The inviscid limit for long time statistics of the one-dimensional
stochastic Ginzburg-Landau equation
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We consider the long time statistics of a one-dimensional stochastic Ginzburg-Landau equation with cubic nonlinearity while being subjected to random perturbations via an additive Gaussian noise. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we find that the dynamics is attractive toward the unique invariant probability measure with a polynomial rate that is independent of the vanishing viscosity. This relies on a coupling technique exploiting a Foias-Prodi argument specifically tailored to the system. Then, in the inviscid regime, we show that the sequence of invariant measures converges toward the invariant measure of the stochastic Schr\"odinger equation in a suitable Wasserstein distance. Together with the uniform polynomial mixing, we obtain the validity of the inviscid limit for the solutions on the infinite time horizon with a log log rate.Keywords:
Inviscid flow
In this paper, we are concerned with global strong solutions and large time behavior for some inviscid Oldroyd-B models. We first establish the energy estimate and B-K-M criterion for the 2-D co-rotation inviscid Oldroyd-B model. Then, we obtain global strong solutions with large data in Sobolev space by proving the boundedness of vorticity. As a corollary, we prove global existence of the corresponding inviscid Hooke model near equilibrium. Furthermore, we present global existence for the 2-D co-rotation inviscid Oldroyd-B model in critical Besov space by a refined estimate in Besov spaces with index $0$. Finally, we study large time behaviour for the noncorotation inviscid Oldroyd-B model. Applying the Fourier splitting method, we prove the $H^1$ decay rate for global strong solutions constructed by T. M. Elgindi and F. Rousset.
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This paper extends a previous modified axisymmetric analog method to predict heating rates on hypersonic vehicles in conjunction with inviscid computational fluid dynamic (CFD) codes which can provide more accurate inviscid solutions and are suited for complex configurations. The major problem is the heating anomalies encountered in the stagnation region, as the quality of the heating solution is very sensitive to the quality of the inviscid solution in the high-gradient stagnation region. To overcome this problem, a hybrid approach is developed to eliminate noise in the inviscid solution in the near-stagnation region by recalculating a noise-free inviscid solution in that region using an engineering method. As a result, there is no need to spend much effort to compute a high-quality inviscid solution in the near-stagnation region when solving the inviscid solution using an inviscid CFD code, thus significantly reducing run times. The proposed method is applied to several typical hypersonic vehicles and compared with existing approaches to validate its effectiveness. The results show that the proposed method can predict surface heating rates on complex configurations with reasonable accuracy but requires much shorter computational times.
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This chapter introduces inviscid flow and potential flow method. Characteristics of inviscid flow is introduced and the rationality of neglecting viscosity in many actual flow cases is discussed. Then the characteristics of rotational flow for inviscid flow is discussed. The three factors that may cause a fluid to change from irrotational to rotational are enumerated and explained, namely the viscous force, baroclinic flow, and non-conservative body force. For irrotational flow, velocity potential is introduced and several elementary flows are taken as an example to illustrate the computational methods for planar potential flow theory. In the end, complex potential is briefly introduced.
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Abstract : A viscous inviscid interaction method for three dimensional flows, in which the partially parabolic Reynolds equations are coupled with an inviscid flow solution procedure in an interactive and iterative manner, is applied to two simple three dimensional bodies for which experimental data are available for comparison. The relative merits of interactive and global solution procedures are evaluation by comparing the viscous inviscid interaction solutions with noninteractive large domain solutions of only the viscous flow equations. Both methods yield satisfactory results, although the interaction solutions are found to be computationally more efficient for the cases considered. Keywords: Thick Three Dimensional Layer; Viscous Inviscid Interaction; Partially parabolic Equations; Computational Fluid Dynamics.
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A criterion of flow instability and turbulent transition in curved shear flows is obtained via the analysis of energy equilibrium of fluid particles. Then, three important theorems for fluid stability are deduced: (1) Potential flow (inviscid and irrotational) is stable. (2) Inviscid rotational (inviscid and nonzero vorticity) flow is unstable. (3) Velocity profile with an inflectional point is unstable when there is no work input or output to the system, for both inviscid and viscous flows. These theorems have significant implications for vortex breakdown and flow transition.
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