Solution Of Navier Research Articles

Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock–Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for the high-order flux reconstruction/correction procedure via reconstruction solution of the unsteady Navier–Stokes equations. Pseudo-transient continuation is employed to solve the nonlinear equation at each stage of ESDIRK (excluding the first stage) and each step of BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method solver is employed with a low storage element-Jacobi preconditioner to solve the linear system at each stage of ROW and each pseudo time iteration of ESDIRK and BDF2. Several numerical experiments, including both laminar and turbulent flow simulations, are conducted to carry out the comparison. We observe that the multistage ROW2 and ESDIRK2 are more efficient than the multistep BDF2, and higher-order implicit time integrators are more efficient than lower-order ones. In general, the ESDIRK method allows a larger physical time step size for unsteady flow simulation than the ROW method when the element-Jacobi preconditioner is employed, especially for wall-bounded flows; and the ROW method can be more efficient than the ESDIRK method when the time step size is refined.

Semiflow selection for the compressible Navier\u2013Stokes system

Although the existence of dissipative weak solutions for the compressible Navier–Stokes system has already been established for any finite energy initial data, uniqueness is still an open problem. The idea is then to select a solution satisfying the semigroup property, an important feature of systems with uniqueness. More precisely, we are going to prove the existence of a semiflow selection in terms of the three state variables: the density, the momentum, and the energy. Finally, we will show that it is possible to introduce a new selection defined only in terms of the initial density and momentum; however, the price to pay is that the semigroup property will hold almost everywhere in time.

Anisotropic VMS solution of advection–diffusion problems by spectral approximation of sub-grid scales

In this article, we extend the Variational Multi-scale method with spectral approximation of the sub-scales to two-dimensional advection–diffusion problems. The spectral VMS method is cast for low-order elements as a standard VMS method with specific stabilized coefficients, that are anisotropic in the sense that they depend on two grid Péclet numbers, each associated to a component of the advection velocity. We compute the stabilized coefficients for grids of isosceles right triangles and right quadrilaterals, based upon the explicit computation of the eigen-pairs of the advection–diffusion operator with Dirichlet boundary conditions. To reduce the computing time, the stabilized coefficients are pre-computed at the nodes of a grid in an off-line step, and then interpolated by a fast procedure in the on-line computation.Finally, we present some numerical tests in order to compare our results with those provided by other stabilization coefficients, both isotropic and anisotropic ones. We consider tests for advection–diffusion equations with constant and anisotropic velocities, as well as tests in structured and unstructured meshes. We also test the method for the solution of Navier–Stokes equations. We observe a relevant accuracy gain for moderately large grid Péclet numbers for variable advection velocity.

Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

AbstractWe investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

Non-uniqueness of weak solutions to hyperviscous Navier\u2013Stokes equations: on sharpness of J.-L. Lions exponent

Using the convex integration technique for the three-dimensional Navier–Stokes equations introduced by Buckmaster and Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier–Stokes equations with fractional hyperviscosity (-Delta )^{theta }, whenever the exponent theta is less than Lions’ exponent 5/4, i.e., when theta < 5/4.

Experimental Study of the Performance of a Novel Vertical-Axis Wind Turbine

Basic equations for estimating the aerodynamic power captured by the Anderson vertical-axis wind turbine (AVAWT) are derived from a solution of Navier–Stokes (N–S) equations for a baroclinic inviscid flow. In a nutshell, the pressure difference across the AVAWT is derived from the Bernoulli’s equation—an upshot of the integration of the Euler’s momentum equation, which is the N–S momentum equation for a baroclinic inviscid flow. The resulting expression for the pressure difference across the AVAWT rotor is plotted as a function of the free-stream speed. Experimentally determined airstream speeds at the AVAWT inlet and outlet, coupled with corresponding free-stream speeds, are used in estimating the aerodynamic power captured. The aerodynamic power of the AVAWT is subsequently used in calculating its aerodynamic power coefficient. The actual power coefficient is calculated from the power generated by the AVAWT at various free-stream speeds and plotted as a function of the latter. Experimental results show that at all free-stream speeds and tip-speed ratios, the aerodynamic power coefficient of the AVAWT is higher than its actual power coefficient. Consequently, the power generated by the AVAWT prototype is lower than the aerodynamic power captured, given the same inflow wind conditions. Besides the foregoing, the main purpose of this experiment is to investigate the technical feasibility of the AVAWT. This proof of concept enables the inventor to commercialize the AVAWT.

Global Small Solutions of Heat Conductive Compressible Navier–Stokes Equations with Vacuum: Smallness on Scaling Invariant Quantity

In this paper, we consider the Cauchy problem to the heat conductive compressible Navier-Stokes equations in the presence of vacuum and with vacuum far field. Global well-posedness of strong solutions is established under the assumption, among some other regularity and compatibility conditions, that the scaling invariant quantity $\|\rho_0\|_\infty(\|\rho_0\|_3+\|\rho_0\|_\infty^2\|\sqrt{\rho_0}u_0\|_2^2)(\|\nabla u_0\|_2^2+\|\rho_0\|_\infty\|\sqrt{\rho_0}E_0\|_2^2)$ is sufficiently small, with the smallness depending only on the parameters $R, \gamma, \mu, \lambda,$ and $\kappa$ in the system. The total mass can be either finite or infinite.

Excitation of Self-Sustained Oscillations by a Flow of Liquid in a Cylindrical Duct with two Diaphragms

We study the flow of a viscous incompressible liquid in a cylindrical duct with two serial diaphragms by finding the numerical solutions of nonstationary Navier–Stokes equations. It is shown that, within a certain range of Reynolds numbers, the flow of liquid in the region between the diaphragms is characterized by the presence of an unstable shear layer. A series of sequential vortices formed in the shear layer generates self-sustained oscillations of the velocity profile in the opening of the second diaphragm. It is shown that the analyzed system also has two stable modes of periodic oscillations with different frequencies. These oscillations may play the role of an acoustic source in the duct.

Investigations of the vortex ring state on a helicopter main rotor using the URANS solver

PurposeThis paper aims to present the novel methodology of computational simulation of a helicopter flight, developed especially to investigate the vortex ring state (VRS) – a dangerous phenomenon that may occur in helicopter vertical or steep descent. Therefore, the methodology has to enable modelling of fast manoeuvres of a helicopter such as the entrance in and safe escape from the VRS. The additional purpose of the paper is to discuss the results of conducted simulations of such manoeuvres.Design/methodology/approachThe developed methodology joins several methods of computational fluid dynamics and flight dynamic. The approach consists of calculation of aerodynamic forces acting on rotorcraft, by solution of the unsteady Reynold-averaged Navier–Stokes (URANS) equations using the finite volume method. In parallel, the equations of motion of the helicopter and the fluid–structure-interaction equations are solved. To reduce computational costs, the flow effects caused by rotating blades are modelled using a simplified approach based on the virtual blade model.FindingsThe developed methodology of computational simulation of fast manoeuvres of a helicopter may be a valuable and reliable tool, useful when investigating the VRS. The presented results of conducted simulations of helicopter manoeuvres qualitatively comply with both the results of known experimental studies and flight tests.Research limitations/implicationsThe continuation of the presented research will primarily include quantitative validation of the developed methodology, with respect to well-documented flight tests of real helicopters.Practical implicationsThe VRS is a very dangerous phenomenon that usually causes a sudden decrease of rotor thrust, an increase of the descent rate, deterioration of manoeuvrability and deficit of power. Because of this, it is difficult and risky to test the VRS during the real flight tests. Therefore, the reliable computer simulations performed using the developed methodology can significantly contribute to increase helicopter flight safety.Originality/valueThe paper presents the innovative and original methodology for simulating fast helicopter manoeuvres, distinguished by the original approach to flight control as well as the fact that the aerodynamic forces acting on the rotorcraft are calculated during the simulation based on the solution of URANS equations.

The optimal decay rate of strong solution for the compressible Navier–Stokes equations with large initial data

In a recent paper (He et al., 2019), it is shown that the upper decay rate of global solution of compressible Navier–Stokes(CNS) equations converging to constant equilibrium state (1,0) in H1−norm is (1+t)−34(2p−1) when the initial data is large and belongs to H2(R3)∩Lp(R3)(p∈[1,2)). Thus, the first result in this paper is devoted to showing that the upper decay rate of the first order spatial derivative converging to zero in H1−norm is (1+t)−32(1p−12)−12. For the case of p=1, the lower bound of decay rate for the global solution of CNS equations converging to constant equilibrium state (1,0) in L2−norm is (1+t)−34 if the initial data satisfies some low frequency assumption additionally. In other words, the optimal decay rate for the global solution of CNS equations converging to constant equilibrium state in L2−norm is (1+t)−34 although the associated initial data is large.

Development of a Low-Friction Radial Shaft Seal: Using CFD Simulations to Optimise the Microstructured Sealing Lip

The sealing of shaft interfaces in machine housings against oil leakage is commonly realized by radial shaft seals, hence they are used millions of times in technical systems. However, with increasing speed they cause significant friction losses, which reduce the efficiency of the system significantly. In addition, the prevailing trend towards higher speed levels in electrified drive trains is already pushing sealing technology to its performance limits. Therefore, friction reduction offers a chance to extend the current performance limits and increase efficiency within existing applications. In this paper, a methodology for friction reduction is proposed, which is based on microstructuring and surface treatment of the seal sliding surface. Since experimental structural design is linked to high costs and time, a simulation-based method is proposed. Computational Fluid Dynamics (CFD) simulations are performed to analyze the influence of the structural geometry on local fluid flow. It is shown that for increasing sliding speeds, the analysis and the subsequent optimization of deterministic microstructures require the numerical solution of the complete Navier–Stokes equation in order to take inertial effects into account. Based on these results, an optimal geometric shape for the microstructure is found depending on the operating conditions.

Explicit unsteady Navier–Stokes solutions and their analysis via local vortex criteria

We construct a class of spatially polynomial velocity fields that are exact solutions of the planar unsteady Navier–Stokes equation. These solutions can be used as simple benchmarks for testing numerical methods or verifying the feasibility of flow-feature identification principles. We use examples from the constructed solution family to illustrate the deficiencies of streamline-based feature detection and those of the Okubo–Weiss criterion, which is the common two-dimensional version of the broadly used Q-, Δ-, λ2-, and λci-criteria for vortex-detection. Our planar polynomial solutions also extend directly to explicit, three-dimensional unsteady Navier–Stokes solutions with a symmetry.

Convergence of the two‐fluid compressible Navier–Stokes–Poisson system to the incompressible Euler equations

In this paper, we consider the combined quasi‐neutral and inviscid limits of the two‐fluid compressible Navier–Stokes–Poisson system in the unbounded domain with the ill‐prepared initial data. We prove that the weak solutions of the compressible Navier–Stokes–Poisson system converge to the strong solution of the incompressible Euler equation as long as the latter exists. Moreover, the convergence rates are also obtained.

Weak Solutions for Navier–Stokes Equations with Initial Data in Weighted $$L^2$$ Spaces

We show the existence of global weak solutions of the 3D Navier-Stokes equations with initial velocity in the weighted spaces L 2 w$\gamma$ , where w $\gamma$ (x) = (1 + |x|) --$\gamma$ and 0 < $\gamma$ $\le$ 2, using new energy controls. As application we give a new proof of the existence of global weak discretely self-similar solutions of the 3D Navier-Stokes equations for discretely self-similar initial velocities which are locally square inte-grable.

Existence of global weak solution for quantum Navier–Stokes system

In this paper, the barotropic compressible quantum Navier–Stokes equations with a density-dependent viscosity in a three-dimensional torus is studied. By introducing a cold pressure to handle the convection term, we prove the global-in-time existence of weak solutions to quantum Navier–Stokes equations for large data in the sense of standard definition.

New effective pressure and existence of global strong solution for compressible Navier–Stokes equations with general viscosity coefficient in one dimension

In this paper we prove the existence of a unique global strong solution for the Cauchy problem associated to the one dimensional Navier–Stokes equations with general degenerate viscosity coefficients. The cornerstone of the proof is the introduction of a new effective pressure which allows to obtain an Oleinik-type estimate for the so called effective velocity. In our proof we make use of additional regularizing effects on the velocity which requires to extend the techniques developed by Hoff for the constant viscosity case.

Global Smooth Solutions for 1D Barotropic Navier–Stokes Equations with a Large Class of Degenerate Viscosities

We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier-Stokes system with degenerate viscosity $\mu(\rho)=\rho^\alpha$. We establish that the smooth solutions have possibly two different far-fields, and the initial density remains positive globally in time, for the initial data satisfying the same conditions. In addition, our result works for any $\alpha>0$, i.e., for a large class of degenerate viscosities. In particular, our models include the viscous shallow water equations. This extends the result of Constantin-Drivas-Nguyen-Pasqualotto \cite[Theorem 1.5]{CDNP} (on the case of periodic domain) to the case where smooth solutions connect possibly two different limits at the infinity on the whole space.

Solutions of Navier–Stokes–Maxwell systems in large energy spaces

Large weak solutions to Navier–Stokes–Maxwell systems are not known to exist in their corresponding energy space in full generality. Here, we mainly focus on the three-dimensional setting of a classical incompressible Navier–Stokes–Maxwell system and—in an effort to build solutions in the largest possible functional spaces—prove that global solutions exist under the assumption that the initial velocity and electromagnetic fields have finite energy, and that the initial electromagnetic field is small in H ˙ s ( R 3 ) \dot H^s\left ({\mathbb R}^3\right ) with s ∈ [ 1 2 , 3 2 ) s\in \left [\frac 12,\frac 32\right ) . We also apply our method to improve known results in two dimensions by providing uniform estimates as the speed of light tends to infinity. The method of proof relies on refined energy estimates and a Grönwall-like argument, along with a new maximal estimate on the heat flow in Besov spaces. The latter parabolic estimate allows us to bypass the use of the so-called Chemin–Lerner spaces altogether, which is crucial and could be of independent interest.

Global existence of weak solutions for Navier–Stokes-BGK system

In this paper, we study the global well-posedness of a coupled system of kinetic and fluid equations. More precisely, we establish the global existence of weak solutions for Navier–Stokes–BGK system consisting of the BGK model of Boltzmann equation and incompressible Navier–Stokes equations coupled through a drag forcing term. This is achieved by combining weak compactness of the particle interaction operator based on Dunford–Pettis theorem, strong compactness of macroscopic fields of the kinetic part relied on velocity averaging lemma and a high order moment estimate, and strong compactness of the fluid part by Aubin–Lions lemma.

Bimodal vortex solutions on a sphere

Steady and time-periodic solutions of the two-dimensional Navier–Stokes equations are studied on a sphere under steady forcings represented by a single spherical harmonic function. This setting is similar to the Kolmogorov problem of the two-dimensional flows under a sinusoidal forcing on a doubly periodic domain (a flat torus), where either simple steady or time-periodic unimodal solutions have been found even at high Reynolds numbers depending on the forcing profiles. In our spherical case, we investigate the bifurcation structure and find either simple steady or time-periodic bimodal solutions, consisting of only two pairs of positive and negative vortices, depending on the forcing profiles at very high Reynolds numbers.