Steady Euler Flows Research Articles

Hypersonic similarity for steady compressible full Euler flows over two-dimensional Lipschitz wedges

We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. Mathematically, it can be formulated as the comparison of the entropy solutions in BV∩L1 between the two initial-boundary value problems for the compressible full Euler equations with parameter τ>0 and the hypersonic small-disturbance equations (the scaled compressible full Euler equations with parameter τ=0) with curved characteristic boundaries. We establish the L1–convergence estimate of these two solutions with the optimal convergence rate, which justifies the Van Dyke's similarity theory rigorously for the compressible full Euler flows. This is the first mathematical result on the comparison of two solutions of the compressible Euler equations with characteristic boundary conditions. To achieve this, we first employ the special structures of the two systems and establish the global existence and the L1–stability of the entropy solutions via the wave-front tracking scheme under the smallness assumption on the total variation of both the initial data and the tangential slope function of the wedge boundary. Based on the L1–stability properties of the approximate solutions to the scaled equations with parameter τ>0, a uniform Lipschtiz continuous map with respect to the initial data and the wedge boundary is obtained, which is the first time for the characteristic boundary conditions. Next, we compare the solutions given by the Riemann solvers of the two systems by taking the boundary perturbations into account case by case. Then, for a given fixed hypersonic similarity parameter, as the Mach number tends to infinity, by employing the Lipschitz continuous properties of the map, we establish the desired L1–convergence estimate with the optimal convergence rate. Finally, we show the optimality of the convergence rate by investigating a special solution.

Smooth Transonic Flows with Nonzero Vorticity to a Quasi Two Dimensional Steady Euler Flow Model

Smooth Transonic Flows with Nonzero Vorticity to a Quasi Two Dimensional Steady Euler Flow Model

Generalized Newton-Busemann Law for Two-dimensional Steady Hypersonic-limit Euler Flows Passing Ramps with Skin-frictions

Generalized Newton-Busemann Law for Two-dimensional Steady Hypersonic-limit Euler Flows Passing Ramps with Skin-frictions

Global stability of supersonic Euler flows in quasi-one-dimensional convergent nozzles

In this paper, we investigate the global well-posedness and stability of steady supersonic Euler flows in a quasi-one-dimensional convergent nozzle. First, we show that there exists a critical nozzle length L1, the global well-posedness of steady supersonic flows can be obtained and the explicit solution is given as long as the nozzle length L < L1. Then we prove the global stability of steady supersonic flows under small perturbations of initial-boundary values by wave decomposition.

Transonic shock solutions of the steady Euler flow in quasi-one-dimensional convergent nozzles

In this paper, we investigate the well-posedness of transonic shock solutions of the steady Euler flow in quasi-one-dimensional convergent nozzles under the physical boundary conditions proposed by Courant-Friedrichs in [12]. Inspired by [20], for any supersonic incoming flow at the entrance, we derive the critical length of the nozzle and show that as long as the nozzle length is less than or equal to the critical length, the existence and uniqueness of transonic shock solutions with the exit pressure lying in a certain range can be obtained. Moreover, the transonic shock position is monotonically determined by the exit pressure.

Nested invariant tori foliating a vector field and its curl: Toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries

This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.

Plasma knots

We present a Lagrangian method for the computation of ideal plasma knots and links. It is based on a variational principle for stable equilibria of an ideal plasma in the case of a free boundary subjected to external magnetic or plasma pressure forces. For this purpose, we introduce a structure preserving discretization of plasma based on decompositions of Riemannian manifolds representing pressure confined plasma regions in magnetohydrostatic equilibrium. Moreover, we show that, by the virtue of an analogy, the method can be used for the approximation of steady Euler-flows of arbitrarily complex topology.

Universality of Euler flows and flexibility of Reeb embeddings

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [38,39] launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, in this article we prove that the stationary Euler equations exhibit several universality features. More precisely, we show that any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. The solutions we construct are of Beltrami type, and being stationary they exist for all time. Using this result, we establish the Turing completeness of the steady Euler flows, i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories. Our proofs deepen the correspondence between contact topology and hydrodynamics, which is key to establish the universality of the Reeb flows and their Beltrami counterparts. An essential ingredient in the proofs, of interest in itself, is a novel flexibility theorem for embeddings in Reeb dynamics in terms of an h-principle in contact geometry, which unveils the flexible behavior of the steady Euler flows. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature.

Stability of two-dimensional steady Euler flows with concentrated vorticity

In this paper, we study the stability of two-dimensional steady Euler flows with sharply concentrated vorticity in a bounded domain. These flows are obtained as maximizers of the kinetic energy on some isovortical surface, under the constraint that the vorticity is compactly supported in a finite number of disjoint regions of small diameter. We prove the nonlinear stability of these flows when the vorticity is sufficiently concentrated in one small region, or in two small regions with opposite signs. The proof is achieved by showing that these flows constitute a compact and isolated set of local maximizers of the kinetic energy on the isovortical surface. The separation property of the stream function plays a crucial role in validating isolatedness.

The emergence of delta shock in the limit process for two‐dimensional steady nonisentropic relativistic Euler flow onto a straight wedge

AbstractIn this paper, we mainly study the limit of solutions to the two‐dimensional steady nonisentropic relativistic Euler flow onto a straight wedge. It turns out that the sequence of shock solutions tends to a delta wave adhering to the wedge surface when the velocity of the incident flow goes to the speed of light and the adiabatic index tends to 1 in turn. Meanwhile, it also verifies that the pressure coefficient in the limit case is consistent with the Newtonian sin‐squared law when the speed of light tends to infinity. To this end, we also derive the generalized Bernoulli equation, Taub adiabat (or the generalized Hugoniot adiabat) and a shock polar, and so forth. Furthermore, we give the construction of the delta wave by the definition of weak solution.

Adjoint Based Aerodynamic Shape Optimisation Using Kinetic Meshfree Method

The gradient based optimisation algorithms combined with the finite volume or element based adjoint approaches have been very successful in aerodynamic shape optimization (ASO). The meshfree least squares kinetic upwind method (LSKUM), which works on a cloud of points and its connectivity, has the advantage in terms of flexibility of generating a cloud of points and having an arbitrary number of points in the connectivity. The LSKUM based meshfree solvers have been successfully applied to problems involving multi-body configurations and moving boundaries. In the present work, the LSKUM based primal and discrete adjoint meshfree solvers for steady Euler flows have been used to perform aerodynamic shape optimisation. It is well-known that the raw shape sensitivities of a defined objective function are highly oscillatory. In this research, we have used the Sobolev gradient smoothing. An approach is developed to find a suitable choice for the smoothing parameter in the Sobolev gradient algorithm. Numerical results are presented for the shape optimisation of the NACA0012 airfoil at transonic and supersonic flows with aerodynamic and geometric constraints. The performance of the discrete adjoint LSKUM solver in accurate computation of shape sensitivities is demonstrated. The advantages of using LSKUM in aerodynamic shape optimisation have been shown.

Steady Euler Flows on $${\mathbb {R}}^3$$ with Wild and Universal Dynamics

Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set $\mathscr G$ of stationary solutions to the Euler equations in $\mathbb R^3$ such that each vector field $X\in \mathscr G$ is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincar\'e map of $X$ at some transverse section. We remark that this universality is approximate but occurs at all scales. In particular, our results establish that a steady Euler flow may exhibit any conservative finite codimensional dynamical phenomenon; this includes the existence of horseshoes accumulated by elliptic islands, increasing union of horseshoes of Hausdorff dimension $3$ or homoclinic tangencies of arbitrarily high multiplicity. The steady solutions we construct are Beltrami fields with sharp decay at infinity. To prove these results we introduce new perturbation methods in the context of Beltrami fields that allow us to import deep techniques from bifurcation theory: the Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte theorems on the geometry of wild hyperbolic sets. These perturbation methods rely on two tools from linear PDEs: global approximation and Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.

Convergence rate of hypersonic similarity for steady potential flows over two-dimensional Lipschitz wedge

This paper is devoted to establishing the convergence rate of the hypersonic similarity for the inviscid steady irrotational Euler flow over a two-dimensional Lipschitz slender wedge in $$BV\cap L^1$$ space. The rate we established is the same as the one predicted by Newtonian-Busemann law (see (3.29) in [2, p 67] for more details) as the incoming Mach number $$\text {M}_{\infty }\rightarrow \infty $$ for a fixed hypersonic similarity parameter K. The hypersonic similarity, which is also called the Mach-number independence principle, is equivalent to the following Van Dyke’s similarity theory: For a given hypersonic similarity parameter K, when the Mach number of the flow is sufficiently large, the governing equations after the scaling are approximated by a simpler equation, that is called the hypersonic small-disturbance equation. To achieve the convergence rate, we approximate the curved boundary by piecewisely straight lines and find a new Lipschitz continuous map $$\mathcal {P}_{h}$$ such that the trajectory can be obtained by piecing together the Riemann solutions near the approximated boundary. Next, we derive the $$L^1$$ difference estimates between the approximate solutions $$U^{(\tau )}_{h,\nu }(x,\cdot )$$ to the initial-boundary value problem for the scaled equations and the trajectories $$\mathcal {P}_{h}(x,0)(U^{\nu }_{0})$$ by piecing together all the Riemann solvers. Then, by the uniqueness and the compactness of $$\mathcal {P}_{h}$$ and $$U^{(\tau )}_{h,\nu }$$ , we can further establish the $$L^1$$ estimates of order $$\tau ^2$$ between the solutions to the initial-boundary value problem for the scaled equations and the solutions to the initial-boundary value problem for the hypersonic small-disturbance equations, if the total variations of the initial data and the tangential derivative of the boundary are sufficiently small. Based on it, we can further establish a better convergence rate by considering the hypersonic flow past a two-dimensional Lipschitz slender wing and show that for the length of the wing with the effect scale order $$O(\tau ^{-1})$$ , that is, the $$L^1$$ convergence rate between the two solutions is of order $$O(\tau ^{\frac{3}{2}})$$ under the assumption that the initial perturbation has compact support.

Beltrami fields and knotted vortex structures in incompressible fluid flows

AbstractThis paper gives a survey on recent results about the existence of knotted vortex structures in incompressible fluids. This includes the proof of Lord Kelvin's conjecture on the existence of knotted vortex tubes in steady Euler flows and a new probabilistic approach to address Arnold's speculation that typical Beltrami fields should exhibit vortex lines of arbitrary topological complexity. We review the key tools to establish these results: the global approximation properties of Beltrami fields, the inverse localization principle in spectral theory, and the theory of Gaussian random Beltrami fields, which permits the introduction of probabilistic considerations in the picture. Finally, we include some applications of these techniques to other problems, such as the existence of vortex reconnections for smooth solutions of the 3D Navier–Stokes equation and the evolution of local hot spots under the heat equation.

$ L^1 $–stability of vortex sheets and entropy waves in steady supersonic Euler flows over Lipschitz walls

<p style='text-indent:20px;'>We study the well-posedness of compressible vortex sheets and entropy waves in two-dimensional steady supersonic Euler flows over Lipschitz walls with <inline-formula><tex-math id="M2">\begin{document}$ BV $\end{document}</tex-math></inline-formula> incoming flows. Both the Lipschitz wall of <inline-formula><tex-math id="M3">\begin{document}$ BV $\end{document}</tex-math></inline-formula> tangential angle function and the <inline-formula><tex-math id="M4">\begin{document}$ BV $\end{document}</tex-math></inline-formula> incoming flow perturb a background strong vortex sheet/entropy wave. In particular, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is small, we prove that the two-dimensional steady supersonic Euler flows containing a strong vortex sheet/entropy wave past the Lipschitz wall are <inline-formula><tex-math id="M5">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula>–stable. The weak waves are reflected after the nonlinear waves interact with the strong vortex sheet/entropy wave and the wall boundary. The existence of solutions in <inline-formula><tex-math id="M6">\begin{document}$ BV $\end{document}</tex-math></inline-formula> over the Lipschitz walls is first shown, when the total variation of the incoming flow perturbation around the background strong vortex sheet/entropy wave is suitably small, by using the wave-front tracking method. Then we establish the <inline-formula><tex-math id="M7">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula>–stability of the solutions with respect to the incoming flows. To achieve this, a Lyapunov functional, equivalent to the <inline-formula><tex-math id="M8">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula>–distance between two solutions containing the strong vortex sheets/entropy waves, is carefully constructed to include the nonlinear waves generated by both the wall boundary and the incoming flow. This Lyapunov functional is then proved to decrease in the flow direction, leading to the <inline-formula><tex-math id="M9">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula>–stability of the solutions. Furthermore, the uniqueness of these solutions extends to a larger class of viscosity solutions.</p>

Location of concentrated vortices in planar steady Euler flows

Location of concentrated vortices in planar steady Euler flows

Computability and Beltrami fields in Euclidean space

In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of R3, however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on T3 (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) computational complexity as high as desired. We also quantify the energetic cost for a Beltrami field on T3 to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded Church-Turing thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability 1. Finally, our proof also yields Turing complete flows and diffeomorphisms on S2 with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity.

Stability of 2D steady Euler flows related to least energy solutions of the Lane-Emden equation

In this paper, we investigate nonlinear stability of planar steady Euler flows related to least energy solutions of the Lane-Emden equation in a smooth bounded domain. We prove the orbital stability of these flows in terms of both the Ls norm of the vorticity for any s∈(1,+∞) and the energy norm. As a consequence, nonlinear stability is obtained when the least energy solution is unique, which actually holds for a large class of domains and exponents. The proofs are based on a new variational characterization of least energy solutions in terms of the vorticity, a compactness argument, and proper use of conserved quantities of the Euler equation.

Conjugate point criteria on the area-preserving diffeomorphism group

This paper answers some questions about conjugate points along the geodesics corresponding to steady 2D Euler flows, posed by a paper of Drivas-Misiołek-Shi-Yoneda. We present a new sufficient criterion for the existence of conjugate points, which improves on the criterion of Misiołek. It applies in any rotational cell of a steady 2D Euler flow, and in case of rotational symmetry it captures all known conjugate points. We give a general construction of the surfaces that admit a steady fluid with given area form, velocity profile, and vorticity profile, and from this we show how to detect conjugate points in a single rotational cell of a steady flow. When the velocity profile has a local extremum, the criterion becomes particularly simple. Several examples are provided, and in an appendix we use the Misiołek criterion to give some new examples of conjugate points along Kolmogorov flows on the torus.

On Admissible Positions of Transonic Shocks for Steady Isothermal Euler Flows in a Horizontal Flat Nozzle under Vertical Gravity

In this paper we are concerned with the existence of transonic shocks for 2-D steady isothermal Euler flows in a horizontal flat nozzle under vertical gravity. In particular, we focus on the contribution of the vertical gravity in determining the position of the shock front. For steady horizontal flows, the existence of normal shocks with the position of the shock front being arbitrary in the nozzle can be easily established. This paper will try to determine the position of the shock front as the state of the flow at the entrance of the nozzle and the pressure at the exit are slightly perturbed. To this end, this paper proposes a free boundary problem of the linearized Euler system with vertical gravity, whose solution could be an initial approximation for the shock solution with the free boundary being the approximation for the shock front. Due to the existence of the vertical gravity, difficulties arise in solving the boundary value problem in the approximate subsonic domain behind the shock front. The linearized Euler system is elliptic-hyperbolic composite for subsonic flows, and the elliptic part and the hyperbolic part are coupled in the $0$-order terms depending on the acceleration of gravity $g$. Moreover, the coefficients are not constants since the unperturbed shock solution depends on the vertical variable. New ideas and techniques are developed to deal with these difficulties and, under certain sufficient conditions on the perturbation, the existence of the solution to the proposed free boundary problem is established as the acceleration of gravity $g>0$ and the perturbation are sufficiently small. Then, with the obtained initial approximation of the shock solution, a nonlinear iteration scheme can be constructed which leads to a transonic shock solution with the position of the shock front being close to the initial approximating position.