Two-dimensional Navier-Stokes Equations Research Articles

A novel variational method for 3D viscous flow in flow channel of turbomachines based on differential geometry

ABSTRACT This paper presents a novel variational method for treating three-dimensional rotational Navier-Stokes equations in flow channel of turbomachines. The proposed method establishes a new semi-geodesic coordinate system on the central surface of blades. From the perspective of differential geometry, the system under concern is split into a set of membrane operator equations on two-dimensional manifolds and bending operator equations along hub circle. The third variable of the new coordinate system is approximated by the central difference scheme. We derive a new formulation of two-dimensional Navier-Stokes equations with three components on the manifolds in the variational sense. The well-posedness of the proposed variational formulation is rigorously justified.

Modeling and analysis of fractional neutral disturbance waves in arterial vessels

The behavior of neutral disturbance in arterial vessels has attracted more and more attention in recent decades because it carries some important information which can be applied to predict and diagnose related heart disease, such as arteriosclerosis and hypertension, etc. Because of the complexity of blood flow in arteries, it is very necessary to construct accurate mathematical model and analyze the mechanical behavior of neutral disturbance in arterial vessels. In this paper, start from the basic equations of blood flow and the two-dimensional Navier–Stokes equation, the vorticity equation describing the disturbance flow is presented. Then, by use of multi-scale analysis and perturbation expansion method, the ZK equation is put forward which can reflect the behavior of the neutral perturbation flow in arterial vessels. Compared with the traditional KdV model, the model established in the paper can show the propagation of the disturbance flow in the radius direction. Furthermore, the time-fractional ZK equation is derived by semi-inverse method and Agrawal’s method, which is more convenient and accurate for discussing the feature of neutral disturbance in arterial vessels and can provide more information for analyzing some related heart disease. Meanwhile, with the help of the modified extended tanh method, the above mentioned equation is solved. The results show that neutral disturbance exists in arterial vessels and propagates in the form of solitary waves. By calculating, we find the relation of the stroke volume with vascular radius, blood flow velocity as well as the fractional order parameterα, which is very meaningful for preventing and treating related heart disease because the stroke volume is closely linked with heart disease.

Calculation of Nonequilibrium Radiation of Shock Waves in the Air Using Two Models

The results of the numerical modeling of the spectral radiating capacity of shock-wave-heated air are presented. Two models of nonequilibrium radiation emitted from the relaxation zone behind the strong shock wave front are used. The first model is based on Euler equations and formulated for the one-dimensional problem of kinetic processes behind the shock front. The second model is based on the two-dimensional Navier-Stokes equations for a viscous shock layer on the nose surface of a blunt body in a hypersonic flow. The problem of the interpretation of the Fire-II flight experiment are discussed using the above-mentioned models.

Mathematical investigation of pressure pulsations characteristics and natural acoustic frequencies in the gas-dynamic channel

Paper presents a numerical simulation of the occurrence of flow instability and pressure self-oscillations for a complex configuration of the gas-dynamic tract in combustion chamber. Unsteady axisymmetric two-dimensional Navier-Stokes equations are used for mathematical modelling of compressible one-phase medium. To simulate turbulence, the k-ε and LES models were used. Fast Fourier Transform (FFT) determined the frequency spectrum of pressure pulsations in the combustion chamber. It is shown that in the case of a simple geometry of the free gas cavity in combustion chamber, both models of turbulence make it possible to determine the spectrum of the natural acoustic frequencies. Using the LES model in the case of complex geometry makes it possible to predict the hydrodynamic structure of a flow accurately. The flow, in this case, has an intensive vortex generation. Formation of small-scale vortex occurs in the near-wall regions and large eddies in the core of a flow. Frequency of large eddies formation can be combined with the natural acoustic frequencies of combustion chamber and can affect the amplitude of pressure pulsations.

A mixed virtual element method for the Navier–Stokes equations

A mixed virtual element method (mixed-VEM) for a pseudostress-velocity formulation of the two-dimensional Navier–Stokes equations with Dirichlet boundary conditions is proposed and analyzed in this work. More precisely, we employ a dual-mixed approach based on the introduction of a nonlinear pseudostress linking the usual linear one for the Stokes equations and the convective term. In this way, the aforementioned new tensor together with the velocity constitute the only unknowns of the problem, whereas the pressure is computed via a postprocessing formula. In addition, the resulting continuous scheme is augmented with Galerkin type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition, all them multiplied by suitable stabilization parameters, so that the Banach fixed-point and Lax–Milgram theorems are applied to conclude the well-posedness of the continuous and discrete formulations. Next, we describe the main VEM ingredients that are required for our discrete analysis, which, besides projectors commonly utilized for related models, include, as the main novelty, the simultaneous use of virtual element subspaces for [Formula: see text] and [Formula: see text] in order to approximate the velocity and the pseudostress, respectively. Then, the discrete bilinear and trilinear forms involved, their main properties and the associated mixed virtual scheme are defined, and the corresponding solvability analysis is performed using again appropriate fixed-point arguments. Moreover, Strang-type estimates are applied to derive the a priori error estimates for the two components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure. As a consequence, the corresponding rates of convergence are also established. Finally, we follow the same approach employed in previous works by some of the authors and introduce an element-by-element postprocessing formula for the fully computable pseudostress, thus yielding an optimally convergent approximation of this unknown with respect to the broken [Formula: see text]-norm.

Large deviations for the two-dimensional stochastic Navier–Stokes equation with vanishing noise correlation

We are dealing with the validity of a large deviation principle for the two-dimensional Navier-Stokes equation, with periodic boundary conditions, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\e$ and $\d(\e)$, respectively, with $0<\e, \d(\e)<<1$. Depending on the relationship between $\e$ and $\d(\e)$ we will prove the validity of the large deviation principle in different functional spaces.

Numerical simulation of incompressible gusty flow past a circular cylinder

Flow around a circular cylinder under the influence of a one dimensional gust is investigated in the present paper. In the present study the gusty flow which impinges on the cylinder is produced by superimposing a single harmonic sinusoidal transverse perturbation velocity component on uniform flow. A gust source region is defined for creating the perturbation velocity. Streamlines, vorticity and iso-velocity contours are used to physically interpret the flow field features. Temporal variations of force coefficients and stagnation-separation points on the cylinder surface are studied. Frequency spectrum of force coefficients reveal interesting facts about the variation of energy content in the gust frequency peak in comparison with vortex shedding frequency peak as a function of Reynolds number (Re) and the angular frequency of the gust (ω). Wherever possible, an attempt has been made to compare the flow features between uniform and gusty flow conditions. The present numerical work has been performed using ‘CFRUNS’ – a numerical scheme developed by Harichandan and Roy (2010) for solving incompressible two-dimensional Navier-Stokes equations using unstructured grid.

Dynamic analysis on fluid-structure interaction of an elastically mounted square cylinder at low Reynolds numbers

Fluid-structure interaction of an elastically mounted rigid square cylinder of low non-dimensional mass immersed in fluid flow is investigated numerically in the Reynolds numbers range of 60≤Re≤250. The square cylinder is allowed to freely vibrate only in the transverse direction perpendicular to the incoming flow. The two-dimensional incompressible Reynolds-averaged Navier-Stokes equations are solved by the finite volume method for the fluid flow. The equation of motion is solved for the vibration of the square cylinder. The results show that abrupt both the frequency and amplitude ratios curves experience sudden change at Re=90 and 168, which marks the onset of the lock-in and galloping regime, respectively. Thus, four regimes can be divided in the present study and which are the initial regime, the lock-in regime, the lower branch and galloping regime. A local peak value is observed in the force coefficients curve and the maximum value is reached at Re=231. It is found that the peak oscillating amplitude of the lock-in regime is reached at 0.22D and the width of lock-in region with sharp corner is very narrow. In the galloping regime, the peak amplitude of the oscillating square cylinder is close to 0.70D at Re=231. Typical 2S vortex structure is observed in the initial regime, the lock-in regime, and the lower branch. While in the galloping regime, 2P, 2P+2S and more complicated vortex patterns are observed as Re increases.

Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations

In this paper, we investigate the asymptotic behavior of the solutions of the two-dimensional stochastic Navier-Stokes equations via the stationary Wong-Zakai approximations given by the Wiener shift. We prove the existence and uniqueness of tempered pullback attractors for the random equations of the Wong-Zakai approximations with a Lipschitz continuous diffusion term. Under certain conditions, we also prove the convergence of solutions and random attractors of the approximate equations when the step size of approximations approaches zero.

Solving the stationary Navier–Stokes equations by using Taylor meshless method

The numerical implementation of the Taylor Meshless Method (TMM) for solving the two-dimensional Navier–Stokes equations is considered in this paper, where the TMM is a true meshless integration-free method based on Taylor series. The performance of the proposed numerical technique is evaluated by three benchmark tests. In the first case, the exact solution ia a high degree polynomial that permits to check the consistency and high accuracy of the proposed numerical method. Then the comparison with the analytical solution of the Kovasznay flow for the Reynolds number in the range [0,1000] is considered. It turns out that the TMM works well for large values of Reynolds number and it converges rapidly with the degree of Taylor series (p-convergence) and the number of sub-domains (h-convergence). Finally the square lid-driven cavity flow problems for the Reynolds number up to Re=1000 are solved and excellent agreements with corresponding benchmark solutions are found.

Chordwise wing flexibility may passively stabilize hovering insects.

Insect wings are flexible, and the dynamically deforming wing shape influences the resulting aerodynamics and power consumption. However, the influence of wing flexibility on the flight dynamics of insects is unknown. Most stability studies in the literature consider rigid wings and conclude that the hover equilibrium condition is unstable. The rigid wings possess an unstable oscillatory mode mainly due to their pitch sensitivity to horizontal velocity perturbations. Here, we show that a flapping wing flyer with flexible wings exhibits stable hover equilibria. The free-flight insect flight dynamics are simulated at the fruit fly scale in the longitudinal plane. The chordwise wing flexibility is modelled as a linear beam. The two-dimensional Navier-Stokes equations are solved in a tight fluid-structure integration scheme. For a range of wing flexibilities similar to live insects, all eigenvalues of the system matrix about the hover equilibrium have negative real parts. Flexible wings appear to stabilize the unstable mode by passively deforming their wing shape in the presence of perturbations, generating significantly more horizontal velocity damping and pitch rate damping. These results suggest that insects may passively stabilize their hover flight via wing flexibility, which can inform designs of synthetic flapping wing robots.

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier-Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (1+1)-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.

Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows

We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : $$ u^\nu(t,x,y) \, = \, \big (U^E(t,y) + U^{BL}(t,\frac{y}{\sqrt{\nu}})\,, \, 0 \big )\, , \quad 0<\nu \ll 1\,. $$ We show that if $U^{BL}$ is monotonic and concave in $Y = y /\sqrt{\nu}$ then $u^\nu$ is stable over some time interval $(0,T)$, $T$ independent of $\nu$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in $x$ and $y$). Moreover, in the case where $U^{BL}$ is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr-Sommerfeld operator.

Strong $L^2$ convergence of time numerical schemes for the stochastic two-dimensional Navier–Stokes equations

Abstract We prove that some time discretization schemes for the two-dimensional Navier–Stokes equations on the torus subject to a random perturbation converge in $L^2(\varOmega )$. This refines previous results that established the convergence only in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier–Stokes equations and convergence of a localized scheme we can prove strong convergence of fully implicit and semiimplicit temporal Euler discretizations and of a splitting scheme. The speed of the $L^2(\varOmega )$ convergence depends on the diffusion coefficient and on the viscosity parameter.

Forced convection heat transfer from a circular cylinder with a flexible fin

Forced convection heat transfer from a circular cylinder with a flexible fin in laminar flow with Re = 200 and Pr = 0.7 is investigated numerically. The two-dimensional incompressible Navier-Stokes equations and energy equation are coupled with the Euler-Bernoulli beam equation to describe the flow-induced vibration (FIV) of the flexible fin considering the convection heat transfer process. The modified characteristic-based split scheme, Galerkin finite element method, semi-torsional spring analogy method and loosely coupled partitioned approach are employed irrespectively for the flow and convection heat transfer, fin vibration, mesh movement and fluid–structure interaction. The accuracy and stability of the proposed numerical method are validated using three benchmark models including the forced convection heat transfer from a stationary cylinder, forced convection heat transfer from a transversely oscillating cylinder and FIV of a flexible plate behind a square cylinder. Finally, forced convection heat transfer characteristics from a circular cylinder with a flexible fin with fin length l = 0.5D–1.5D (D is the cylinder diameter) and elastic modulus E = 104 - 5 × 105 are analyzed in detail. The numerical results show that, when the vortex shedding frequency approaches the natural frequency of the flexible fin, the FIV frequency is locked on the natural frequency and the fin exhibits large-amplitude vibration. As a result, the ‘dead water’ region behind the cylinder is reduced and the convection heat transfer is improved. In the combinations of parameters considered, a maximum of 11.07% enhancement in heat transfer is obtained by the flexible fin.

Investigation on gaseous jet in forebody/inlet for shock-induced combustion ramjet (shcramjet) engines

As one of the most promising propulsion systems in the future, there are unusual advantages of the shock-induced combustion ramjet (shcramjet) engine. In the current study, the slot is placed on the second ramp of the shcramjet inlet to promote the mixing process between the fuel and the hypersonic crossflow. The influences of the injection angle and the jet-to-crossflow pressure ratio have been investigated numerically based on the grid independency analysis and the code validation. The obtained results predicted by the two-dimensional Reynolds-average Navier-Stokes (RANS) equations coupled with the two equation RNG k-ε turbulence model show that the grid scale makes only a slight difference to the wall pressure profiles for all jet-to-crossflow pressure ratios employed in this article. With the smaller injection angle, higher fuel/air mixing efficiency, less loss of the stagnation pressure and lower fuel penetration depth are shown. Low jet-to-crossflow pressure ratio can increase the mixing efficiency and decrease the loss of the stagnation pressure. High jet-to-crossflow pressure ratio results in higher penetration depth. There is an optimum injection angle for each jet-to-crossflow pressure ratio to achieve the maximization of the fuel/air mixing in hypersonic flows.

Slip flow in a microchannel driven by rhythmic wall contractions

We adopt a recent minimal mathematical model of a pumping mechanism in entomological respiratory systems and consider the model’s behavior in the slip flow regime, which occurs naturally in the distalmost portions of insect respiratory systems. In the model, a phase lag in the timing of two neighboring wall contractions in a rectangular microchannel produces a unidirectional flow. The current study investigates the results of incorporating slip effects into the model by introducing first-order accurate slip boundary conditions to investigate the method’s performance for slip flows at the microscale in the slip flow regime. The two-dimensional Navier–Stokes equations are solved with microscale and lubrication theory assumptions, and the tangential momentum accommodation coefficient is assumed to be one, so that the slip flow parameter $$\beta $$ is identically equivalent to the Knudsen number, Kn. The variations of the axial velocity, pressure gradient, and total pressure along the channel are determined for three representative Knudsen numbers that span the continuum and slip flow regimes. It was observed that for the shear-driven flow investigated here, the overall effect of increasing the amount of slip is to decrease the volumetric flow rate and that the phase lag for producing maximum flow is in the range of $$63^\circ $$ – $$67^\circ $$ , while in the no-slip case the optimum phase lag is approximately $$63^\circ $$ . The results suggest that shear-driven flows at the microscale in the slip flow regime may see a reduction in flow rate in contrast to pressure-driven microscale gas flows in the slip flow regime.

Numerical investigation on VIV energy harvesting of four cylinders in close staggered formation

Marine hydrokinetic energy in ocean currents can be harvested using Vortex Induced Vibration (VIV). The purpose of this paper is to investigate the VIV energy harvesting and maximize the power-to-volume density by arranging four staggered cylinders reasonably. Consequently, the effect of spacing between the cylinders is the focus considered throughout this paper. The in-flow spacing X varies from 1.2D to 10.0D, where D is the Diameter of the cylinder; the transverse spacing Y varies from 2.0D to 8.0D. A series of numerical simulations are conducted using two-dimensional Reynolds-Averaged Navier-Stokes equations in conjunction with the k-ω SST turbulence model. Results indicate that the interference between the tandem cylinders is stronger than between the side-by-side arranged cylinders. The multi-cylinder interference is divided into three regions: Synchronization Region, Blocking/Encouraged Region and Recovery Region. Interaction effects can induce much more vigorous VIV response of the downstream cylinder in Encouraged Region, where the maximum converted power ratio reaches up to 3.9 with in-flow spacing X of 2.0D and transverse spacing Y of 4.0D. Moreover, the average power-to-volume density will keep increasing as the spacing X decreases if there are no other physical limitations. These results lead the way for future layout optimization of multiple cylinders.

Turnpike Property for Two-Dimensional Navier–Stokes Equations

In this paper we study the turnpike phenomenon arising in the optimal distributed control tracking-type problem for the Navier–Stokes equations. We obtain a positive answer to this property in the case when the control is time-dependent function and also when it is independent of time. In both cases we prove an exponential turnpike property assuming that the stationary optimal state satisfies certain properties of smallness.

A multidomain multigrid pseudospectral method for incompressible flows

Pseudospectral method has the merit of high accuracy and the defects of simple geometry suitability and low computational efficiency. To remedy the two defects, a multidomain multigrid Chebyshev pseudospectral method is proposed and validated through the numerical solution of two-dimensional incompressible Navier-Stokes equations in the primitive variable formulation. To facilitate the implementation of the multidomain multigrid method, the IPN-IPN method is utilized to approximate the velocity and pressure with the same degree of Chebyshev polynomials within each subdomain, and an interface/boundary condensation method is developed to implement the pseudospectral operators of multigrid at the interface/boundary of subdomains. The accuracy and efficiency of the proposed method are first validated by numerical solutions of the lid-driven cavity problem. The numerical results are in good agreement with the benchmark solutions, and the speeding up of multigrid is 4–9 compared against the single grid. Then the capability of the proposed method for even more complex geometries with a close/open boundary is demonstrated by numerical solutions of several typical problems. The proposed method is quite generic and can be extended to the high accuracy and efficiency solution of three-dimensional incompressible/compressible, unsteady/steady fluid flows and heat transfer problems.