Two-dimensional Incompressible Navier-Stokes Equations Research Articles

Numerical investigation of flow around a slotted circular cylinder at low Reynolds number

Herein, a circular cylinder with a slit is implemented as an effective passive flow control technique for flow control and other relevant industrial practices. Two-dimensional incompressible Navier–Stokes equations are numerically solved using the high-resolution spectral element method at a Reynolds number of 200. The slit width ratio s/d is varied from 0 to 0.3, and the slit attack angle is between 0° to 90°. The findings show that, for attack angle 0° ≤ β ≤ 45°, a larger slit width ratio and a smaller attack angle tend to induce a lower time-mean drag coefficient and root-mean-square lift coefficient. The relative reduction ratio of the time-mean drag coefficient reaches a maximum of 28.6% at s/d = 0.30 and β = 0°. For attack angle 45° <β < 60°, the slotted cylinder has only a slight effect on the force characteristics and flow patterns. For attack angle 60° ≤ β ≤ 90°, the mean drag coefficient and root-mean-square lift coefficient are greater than the normal cylinder for each slit width ratio. The strength of the vortices is found to be strengthened and more stable than that of a normal cylinder.

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.

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.

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.

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.

A Comparison between the Reduced Differential Transform Method and Perturbation-Iteration Algorithm for Solving Two-Dimensional Unsteady Incompressible Navier-Stokes Equations

In this work, approximate analytical solutions to the lid-driven square cavity flow problem, which satisfied two-dimensional unsteady incompressible Navier-Stokes equations, are presented using the kinetically reduced local Navier-Stokes equations. Reduced differential transform method and perturbation-iteration algorithm are applied to solve this problem. The convergence analysis was discussed for both methods. The numerical results of both methods are given at some Reynolds numbers and low Mach numbers, and compared with results of earlier studies in the review of the literatures. These two methods are easy and fast to implement, and the results are close to each other and other numerical results, so it can be said that these methods are useful in finding approximate analytical solutions to the unsteady incompressible flow problems at low Mach numbers.

COMPUTATIONAL FLUID DYNAMICS STUDY OF DUSTY AIR FLOW OVER NACA 63415 AIRFOIL FOR WIND TURBINE APPLICATIONS

Gulf and South African countries have enormous potential for wind energy. However, the emergence of sand storms in this region postulates performance and reliability challenges on wind turbines. This study investigates the effects of debris flow on wind turbine blade performance. In this paper, two-dimensional incompressible Navier-Stokes equations and the transition SST turbulence model are used to analyze the aerodynamic performance of NACA 63415 airfoil under clean and sandy conditions. The numerical simulation of the airfoil under clean surface condition is performed at Reynolds number 460×103, and the numerical results have a good consistency with the experimental data. The Discrete Phase Model has been used to investigate the role sand particles play in the aerodynamic performance degradation. The pressure and lift coefficients of the airfoil have been computed under different sand particles flow rates. The performance of the airfoil under different angle of attacks has been studied. Results showed that the blade lift coefficient can deteriorate by 28% in conditions relevant to the Gulf and South African countries sand storms. As a result, the numerical simulation method has been verified to be economically available for accurate estimation of the sand particles effect on the wind turbine blades.

The vanishing viscosity limit for 2D Navier–Stokes in a rough domain

We study the high Reynolds number limit of a viscous fluid in the presence of a rough boundary. We consider the two-dimensional incompressible Navier–Stokes equations with Navier slip boundary condition, in a domain whose boundaries exhibit fast oscillations in the form x2=ε1+αη(x1/ε), α>0. Under suitable conditions on the oscillating parameter ε and the viscosity ν, we show that solutions of the Navier–Stokes system converge to solutions of the Euler system in the vanishing limit of both ν and ε. The main issue is that the curvature of the boundary is unbounded as ε→0, which precludes the use of standard methods to obtain the inviscid limit. Our approach is to first construct an accurate boundary layer approximation to the Euler solution in the rough domain, and then to derive stability estimates for this approximation under the Navier–Stokes evolution.

A study on the aerodynamic characteristics of airfoil in the flapping adjustment stage during forward flight

The aim of this study is to investigate the aerodynamic characteristics of a flapping airfoil in the adjustment stage between two specific flight patterns during the forward flight. Four flapping movement models in adjustment stage are firstly established by using the multi-objective optimization algorithm. Then, a numerical experiment is carried out by using finite volume method to solve the two-dimensional time-dependent incompressible Navier-Stokes equations. The attack angles are selected from −5° to 7.5° with an increase of 2.5°. The results are systematically analyzed and special attention is paid to the corresponding changes of aerodynamic forces, vortex shedding mechanism in the wake structure and thrust efficiency. Present results show that output aerodynamic performance of flapping airfoil can be improved by the increasement of amplitude and frequency in the flapping adjustment stage, which further validates and complements previous studies. Moreover, it is also show that the manner using multi-objective optimization algorithm to generate a movement model in adjustment stage, to connect other two specific plunging motions, is a feasible and effective method. Current study is dedicated to providing some helpful references for the design and control of artificial flapping wing air vehicles.

Applications of parallel computing technologies for modeling the mixed convection in backward-facing step flows with the vertical buoyancy forces

This paper presents numerical solutions of the mixed convection in backward-facing step flows with the vertical buoyancy forces. A two-dimensional incompressible Navier-Stokes equation is used to describe this process. This system is approximated by the control volume method and solved numerically by the projection method. The two-dimensional Poisson equation satisfying the discrete continuity equation that is solved by the Jacobi iterative method at each time step. The numerical solutions of the laminar flow behind the backward-facing step with the vertical buoyancy forces are compared with the numerical results of other authors. This numerical algorithm is completely parallelized using various geometric domain decompositions (1D, 2D and 3D). Preliminary theoretical analysis of the various decomposition methods effectiveness of the computational domain and real computational experiments for this problem were made and the best domain decomposition method was determined. In the future, a proven mathematical model and parallelized numerical algorithm with the best domain decomposition method can be applied for various complex flows with the vertical buoyancy forces. G M T Detect language Afrikaans Albanian Arabic Armenian Azerbaijani Basque Belarusian Bengali Bosnian Bulgarian Catalan Cebuano Chichewa Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Esperanto Estonian Filipino Finnish French Galician Georgian German Greek Gujarati Haitian Creole Hausa Hebrew Hindi Hmong Hungarian Icelandic Igbo Indonesian Irish Italian Japanese Javanese Kannada Kazakh Khmer Korean Lao Latin Latvian Lithuanian Macedonian Malagasy Malay Malayalam Maltese Maori Marathi Mongolian Myanmar (Burmese) Nepali Norwegian Persian Polish Portuguese Punjabi Romanian Russian Serbian Sesotho Sinhala Slovak Slovenian Somali Spanish Sundanese Swahili Swedish Tajik Tamil Telugu Thai Turkish Ukrainian Urdu Uzbek Vietnamese Welsh Yiddish Yoruba Zulu Afrikaans Albanian Arabic Armenian Azerbaijani Basque Belarusian Bengali Bosnian Bulgarian Catalan Cebuano Chichewa Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Esperanto Estonian Filipino Finnish French Galician Georgian German Greek Gujarati Haitian Creole Hausa Hebrew Hindi Hmong Hungarian Icelandic Igbo Indonesian Irish Italian Japanese Javanese Kannada Kazakh Khmer Korean Lao Latin Latvian Lithuanian Macedonian Malagasy Malay Malayalam Maltese Maori Marathi Mongolian Myanmar (Burmese) Nepali Norwegian Persian Polish Portuguese Punjabi Romanian Russian Serbian Sesotho Sinhala Slovak Slovenian Somali Spanish Sundanese Swahili Swedish Tajik Tamil Telugu Thai Turkish Ukrainian Urdu Uzbek Vietnamese Welsh Yiddish Yoruba Zulu Text-to-speech function is limited to 200 characters Options : History : Feedback : Donate Close

Numerical approximation of statistical solutions of planar, incompressible flows

We present a finite difference-(multi-level) Monte Carlo algorithm to efficiently compute statistical solutions of the two-dimensional incompressible Navier–Stokes equations (NSE), with periodic boundary conditions and for arbitrary high Reynolds number. We propose a reformulation of statistical solutions in the sense of Foiaş and Prodi in the vorticity-stream function form. The vorticity-stream function formulation of the NSE in two-space dimensions is discretized with a finite difference scheme. We obtain a convergence rate error estimate for this approximation which is explicit in the viscosity parameter [Formula: see text], under realistic assumptions on the solution regularity. We also prove convergence and complexity estimates, for the (multi-level) Monte Carlo finite difference algorithm to compute statistical solutions. Numerical experiments illustrating the validity of our estimates are presented. They show that the multi-level Monte Carlo algorithm can significantly accelerate the computation of statistical solutions in the sense of Foiaş and Prodi, even for very high Reynolds numbers.

Numerical investigation into the effects of stroke trajectory on the aerodynamic performance of insect hovering flight

Various stroke trajectories may be observed in insect hovering flight in nature; however their influences on the flight performance of insect are not well estimated. In this study, a numerical investigation into the effects of stroke trajectories on the aerodynamic performance of insect hovering flight is carried out through the solution of the two-dimensional unsteady incompressible Navier-Stokes equations. An insect wing model with ellipse cross section in hovering flight is considered for the purpose and four types of idealized trajectories (Named linear, oval, figure-eight and double-eight) which possess different deviation characteristics from the stroke plane are examined. The influences of the deviation amplitude of trajectory, the attack angle of wing and the inclined angle of stroke plane on the aerodynamic characteristics of hovering wing are systematically analyzed. The results show that in the case of the wing in a normal hovering flight where the stroke plane is horizontal, the trajectory deviation from the stroke plane weakens the aerodynamic performance for each trajectory case considered, and this deteriorative effect becomes more serious as the amplitude of deviation increases. With regard to the influence of the angle of attack, the results show that the time-averaged drag force and power consumption increase monotonically with the angle, whereas the time-averaged lift force and the lifting efficiency increase first and then decrease as the angle increases further. In the case of a hovering flight with an inclined stroke plane, distinctly different trends from a normal hovering flight are obtained.

A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier-Stokes Equations

AbstractWe study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier-Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time coarse-mesh nodal-point observational measurements of the velocity field of this reference solution (subsampling), as might be measured by an array of weather vane anemometers. Our calculations show that the required nodal observation density is remarkably less than what is suggested by the analytical study; and is in fact comparable to thenumber of numerically determining Fourier modes, which was reported in an earlier computational study by the authors. Thus, this method is computationally efficient and performs far better than the analytical estimates suggest.

Numerical Simulation of Flow over an Open Cavity with Self-Sustained Oscillation Mode Switching

Numerical simulations are used to investigate the self-sustained oscillating flows past an open cavity. The two-dimensional incompressible Navier-Stokes equations are solved directly by using the finite difference method for cavities with an upstream laminar boundary layer. A series of simulations are performed for a variety of cavity length-to-depth ratio. The results show the switching among some flow modes including non-oscillation mode, shear layer mode and wake mode. The variation of the Strouhal number is in favorable agreement with available experimental data. The results of flow fields in the cavity reveal the relationship between the cavity shear layer oscillation modes and recirculating vortices in the cavity.

Local RBFs Based Collocation Methods for Unsteady Navier-Stokes Equations

AbstractThe local RBFs based collocation methods (LRBFCM) is presented to solve two-dimensional incompressible Navier-Stokes equations. In avoiding the ill-conditioned problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear systems within local supporting domain. Then, we reformulate local matrix in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The numerical experiments have shown that the developed LRBFCM is suitable for solving the incompressible Navier-Stokes equations with high accuracy and efficiency.

Effect of flapping frequency on aerodynamics of wing in freely hovering flight

Hovering flight is investigated numerically when the wing is free to vibrate in the vertical direction under the action of wing lift and insect weight in low Reynolds number regime. The two-dimensional incompressible Navier–Stokes equations are solved using the immersed boundary method. The wing is driven to translate in the horizontal direction and rotate periodically to emulate the wing motion of a fruit fly in normal hovering flight, while the motion in the vertical direction responds passively to the action of the wing aerodynamic lift and weight of the insect body. The insect body is modeled by a point mass. It is shown that flapping wing cannot produce required lift to maintain stable hovering flight in specified range with low flapping frequencies, if the insect weight is equivalent to the averaged wing lift in one cycle on the assumption of zero vertical velocity. The vertical velocity influences the instantaneous angle of attack of the hovering wing, which results in the variation in aerodynamics of the wing. The insect may experience fluctuating hovering flight with a reduced weight when the flapping frequency is low. The fluctuating amplitude decreases with increasing flapping frequency. The efficiency of hovering flight is also a problem of concern.

Simulation of solid body motion in a Newtonian fluid using a vorticity-based pseudo-spectral immersed boundary method augmented by the radial basis functions

The moving boundary conditions are implemented into the Fourier pseudo-spectral solution of the two-dimensional incompressible Navier–Stokes equations (NSE) in the vorticity-velocity form, using the radial basis functions (RBF). Without explicit definition of an external forcing function, the desired immersed boundary conditions are imposed by direct modification of the convection and diffusion terms. At the beginning of each time-step the solenoidal velocities, satisfying the desired moving boundary conditions, along with a modified vorticity are obtained and used in modification of the convection and diffusion terms of the vorticity evolution equation. Time integration is performed by the explicit fourth-order Runge–Kutta method and the boundary conditions are set at the beginning of each sub-step. The method is applied to a couple of moving boundary problems and more than second-order of accuracy in space is demonstrated for the Reynolds numbers up to Re = 550. Moreover, performance of the method is shown in comparison with the classical Fourier pseudo-spectral method.

Flow-induced vibrations of square and rectangular cylinders at low Reynolds number

One-degree-of-freedom (1-dof) and two-degrees-of-freedom (2-dof) flow-induced vibrations (FIVs) of square and rectangular cylinders at a mass ratio of 10 and a low Reynolds number of 200 are studied numerically by solving the two-dimensional incompressible Navier–Stokes equations. The aim of the study is to identify the effects of the aspect ratio α, defined to be the ratio of the cylinder dimension in the cross-flow direction to that in the inline direction, on the vortex-induced vibration (VIV) and galloping responses. Simulations are conducted for aspect ratios of 0.3, 0.5, 0.7, 1 and 1.25 and reduced velocities ranging from 1 to 30. Distinct VIV lock-in and galloping regimes are found for all the aspect ratios except α = 0.3, for which only VIV lock-in is found. The VIV lock-in regime and the galloping regime are separated by a reduced velocity range, where the response amplitude is very small and the response frequency is a linear function of the reduced velocity. It is found that the maximum amplitude in the VIV lock-in regime decreases with increasing aspect ratio. Galloping does not start until the reduced velocity exceeds a critical value. The critical reduced velocity for galloping increases with increasing aspect ratio. For α = 0.5, galloping starts at Vr = 7 and 6 for 1-dof and 2-dof vibrations, respectively. The critical reduced velocity for galloping is increased to 17 at α = 1.25 for both 1-dof and 2-dof vibrations. Because the response amplitude in the inline direction is much smaller than that in the cross-flow direction, the response amplitude and frequency in 2-dof vibration are very similar to their counterparts in 1-dof vibration. However, the response amplitude in 2-dof galloping is greater than that in 1-dof galloping. A 2T vortex shedding mode is observed in the VIV lock-in regime for α = 0.3 and 0.5.

Continuous data assimilation with stochastically noisy data**Dedicated to Professor Ciprian Foias on the occasion of his 80th birthday.

We analyse the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimensional incompressible Navier–Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the data-assimilation algorithm consists of a system of stochastically forced Navier–Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the limit supremum, as the time tends to infinity, of the expected value of the L2-norm and of the H1 Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated.