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1.
2.
Within the Lagrangian framework we present an approach yielding some explicit solutions to the incompressible two-dimensional
Euler equations, generalizing the celebrated Gerstner flow. The solutions so obtained, for which explicit formulas of each
particle trajectory are provided, represent either flows in domains with a rigid boundary or free-surface flows for a fluid
of infinite depth. For some of these solutions the trajectories are epitrochoids or hypotrochoids. Possibilities for obtaining
further flows of this type are indicated. 相似文献
3.
We construct semi-hyperbolic patches of solutions, in which one family out of two nonlinear families of characteristics starts
on sonic curves and ends on transonic shock waves, to the two-dimensional Euler equations. This type of solution appears in
the transonic flow over an airfoil and Guderley reflection, and is common in the numerical solutions of Riemann problems. 相似文献
4.
E. Yu. Meshcheryakova 《Journal of Applied Mechanics and Technical Physics》2003,44(4):455-460
Exact steady and self-similar solutions of the Euler equations are considered, which possess the property of partial invariance with respect to a certain six-parameter Lie group. New examples of vortex motion of a swirled liquid in curved channels are presented. A classification is given for self-similar solutions of the reduced system with two independent variables, which admits a three-parameter group of extensions, whereas the initial system of the Euler equations possesses a two-parameter group. 相似文献
5.
We construct rigorously a three‐parameter family of self‐similar, globally bounded, and continuous weak solutions in two space
dimensions for all positive time to the Euler equations with axisymmetry for polytropic gases with a quadratic pressure‐density
law. We use the axisymmetry and self‐similarity assumptions to reduce the equations to a system of three ordinary differential
equations, from which we obtain detailed structures of solutions besides their existence. These solutions exhibit familiar
structures seen in hurricanes and tornadoes. They all have finite local energy and vorticity with well‐defined initial and
boundary values. These solutions include the one‐parameter family of explicit solutions reported in a recent article of ours.
(Accepted October 29, 1996) 相似文献
6.
Camillo de Lellis László Székelyhidi Jr. 《Archive for Rational Mechanics and Analysis》2010,195(1):225-260
We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements,
like the global and local energy inequalities. Using some techniques introduced in an earlier paper, we show that, for some
bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct,
in more than one space dimension, we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique. 相似文献
7.
Numerical simulation by a finite element method is used to examine the problem of the rotating flow of a viscoelastic fluid in a cylindrical vessel agitated with a paddle impeller. The mathematical model consists of a viscoelastic constitutive equation of Oldroyd B type coupled to the hydrodynamic equations expressed in a rotating frame. This system is solved by using an unsteady approach for velocity, pressure and stress fields. For Reynolds numbers in the range 0.1–10, viscoelastic effects are taken into account up to a Deborah number De of 1.33 and viscoelasticity and inertia cross-effects are studied. Examining the velocity and stress fields as well as the power consumption, it is found that their evolutions are significantly different for low and moderate inertia. These results confirm the trends of experimental studies and show the specific contribution of elasticity without interference of the pseudoplastic character found in actual fluids. 相似文献
8.
Gui-Qiang Chen Fei-Min Huang Tian-Yi Wang 《Archive for Rational Mechanics and Analysis》2016,219(2):719-740
A compactness framework is established for approximate solutions to subsonic-sonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework we develop applies for both non-homentropic and rotational flows. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional subsonic-sonic full Euler flows through infinitely long nozzles. 相似文献
9.
Xinyu He 《Journal of Mathematical Fluid Mechanics》2004,6(4):389-404
A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined byu(x,t) =U(y)/t*-t)
α, y = x/(t* ~ t)β,α,β> 0, whereU(y) satisfiesζU + βy. ΔU + U. VU + VP = 0,divU = 0. For α = β = 1/2, which is the limiting case of Leray’s self-similar Navier—Stokes equations, we prove the existence of(U,P) ε H3(Ω,R3 X R) in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that
solutions of the Euler equations blow up at a timet = t*, t* < +∞. 相似文献
10.
应用有限体积法求解Euler方程计算了真实飞机外形的跨音速大迎角绕流流场及空气动力特性。本方法能自动捕获大后掠机翼前缘脱体涡,侧缘涡,机身体涡以及它们间的相互干扰。计算的机翼压力分布及全机气动力系数与实验值符合良好。表明了本文所采用的计算网格生成技术及流场计算方法是有效、可行的。 相似文献
11.
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 C
2,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 u
2 with a value given at a point, and an algebraic equation on (ρ, u
1, u
2) 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.
This research was supported in part by the Zheng Ge Ru Foundation when Yin Huicheng was visiting The Institute of Mathematical
Sciences, The Chinese University of Hong Kong. Xin is supported in part by Hong Kong RGC Earmarked Research Grants CUHK-4028/04P,
CUHK-4040/06P, and Central Allocation Grant CA05-06.SC01. Yin is supported in part by NNSF of China and Doctoral Program of
NEM of China. 相似文献
12.
Gui-Qiang Chen Cleopatra Christoforou Yongqian Zhang 《Archive for Rational Mechanics and Analysis》2008,189(1):97-130
We establish the L
1-estimates for continuous dependence of entropy solutions to the full Euler equations away from the vacuum on two physical
parameters: the adiabatic exponent γ → 1 that passes from the non-isentropic to isothermal Euler equations and the Mach number
that passes from the compressible to incompressible Euler equations. Our analysis involves the effective approach developed
in our earlier work and additional new techniques that generalize this approach to the setting of the full Euler equations. 相似文献
13.
We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise C 1 smooth functions, under appropriate conditions on the downstream subsonic flows: (i) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (ii) the oblique transonic shocks attached to an infinite wedge; (iii) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing a point where the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure p and the tangent of the flow angle w = v/u obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the L ∞ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of the shock polar applied on the (maybe curved) shock-fronts. 相似文献
14.
Antoine Choffrut 《Archive for Rational Mechanics and Analysis》2013,210(1):133-163
In Dissipative Euler Flows and Onsager’s Conjecture. arxiv.1205.3626, preprint, 2012, De Lellis and Székelyhidi construct Hölder continuous, dissipative (weak) solutions to the incompressible Euler equations in the torus ${{\mathbb T}^3}$ . The construction consists of adding fast oscillations to the trivial solution. We extend this result by establishing optimal h-principles in two and three space dimensions. Specifically, we identify all subsolutions (defined in a suitable sense) which can be approximated in the H ?1-norm by exact solutions. Furthermore, we prove that the flows thus constructed on ${{\mathbb T}^3}$ are genuinely three-dimensional and are not trivially obtained from solutions on ${{\mathbb T}^2}$ . 相似文献
15.
16.
ANGELO lOLLO LUCA ZANNETTI 《International Journal of Computational Fluid Dynamics》2013,27(4):393-396
The shape optimization problem governed by the Euler equations is posed in a fixed reference plane. The boundary control is exerted by a parametric mapping from the physical plane to the reference fixed plane. The adjoint equations are derived in such fixed plane. By using this approach remeshing is unnecessary; furthermore, as in many practical applications the parametric mapping can be easily differentiated, the computation of mesh sensitivities is avoided. 相似文献
17.
We establish an existence theorem for entropy solutions to the Euler equations modeling isentropic compressible fluids. We
develop a new approach for constructing mathematical entropies for the Euler equations, which are singular near the vacuum.
In particular, we identify the optimal assumption required on the singular behavior on the pressure law at the vacuum in order to validate the two-term asymptotic expansion of the entropy kernel we proposed earlier. For more general pressure laws, we introduce a new multiple-term expansion based on the Bessel functions with suitable exponents, and we also identify the optimal assumption needed to validate
the multiple-term expansion and to establish the existence theory. Our results cover, as a special example, the density-pressure
law where and are arbitrary constants.
(Accepted June 10, 2002) Published online December 3, 2002
Communicated by C. M. DAFERMOS 相似文献
18.
In this paper we study the initial value problem of the incompressible Euler equations in
n
for initial data belonging to the critical Triebel-Lizorkin spaces, i.e., v
0
F
n+1
1,q
, q[1, ]. We prove the blow-up criterion of solutions in F
n+1
1,q
for n=2,3. For n=2, in particular, we prove global well-posedness of the Euler equations in F
3
1,q
, q[1, ]. For the proof of these results we establish a sharp Moser-type inequality as well as a commutator-type estimate in these spaces. The key methods are the Littlewood-Paley decomposition and the paradifferential calculus by J. M. Bony. 相似文献
19.
20.
We study the Euler equations for slightly compressible fluids, that is, after rescaling, the limits of the Euler equations
of fluid dynamics as the Mach number tends to zero. In this paper, we consider the general non-isentropic equations and general
data. We first prove the existence of classical solutions for a time independent of the small parameter. Then, on the whole
space ℝ
d
, we prove that the solution converges to the solution of the incompressible Euler equations.
Accepted December 1, 2000?Published online April 23, 2001 相似文献