Two-dimensional Navier-Stokes Flow Research Articles

Numerical comparison of two-dimensional Navier-Stokes flows on the whole plane and the periodic domain

Numerical comparison of two-dimensional Navier-Stokes flows on the whole plane and the periodic domain

Some periodic orbits of chaotic motions for time-periodic forced two-dimensional Navier–Stokes flows

In this study, we study the two-dimensional Navier–Stokes flows with time-periodic external forces. Invariant solutions, including periodic orbits and relative periodic orbits, are extracted with the recurrent flow analysis, while low-dimensional projections based on the dynamic mode decomposition algorithm are used to reduce the cost of searching nearly recurrences. When the period of forces gets a constant increase, the flows change from the stable time-periodic state to oscillate and even turbulent flows. In all cases, one periodic orbit is identified near the initial stage. This orbit represents the stable/unstable base state, and the trajectories of vorticity fields are trapped inside it or escape away from it leading to oscillating/turbulent motions. For the oscillating flows, periodic orbits without any symmetries play the role that the flows visit them and then move away from them to other orbits. In addition, for a moderate period of forces, a bursting phenomenon occurs and the state of oscillating flows turns to turbulent flows with the rapid increase in energy. For the turbulent motions, one unstable periodic, which qualitatively represents the shapes of a large vortex dipole that exists in the turbulent motions, is obtained. Its statistical significance is shown by the frequency of that flows visit it.

English

We examine the viscous effects of slip boundary conditions for the model describing two-dimensional Navier-Stokes flows in a plane diffuser. It is shown that the velocity profile is related to a half period shifted Weierstrass function with two parameters. This allows to approximate the explicit solution by a Taylor series expansion with two new micro- parameters, that can be measured in physical experiments. It is shown that the assumption for no-slip boundary conditions is stable in the sense that a small perturbation of the boundary values result in a small perturbation in the solutions.

Fast divergence-conforming reduced basis methods for steady Navier–Stokes flow

Reduced-basis methods (RB methods or RBMs) form one of the most promising techniques to deliver numerical solutions of parametrized PDEs in real-time with reasonable accuracy. For incompressible flow problems, RBMs based on LBB stable velocity–pressure spaces do not generally inherit the stability of the underlying high-fidelity model and, instead, additional stabilization techniques must be introduced. One way of bypassing the loss of LBB stability in the RBM is to inflate the velocity space with supremizer modes. This however deteriorates the performance of the RBM in the performance-critical online stage, as additional DOFs must be introduced to retain stability, while these DOFs do not effectively contribute to accuracy of the RB approximation. In this work we consider a velocity-only RB approximation, exploiting a solenoidal velocity basis. The solenoidal reduced basis emerges directly from the high-fidelity velocity solutions in the offline stage. By means of Piola transforms, the solenoidality of the velocity space is retained under geometric transformations, making the proposed RB method suitable also for the investigation of geometric parameters. To ensure exact solenoidality of the high-fidelity velocity solutions that constitute the RB, we consider approximations based on divergence-conforming compatible B-splines. We show that the velocity-only RB method leads to a significant improvement in computational efficiency in the online stage, and that the pressure solution can be recovered a posteriori at negligible extra cost. We illustrate the solenoidal RB approach by modeling steady two-dimensional Navier–Stokes flow around a NACA0015 airfoil at various angles of attack.

Unimodal patterns appearing in the two-dimensional Navier–Stokes flows under general forcing at large Reynolds numbers

We consider the Navier–Stokes equations in 2D flat tori. With various external forces and aspect ratios of the tori, we compute steady-states and time-periodic solutions at large Reynolds numbers. Our external forces are more general than those considered previously. We demonstrate, by numerical experiments, that certain solutions having streamline patterns of simple topology appear at large Reynolds numbers.

Fast chemical reaction in two-dimensional Navier-Stokes flow: Initial regime

This paper studies an infinitely fast bimolecular chemical reaction in a two-dimensional biperiodic Navier-Stokes flow. The reactants in stoichiometric quantities are initially segregated by infinite gradients. The focus is placed on the initial stage of the reaction characterized by a well-defined one-dimensional material contact line between the reactants. Particular attention is given to the effect of the diffusion κ of the reactants. This study is an idealized framework for isentropic mixing in the lower stratosphere and is motivated by the need to better understand the effect of resolution on stratospheric chemistry in climate-chemistry models. Adopting a Lagrangian straining theory approach, we relate theoretically the ensemble mean of the length of the contact line, of the gradients along it, and of the modulus of the time derivative of the space-average reactant concentrations (here called the chemical speed) to the joint probability density function of the finite-time Lyapunov exponent λ with two times τ and τ[over ̃]. The time 1/λ measures the stretching time scale of a Lagrangian parcel on a chaotic orbit up to a finite time t, while τ measures it in the recent past before t, and τ[over ̃] in the early part of the trajectory. We show that the chemical speed scales like κ(1/2) and that its time evolution is determined by rare large events in the finite-time Lyapunov exponent distribution. The case of smooth initial gradients is also discussed. The theoretical results are tested with an ensemble of direct numerical simulations (DNSs) using a pseudospectral model.

Enhancement of wind turbine aerodynamic performance by a numerical optimization technique

Sectional aerodynamic design optimization was performed to enhance the aerodynamic performance of horizontal axis wind turbine rotor blades based on a computational fluid dynamics technique. The proposed sectional optimization framework consists of airfoil section contour modeling by the PARSEC shape function and a modified feasible direction search algorithm. To enhance the aerodynamic performance of wind turbine rotor blades, the objective of the design framework was set to maximize the lift-over-drag ratio for each design section. A two-dimensional Navier-Stokes flow solver coupled with a transition turbulence model was used to evaluate the aerodynamic performance during the iterative design optimization procedure. The sectional flow conditions were extracted from the flow of a three-dimensional rotor blade configuration. The design framework was applied to the National Renewable Energy Laboratory Phase VI rotor blade. The design optimization was conducted at nine spanwise sections of the rotor blade. To validate the present methodology, the aerodynamic performances of the original baseline rotor and the rotor after the design optimization were compared by using a three-dimensional Navier-Stokes flow solver. It was found that approximately 11% of torque enhancement was achieved after the aerodynamic shape design optimization.

Quantification of topological changes of vorticity contours in two-dimensional Navier-Stokes flow

A characterization of reconnection of vorticity contours is made by direct numerical simulations of the two-dimensional Navier-Stokes flow at a relatively low Reynolds number. We identify all the critical points of the vorticity field and classify them by solving an eigenvalue problem of its Hessian matrix on the basis of critical-point theory. The numbers of hyperbolic (saddles) and elliptic (minima and maxima) points are confirmed to satisfy Euler's index theorem numerically. Time evolution of these indices is studied for a simple initial condition. Generally speaking, we have found that the indices are found to decrease in number with time. This result is discussed in connection with related works on streamline topology, in particular, the relationship between stagnation points and the dissipation. Associated elementary procedures in physical space, the merging of vortices, are studied in detail for a number of snapshots. A similar analysis is also done using the stream function.

Mechanisms Influencing the Efficiency of Oscillating Airfoil Propulsion

A NACA0012 airfoil undergoing pitching and plunging motion at Re = 20,000-40,000 was simulated using a two-dimensional Navier-Stokes flow solver. Results were compared with experimental measurements in the literature and those from an inviscid analytical method and an unsteady panel method code. Although the peak in propulsive efficiency with Strouhal number demonstrated in the experimental results was predicted by the inviscid methods, it was found to be significantly modified by leading-edge vortex shedding and viscous drag at low Strouhal numbers. The occurrence and influence of vortex shedding is controlled by both the motion of the airfoil (amplitudes and phases of plunging and pitching) and the flapping frequency, which limits the time available for vortex formation and convection over the airfoil surface. Thus, Strouhal number alone is insufficient to characterize the efficiency of flapping-foil propulsion.

On Spatial Decay Estimates for Derivatives of Vorticities with Application to Large Time Behavior of the Two-Dimensional Navier–Stokes Flow

In this paper we establish spatial decay estimates for derivatives of vorticities solving the two-dimensional vorticity equations equivalent to the Navier–Stokes equations. As an application we derive asymptotic behaviors of derivatives of vorticities at time infinity. It is well known by now that the vorticity behaves asymptotically as the Oseen vortex provided that the initial vorticity is integrable. We show that each derivative of the vorticity also behaves asymptotically as that of the Oseen vortex.

Streamline topologies near a non-simple degenerate critical point close to a stationary wall using normal forms

Streamline patterns and their bifurcations in two-dimensional Navier–Stokes flow of an incompressible fluid near a non-simple degenerate critical point close to a stationary wall are investigated from the topological point of view by considering a Taylor expansion of the velocity field. Using a five-order normal form approach we obtain a much simplified system of differential equations for the streamlines. Careful analysis of the simplified system gives possible bifurcations for non-simple degeneracies of codimension three. Three heteroclinic connections from three on-wall separation points merge at an in-flow saddle point to produce two separation bubbles with opposite rotations which occur only near a non-simple degenerate critical point. The theory is applied to the patterns and bifurcations found numerically in the studies of Stokes flow in a double-lid-driven rectangular cavity.

Polymer Transport in Random Flow

The dynamics of polymers in a random smooth flow is investigated in the framework of the Hookean dumbbell model. The analytical expression of the time-dependent probability density function of polymer elongation is derived explicitly for a Gaussian, rapidly changing flow. When polymers are in the coiled state the pdf reaches a stationary state characterized by power-law tails both for small and large arguments compared to the equilibrium length. The characteristic relaxation time is computed as a function of the Weissenberg number. In the stretched state the pdf is unstationary and exhibits multiscaling. Numerical simulations for the two-dimensional Navier-Stokes flow confirm the relevance of theoretical results obtained for the delta-correlated model.

Spontaneous Formation of Zonal Currents in Two-Dimensional Navier-Stokes Flow on a Rotating Hemisphere

Spontaneous Formation of Zonal Currents in Two-Dimensional Navier-Stokes Flow on a Rotating Hemisphere

PHYSALIS: a new method for particle simulation: Part II: two-dimensional Navier–Stokes flow around cylinders

This paper presents a new approach to the direct numerical simulation of particle flows. The basic idea is to use a local analytic representation valid near the particle to “transfer” the no-slip condition from the particle surface to the adjacent grid nodes. In this way the geometric complexity arising from the irregular relation between the particle boundary and the underlying mesh is avoided and fast solvers can be used. The results suggest that the computational effort increases very slowly with the number of particles so that the method is efficient for large-scale simulations. The focus here is on the two-dimensional case (cylindrical particles), but the same procedure, to be developed in forthcoming papers, applies to three dimensions (spherical particles). Several extensions are briefly discussed.

Time-accurate Navier–Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier–Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss–Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart–Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade–vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.

Global Existence of Two-Dimensional Navier—Stokes Flow with Nondecaying Initial Velocity

A global-in-time unique smooth solution is constructed for the Cauchy problem of the Navier—Stokes equations in the plane when initial velocity field is merely bounded not necessary square-integrable. The proof is based on a uniform bound for the vorticity which is only valid for planar flows. The uniform bound for the vorticity yields a coarse globally-in-time a priori estimate for the maximum norm of the velocity which is enough to extend a local solution. A global existence of solution for a q-th integrable initial velocity field is also established when \( q > 2 \).

Time evolution of the eddy viscosity in two-dimensional navier-stokes flow

The time evolution of the eddy viscosity associated with an unforced two-dimensional incompressible Navier-Stokes flow is analyzed by direct numerical simulation. The initial condition is such that the eddy viscosity is isotropic and negative. It is shown by concrete examples that the Navier-Stokes dynamics stabilizes negative eddy viscosity effects. In other words, this dynamics moves monotonically the initial negative eddy viscosity to positive values before relaxation due to viscous term occurs.

A high-order adaptive numerical method for recirculating flows at large Reynolds number

A numerical method is constructed for two-dimensional Navier–Stokes flows in a circular domain. Adaptive computational grids are employed to resolve the boundary layer and the boundary vorticity values are obtained in high order of accuracy. Numerical results show a quite good agreement with the asymptotic expansion calculation (Kim, SIAM J. Appl. Math. 58(5) (1998) 1394–1413). Extension to general shape domains is briefly addressed.

Batchelor–Wood formula for negative wall velocity

Extension of a formula giving the constant vorticity of the inviscid limit of steady two-dimensional Navier–Stokes flows in a circular domain is considered for slightly negative wall data. In the marginal separation case the formula is shown to be applicable with relatively small error.

Asymptotic states of decaying turbulence in two-dimensional incompressible flows.

We investigate the relaxation of a strongly turbulent fluid to metastable states, in times much shorter than the dissipation time scale. Several simulations of decaying two-dimensional Navier-Stokes flows were performed, which show the relaxation to organized states dominated by coherent vortex structures of length scales comparable to the size of the system. In the case of periodic boundary conditions, the organized state is characterized by a strong correlation between vorticity and stream function, the second of which satisfies a sinh-Poisson equation quite accurately. However, in the case of free-slip boundary conditions the relaxed state does not display any significant correlation between its vorticity and its stream function. Notwithstanding, in both cases the role of nonlinearities is found to be essential even at these late stages of the evolution. However, we show that even severe truncations of a few (short wave number) nonlinearly coupled Fourier modes provide an accurate description of the long-term dynamics of the fluid. Therefore the dynamics of the flow in these metastable states is somewhere in between a strong turbulent regime and a (mostly linear) dissipative relaxation stage. \textcopyright{} 1996 The American Physical Society.