic state of the downstream subsonic solution is determined.

Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone

For an upstream supersonic flow past a straight-sided cone in $\R^3$ whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. We establish that the self-similar transonic shock-front solution is conditionally stable in structure with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. In particular, it is proved that the slope of the shock-front tends asymptotically to the slope of the unperturbed self-similar shock-front downstream at infinity.

The transonic shock in a nozzle, 2-D and 3-D complete Euler systems

In this paper, we study a transonic shock problem for the Euler flows through a class of 2-D or 3-D nozzles. The nozzle is assumed to be symmetric in the diverging (or converging) part. If the supersonic incoming flow is symmetric near the divergent (or convergent) part of the nozzle, then, as indicated in Section 147 of [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publ., New York, 1948], there exist two constant pressures P 1 and P 2 with P 1 < P 2 such that for given constant exit pressure P e ∈ ( P 1 , P 2 ) , a symmetric transonic shock exists uniquely in the nozzle, and the position and the strength of the shock are completely determined by P e . Moreover, it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting. Furthermore, we establish the global existence, stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure P e when the initial unsteady shock lies in the symmetric diverging part of the 2-D or 3-D nozzle. On the other hand, it is shown that an unsteady symmetric transonic shock is structurally unstable in a global-in-time sense if it lies in the symmetric converging part of the nozzle.

Systems of hyperbolic conservation laws with a resonant moving source

In this paper, we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source. Compared with the previous results [W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098; T.P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. 83 (2) (1982) 243–260] on this stability problem, in this paper, the transonic ith shock is assumed to be relatively strong and stable in the sense of Majda. Then the framework of [M. Lewicka, L 1 stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J. 49 (4) (2000) 1515–1537; M. Lewicka, Stability conditions for patterns of noninteracting large shock waves, SIAM J. Math. Anal. 32 (5) (2001) 1094–1116 (electronic)] can be applied. A new criterion is obtained to test whether such a shock is time asymptotically stable or not. And by constructing the Liu–Yang functional, one can prove the L 1 stability of the shock under the stability condition. This is an extension of the result [S.-Y. Ha, T. Yang, L 1 stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (5) (2003) 1226–1251 (electronic); W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098] to a more general case.

A fast and accurate algorithm for computing radial transonic flows

An efficient algorithm is described for calculating stationary one-dimensional transonic outflow solutions of the compressible Euler equations with gravity and heat source terms. The stationary equations are solved directly by exploiting their dynamical system form. Transonic expansions are the stable manifolds of saddle-point-type critical points, and can be obtained efficiently and accurately by adaptive integration outward from the critical points. The particular transonic solution and critical point that match the inflow boundary conditions are obtained by a two-by-two Newton iteration which allows the critical point to vary within the manifold of possible critical points. The proposed Newton Critical Point (NCP) method typically converges in a small number of Newton steps, and the adaptively calculated solution trajectories are highly accurate. A sample application area for this method is the calculation of transonic hydrodynamic escape flows from extrasolar planets and the early Earth. The method is also illustrated for an example flow problem that models accretion onto a black hole with a shock.

Horizon-penetrating transonic accretion discs around rotating black holes

The stationary hydrodynamic equations for the transonic accretion disks and flows around rotating black holes are presented by using the Kerr-Schild coordinate where there is no coordinate singularity at the event horizon. We use two types of the causal viscosity prescription, and the boundary conditions for the transonic accretion flows are given at the sonic point. For one type of the causal viscosity prescription we also add the boundary conditions at the viscous point where the accreting radial velocity is nearly equal to the viscous diffusion velocity. Based on the formalism for the transonic accretion disks, after we present the calculation method of the transonic solutions, the horizon-penetrating transonic solutions which smoothly pass the event horizon are calculated for several types of the accretion flow models: the ideal isothermal flows, the ideal and the viscous polytropic flows, the advection dominated accretion flows (ADAFs) with the relativistic equation of state, the adiabatic accretion disks, the standard accretion disks, the supercritical accretion disks. These solutions are obtained for both non-rotating and rotating black holes. The calculated accretion flows plunge into black hole with finite three velocity smaller than the speed of light even at the event horizon or inside the horizon, and the angular velocities of the accretion flow at the horizon are generally different from the angular velocity of the frame-dragging due to the black hole's rotation. These features contrast to the results obtained by using the Boyer-Lindquist coordinate with the coordinate singularity at the horizon.

Parametric decay of circularly polarized Alfvén waves in the radially expanding solar wind

[1] We present a magnetohydrodynamic (MHD) simulation study of the parametric decay of the circularly polarized Alfven waves propagating in the radial outflow of the solar wind. Assuming the transonic wind solution as an initial condition, we continuously injected monochromatic circularly polarized Alfven waves from the inner boundary at the lower corona, and simulated the wave propagation. The injected Alfven waves are subject to the parametric decay, and density fluctuations in the solar wind plasma grow rapidly at a specific region. The location of the most unstable region depends on the amplitude and frequency of injected Alfven waves. We found that the unstable region of simulation results can be well estimated by considering a localized dispersion relation in the frame of reference moving with the background solar wind.

Critical properties of spherically symmetric accretion in a fractal medium

Spherically symmetric transonic accretion of a fractal medium has been studied in both the stationary and the dynamic regimes. The stationary transonic solution is greatly sensitive to infinitesimal deviations in the outer boundary condition, but the flow becomes transonic and stable, when its evolution is followed through time. The evolution towards transonicity is more pronounced for a fractal medium than what is it for a continuum. The dynamic approach also shows that there is a remarkable closeness between an equation of motion for a perturbation in the flow, and the metric of an analogue acoustic black hole. The stationary inflow solutions of a fractal medium are as much stable under the influence of linearised perturbations, as they are for the fluid continuum.

Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles

We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock for the steady, full Euler equations in two-dimensional infinite nozzles of slowly varying cross-sections. Given a smooth incoming flow that is close to a uniform supersonic state at the entrance, we prove that there exists a transonic flow whose infinite downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow. The solution is unique within the class of transonic solutions that are close to the background solution. This problem is approached by a free boundary problem in which the transonic shock is formulated as a free boundary. An iteration scheme for the free boundary is developed and its fixed point is shown to exist, which is a solution of the free boundary problem, by combining some delicate estimates for a second-order nonlinear elliptic equation on a Lipschitz domain.

Steady Models of Optically Thin, Magnetically Supported Black Hole Accretion Disks

Abstract We obtained steady solutions of optically thin, single-temperature, magnetized black hole accretion disks, assuming thermal bremsstrahlung cooling. Based on the results of 3D MHD simulations of accretion disks, we assumed that the magnetic fields inside the disk are turbulent and dominated by an azimuthal component. We decomposed the magnetic fields into an azimuthally averaged mean field and fluctuating fields. We also assumed that the azimuthally averaged Maxwell stress is proportional to the total pressure. The radial advection rate of the azimuthal magnetic flux, $\dot \Phi$, is prescribed as being proportional to $\varpi^{-\zeta}$, where $\varpi$ is the radial coordinate and $\zeta$ is a parameter that parameterizes the radial variation of $\dot \Phi$. We found that when the accretion rate, $\dot M$, exceeds the threshold for the onset of the thermal instability, a magnetic pressure-dominated new branch appears. Thus, the thermal-equilibrium curve of an optically thin disk has a ‘Z’-shape in the plane of the surface density and temperature. This indicates that as the mass accretion rate increases, a gas pressure-dominated optically thin hot accretion disk undergoes a transition to a magnetic pressure dominated, optically thin cool disk. This disk corresponds to the X-ray hard, luminous disk in black hole candidates observed during the transition from a low/hard state to a high/soft state. We also obtained global steady transonic solutions containing such a transition layer.

Evolution of transonicity in an accretion disc

For inviscid, rotational accretion flows driven by a general pseudo-Newtonian potential on to a Schwarzschild black hole, the only possible fixed points are saddle points and centre-type points. For a specific choice of the Newtonian potential, the flow has only two critical points, of which the outer one is a saddle point while the inner one is a centre-type point. A restrictive upper bound is imposed on the admissible range of values of the angular momentum of sub-Keplerian flows through a saddle point. These flows are very unstable to any deviation from a necessarily precise boundary condition. The difficulties against the physical realizability of a solution passing through the saddle point have been addressed through a temporal evolution of the flow, which gives a non-perturbative mechanism for selecting a transonic solution passing through the saddle point. An equation of motion for a real-time perturbation about the stationary flows reveals a very close correspondence with the metric of an acoustic black hole, which is also an indication of the primacy of transonicity.

Advection-dominated accretion flows with causal viscosity

We present the basic equations and sample solutions for the steady-state global transonic solutions of the advection-dominated accretion flows (ADAFs) with a causal viscosity prescription. The procedures for the stable numerical calculations and all explicit formula for obtaining the solutions of ADAFs are also presented. We solve the transonic solutions of ADAFs by using the explicit numerical integrations, as in the Runge-Kutta method. In this calculation method, we first solve the physical values at the sonic radius where L'Hopital's rule is used. Then, we numerically solve the coupled differential equations of the radial velocity, the angular momentum, and the sound speed from the sonic radius in the inward and outward directions. By the calculation procedures presented in this paper, we can cover all parameter spaces of the transonic solutions of ADAFs. Sample transonic solutions for ADAF-thick disk and ADAF-thin disk are presented. The explicit formula for the analytical expansion around the singular points, the sonic and the viscous points, are presented. If we set the diffusion timescale to null, the formalism in this study becomes the formalism of the acausal viscosity that is usually used in past studies for calculating of the ADAF structure.

Simulations of galactic winds and starbursts in galaxy clusters

We present an investigation of the metal enrichment of the intra-cluster medium (ICM) by galactic winds and merger-driven starbursts. We use combined N-body/hydrodynamic simulations with a semi-numerical galaxy formation model. The mass loss by galactic winds is obtained by calculating transonic solutions of steady state outflows, driven by thermal, cosmic ray and MHD wave pressure. The inhomogeneities in the metal distribution caused by these processes are an ideal tool to reveal the dynamical state of a galaxy cluster. We present surface brightness, X-ray emission weighted temperature and metal maps of our model clusters as they would be observed by X-ray telescopes like XMM-Newton. We show that X-ray weighted metal maps distinguish between pre- or post-merger galaxy clusters by comparing the metallicity distribution with the galaxy-density distribution: pre-mergers have a metallicity gap between the subclusters, post-mergers a high metallicity between subclusters. We apply our approach to two observed galaxy clusters, Abell 3528 and Abell 3921, to show whether they are pre- or post-merging systems. The survival time of the inhomogeneities in the metallicity distribution found in our simulations is up to several Gyr. We show that galactic winds and merger-driven starbursts enrich the ICM very efficiently after in the central (~3 Mpc radius) region of a galaxy cluster.

On the Theory of Mixed-Type Equations

We briefly review different well-posed boundary-value problems for mixed-type equations and their applications in transonic gas dynamics. We present barotropic relations for plane-parallel flow of a compressible gas that leads to mixed-type model equations on the hodograph plane. We also discuss the question on the existence of transonic solutions of profile flow problems, which is closely related to Dirichlet problems in mixed-type domains.

Large‐Scale Properties in Turbulent Spherically Symmetric Accretion

The role of turbulence in a spherically symmetric accreting system has been studied on very large spatial scales of the system. This is also a highly subsonic flow region and here the accreting fluid has been treated as nearly incompressible. It has been shown here that the coupling of the mean flow and the fluctuations, gives rise to a scaling relation for an effective turbulent viscosity. This in turn leads to a dynamic scaling for sound propagation in the accretion process. As a consequence of this scaling, the sonic horizon of the transonic inflow solution is shifted inwards, in comparison with the inviscid flow.

A phase analysis of transonic solutions for the hydrodynamic semiconductor model

In the present paper we present a phase plane analysis of transonic solutions of the steady state one-dimensional unipolar hydrodynamic model for semiconductors, taking also into consider shocks.

Basic properties of anisotropic stellar wind expansion in the fluid approach

[1] Whereas in an isotropic temperature atmosphere both the hydrostatic equation and the momentum equation give the same conditions for hydrostatic equilibrium, in the anisotropic case the situation is ambiguous. It is found that for an anisotropic temperature the hydrostatic equilibrium conditions have to be deduced from the momentum equation. The condition for hydrostatic equilibrium is that both the parallel and the perpendicular temperatures, to the radial direction, decrease more rapidly than 1/r in the general case and that the perpendicular temperature decreases more rapidly than 1/r when the parallel temperature is constant. The momentum equation displays a transonic solution when at least one of the temperatures, parallel or perpendicular to the radial direction, decreases less rapidly than 1/r in the general case and when the perpendicular temperature decreases less rapidly than 1/r when the parallel temperature is constant. In an anisotropic atmosphere the parallel thermal velocity is the critical velocity. The properties of the transonic expansion in an isothermal and anisotropic atmosphere are studied. The initial velocity, the critical distance position, the terminal velocity, and the density profile are significantly different from the isotropic case. These properties are opposite with respect to the value of the anisotropy T⊥/T∥ > 1.0 and T⊥/T∥ 2.0. The extension of this study to multimoment anisotropic models can be useful for the interpretation of particle simulation of rarefied stellar atmosphere expansion.

The structural instability of transonic flow associated with amalgamation/splitting of supersonic regions

We study inviscid transonic flow in a channel with a bump modeling an airfoil at zero incidence. A few simple bump configurations are considered. Numerical simulations show the existence of singular free-stream Mach numbers, which trigger the splitting/amalgamation of local supersonic regions. An analysis of this phenomenon contributes to the understanding of the nonuniqueness of transonic solutions at certain free-stream conditions.