Navier-Stokes Equations In Space Research Articles

Well-chosen weak solutions of the instationary Navier-Stokes system and their uniqueness

We clarify the notion of well-chosen weak solutions of the instationary Navier-Stokes system recently introduced by the authors and P.-Y. Hsu in the article {\em Initial values for the Navier-Stokes equations in spaces with weights in time, Funkcialaj Ekvacioj} (2016). Well-chosen weak solutions have initial values in $L^{2}_{\sigma}(\Omega)$ contained also in a quasi-optimal scaling-invariant space of Besov type such that nevertheless Serrin's Uniqueness Theorem cannot be applied. However, we find universal conditions such that a weak solution given by a concrete approximation method coincides with the strong solution in a weighted function class of Serrin type.

Direct numerical simulation of high pressure turbulent lean premixed CH4/H2 – Air slot flames

A numerical study on lean turbulent premixed methane/hydrogen-air slot flames at high pressures is conducted through two-dimensional Direct Numerical Simulation (DNS). A single equivalence ratio flame at Φ=0.7 and 50% of hydrogen content is explored for three different pressures (0.1,1,4 MPa respectively). Due to the decreased kinematic viscosity with increasing pressure, the turbulent Reynolds numbers increase with an associated decrease of the smallest turbulence scales that wrinkle the flame front.Finite difference schemes were adopted to solve the compressible Navier-Stokes equations in space (compact sixth-order in staggered formulation) and time (third-order Runge-Kutta). Accurate molecular transport properties were also taken into account and, a detailed skeletal chemical mechanism for methane/hydrogen-air combustion, consisting of 23 transported species and 124 elementary reactions, was used.A general description of the three flames is provided, evidencing their macroscopic differences in terms of turbulent displacement speed, flame surface areas and mean flame brush thickness. Furthermore, topological features of the flames are explored by analyzing the probability density functions of several quantities: curvature, curvature shape factor, alignment between vorticity and principal strain rate vectors with flame surface normal, displacement speed and its components. Finally the differential diffusivity effect on the local equivalence ratio in the three flames is investigated showing a strong effect of the turbulent flame thickness of the high pressure flames on light species differential diffusivity.

Initial Values for the Navier-Stokes Equations in Spaces with Weights in Time

We consider the nonstationary Navier-Stokes system in a smooth bounded domain Ω ⊂ R3 with initial value u0 ∈ Lσ(Ω). It is an important question to determine the optimal initial value condition in order to prove the existence of a unique local strong solution satisfying Serrin’s condition. In this paper, we introduce a weighted Serrin condition that yields a necessary and sufficient initial value condition to guarantee the existence of local strong solutions u(·) contained in the weighted Serrin class ∫ T 0 (τ ‖u(τ)‖q) dτ < ∞ with 2s + 3 q = 1 − 2α, 0 < α < 1 2 . Moreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class. 2010 Mathematics Subject Classification: 35Q30; 76D05

Parameter Optimization Research on Lift-enhancing of Multi-element Airfoil Using Air-blowing

Based on the response surface methodology and the grid method combining with numerical simulation of Reynolds-averaged Navier-Stokes equations,the lift-enhancing optimizations of three-element airfoil parameters and its air-blowing flow control parameters are carried out. The finite volume method is adopted to discretize the Reynolds-averaged Navier-Stokes equations in space, the second-order upwind scheme is used in time advance, and SST k – ω turbulence model is used to compute turbulence viscosity. The objective function is obtained with the response surface methodology,and the optimal solution is made using nonlinear programming with the grid method. On the base of multi-element airfoil parameter optimization, the influencing parameters ofair-blowing effect are analyzed and selected, the optimization design variables are determinedas the air-blowing slotlocations, angles and air-blowing momentumcoefficients, and lift-enhancing aerodynamic optimization ofair-blowing flow control parameters on multi-element airfoil is completed. It is shown from the results that the optimization method combining response surface methodology with the grid method is fit for parameter optimizationof multi-element airfoil and its flow control with air-blowing on the flap forlift-enhancing; the flow around the multi-element airfoil can be well controlled and its aerodynamic performance is greatly improved, then the maximum lift coefficient achieves 4.9589which is usually the maximum lift coefficient of the five-element airfoil; after optimization the air-blowing slot anglesare of about 20° on the trailing edge flapsand the air-blowing momentum coefficientsof slotsexist an optimum value.

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp,λ*(ℝn) on the basis of Lorentz space Lp,∞ = Lp*(ℝn) (in particular, Mp,0*(ℝn) = Lp,∞, if p > 1), and study some fundamental properties of them; Second, we prove that the heat operator U(t) = etΔ. and Calderon-Zygmund singular integral operators are bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato’s method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp,λ*(ℝn) (1 < p ⩽ n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp,n−p*(ℝn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato’s results.

Global solvability of the Navier-Stokes equations in spaces based on sum-closed frequency sets

We prove existence of global regular solutions for the $3$D Navier-Stokes equations with (or without) Coriolis force for a class of initial data $u_0$ in the space ${{{\mathrm{FM}}}_{\sigma,\delta}}$, i.e., for functions whose Fourier image ${\widehat{u}}_0$ is a vector-valued Radon measure and that are supported in sum-closed frequency sets with distance $\delta$ from the origin. In our main result we establish an upper bound for admissible initial data in terms of the Reynolds number, uniform on the Coriolis parameter $\Omega$. In particular this means that this upper bound is linearly growing in $\delta$. This implies that we obtain global-in-time regular solutions for large (in norm) initial data $u_0$ which may not decay at space infinity, provided that the distance $\delta$ of the sum-closed frequency set from the origin is sufficiently large.

Note: A family of analytical solutions of the Navier-Stokes equations in space

The author gives exact analytical solutions of the Navier-Stokes equations for steady and unsteady laminar plane flows of the second kind near a plane z = 0, in the form of power series in z (up to a power of 6) of the three Cartesian velocity components and of the piezometric gradient, whose coefficients depend on arbitrary values of the velocity gradients in the x- and z-directions and of the piezometric function and its y-gradient at the boundary. An important application is that of flow in straight flat-bottomed channels or conduits of constant cross-section with transverse flows.