Four-dimensional turbulence in a plane channel

2011 ◽  
Vol 680 ◽  
pp. 67-79 ◽  
Author(s):  
NIKOLAY NIKITIN
Keyword(s):  
Reynolds Number ◽  
Channel Flow ◽  
Stokes Equations ◽  
Plane Channel ◽  
Turbulent Channel Flow ◽  
Reynolds Numbers ◽  
Wall Friction ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Number Range

The four-dimensional (4D) incompressible Navier–Stokes equations are solved numerically for the plane channel geometry. The fourth spatial coordinate is introduced formally to be homogeneous and mathematically orthogonal to the others, similar to the spanwise coordinate. Exponential growth of small 4D perturbations superimposed onto 3D turbulent solutions was observed in the Reynolds number range from Re = 4000 to Re = 10000. The growth rate of small 4D perturbations expressed in wall units was found to be λ+4D = 0.016 independent of Reynolds number. Nonlinear evolution of 4D perturbations leads either to attenuation of turbulence and relaminarization or to establishment of a self-sustained 4D turbulent solution (4D turbulent flow). Both results on flow evolution were obtained at the lowest Reynolds number, depending on the grid resolution, pointing to the proximity of Re = 4000 as the critical Reynolds number for 4D turbulence. Self-sustained 4D turbulence appeared to be less intense compared with 3D turbulence in terms of mean wall friction, which is about 55% of that predicted by the empirical Dean law for turbulent channel flow at all Reynolds numbers considered. Thus, the law of resistance of 4D turbulent channel flow can be expressed as Cf = 0.04Re−0.25.

Burgers turbulence model for a channel flow

1971 ◽  
Vol 47 (2) ◽  
pp. 321-335 ◽  
Author(s):  
Jon Lee
Keyword(s):  
Channel Flow ◽  
Stokes Equations ◽  
Turbulent Channel Flow ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Amplitude Equations ◽  
Fourier Amplitude ◽  
Navier Stokes Equations ◽  
Burgers Model ◽  
Model Dynamics

The truncated Burgers models have a unique equilibrium state which is defined continuously for all the Reynolds numbers and attainable from a realizable class of initial disturbances. Hence, they represent a sequence of convergent approximations to the original (untruncated) Burgers problem. We have pointed out that consideration of certain degenerate equilibrium states can lead to the successive turbulence-turbulence transitions and finite-jump transitions that were suggested by Case & Chiu. As a prototype of the Navier–Stokes equations, Burgers model can simulate the initial-value type of numerical integration of the Fourier amplitude equations for a turbulent channel flow. Thus, the Burgers model dynamics display certain idiosyncrasies of the actual channel flow problem described by a truncated set of Fourier amplitude equations, which includes only a modest number of modes due to the limited capability of the computer at hand.

Unsteady flow past a rotating circular cylinder at Reynolds numbers 103 and 104

1990 ◽  
Vol 220 ◽  
pp. 459-484 ◽  
Author(s):  
H. M. Badr ◽  
M. Coutanceau ◽  
S. C. R. Dennis ◽  
C. Ménard
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Unsteady Flow ◽  
Rotation Rate ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Number Range ◽  
Flow Past

The unsteady flow past a circular cylinder which starts translating and rotating impulsively from rest in a viscous fluid is investigated both theoretically and experimentally in the Reynolds number range 103 [les ] R [les ] 104 and for rotational to translational surface speed ratios between 0.5 and 3. The theoretical study is based on numerical solutions of the two-dimensional unsteady Navier–Stokes equations while the experimental investigation is based on visualization of the flow using very fine suspended particles. The object of the study is to examine the effect of increase of rotation on the flow structure. There is excellent agreement between the numerical and experimental results for all speed ratios considered, except in the case of the highest rotation rate. Here three-dimensional effects become more pronounced in the experiments and the laminar flow breaks down, while the calculated flow starts to approach a steady state. For lower rotation rates a periodic structure of vortex evolution and shedding develops in the calculations which is repeated exactly as time advances. Another feature of the calculations is the discrepancy in the lift and drag forces at high Reynolds numbers resulting from solving the boundary-layer limit of the equations of motion rather than the full Navier–Stokes equations. Typical results are given for selected values of the Reynolds number and rotation rate.

Turbulence statistics in fully developed channel flow at low Reynolds number

1987 ◽  
Vol 177 ◽  
pp. 133-166 ◽  
Author(s):  
John Kim ◽  
Parviz Moin ◽  
Robert Moser
Keyword(s):  
Experimental Data ◽  
Reynolds Number ◽  
Channel Flow ◽  
Stokes Equations ◽  
Turbulent Channel Flow ◽  
Navier Stokes ◽  
Turbulence Statistics ◽  
Navier Stokes Equations ◽  
Statistical Correlations ◽  
Grid Points

A direct numerical simulation of a turbulent channel flow is performed. The unsteady Navier-Stokes equations are solved numerically at a Reynolds number of 3300, based on the mean centreline velocity and channel half-width, with about 4 × 106 grid points (192 × 129 × 160 in x, y, z). All essential turbulence scales are resolved on the computational grid and no subgrid model is used. A large number of turbulence statistics are computed and compared with the existing experimental data at comparable Reynolds numbers. Agreements as well as discrepancies are discussed in detail. Particular attention is given to the behaviour of turbulence correlations near the wall. In addition, a number of statistical correlations which are complementary to the existing experimental data are reported for the first time.

scholarly journals Nonlinear amplification in hydrodynamic turbulence

2021 ◽  
Vol 930 ◽  
Author(s):  
Kartik P. Iyer ◽  
Katepalli R. Sreenivasan ◽  
P.K. Yeung
Keyword(s):  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Developed Turbulence ◽  
Advection Term ◽  
Nonlinear Amplification

Using direct numerical simulations performed on periodic cubes of various sizes, the largest being $8192^3$ , we examine the nonlinear advection term in the Navier–Stokes equations generating fully developed turbulence. We find significant dissipation even in flow regions where nonlinearity is locally absent. With increasing Reynolds number, the Navier–Stokes dynamics amplifies the nonlinearity in a global sense. This nonlinear amplification with increasing Reynolds number renders the vortex stretching mechanism more intermittent, with the global suppression of nonlinearity, reported previously, restricted to low Reynolds numbers. In regions where vortex stretching is absent, the angle and the ratio between the convective vorticity and solenoidal advection in three-dimensional isotropic turbulence are statistically similar to those in the two-dimensional case, despite the fundamental differences between them.

Downstream decay of fully developed Dean flow

2015 ◽  
Vol 777 ◽  
pp. 219-244 ◽  
Author(s):  
Jesse T. Ault ◽  
Kevin K. Chen ◽  
Howard A. Stone
Keyword(s):  
Reynolds Number ◽  
Linear Dependence ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Curvature Radius ◽  
Navier Stokes Equations ◽  
Curved Pipe ◽  
Dean Flow ◽  
Junction Flow

Direct numerical simulations were used to investigate the downstream decay of fully developed flow in a $180^{\circ }$ curved pipe that exits into a straight outlet. The flow is studied for a range of Reynolds numbers and pipe-to-curvature radius ratios. Velocity, pressure and vorticity fields are calculated to visualize the downstream decay process. Transition ‘decay’ lengths are calculated using the norm of the velocity perturbation from the Hagen–Poiseuille velocity profile, the wall-averaged shear stress, the integral of the magnitude of the vorticity, and the maximum value of the $Q$-criterion on a cross-section. Transition lengths to the fully developed Poiseuille distribution are found to have a linear dependence on the Reynolds number with no noticeable dependence on the pipe-to-curvature radius ratio, despite the flow’s dependence on both parameters. This linear dependence of Reynolds number on the transition length is explained by linearizing the Navier–Stokes equations about the Poiseuille flow, using the form of the fully developed Dean flow as an initial condition, and using appropriate scaling arguments. We extend our results by comparing this flow recovery downstream of a curved pipe to the flow recovery in the downstream outlets of a T-junction flow. Specifically, we compare the transition lengths between these flows and document how the transition lengths depend on the Reynolds number.

scholarly journals Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels

2013 ◽  
Vol 734 ◽  
pp. 275-316 ◽  
Author(s):  
Rashad Moarref ◽  
Ati S. Sharma ◽  
Joel A. Tropp ◽  
Beverley J. McKeon
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Low Rank ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Self Similar ◽  
The Mean ◽  
High Reynolds

AbstractWe study the Reynolds-number scaling and the geometric self-similarity of a gain-based, low-rank approximation to turbulent channel flows, determined by the resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in order to obtain a description of the streamwise turbulence intensity from direct consideration of the Navier–Stokes equations. Under this formulation, the velocity field is decomposed into propagating waves (with single streamwise and spanwise wavelengths and wave speed) whose wall-normal shapes are determined from the principal singular function of the corresponding resolvent operator. Using the accepted scalings of the mean velocity in wall-bounded turbulent flows, we establish that the resolvent operator admits three classes of wave parameters that induce universal behaviour with Reynolds number in the low-rank model, and which are consistent with scalings proposed throughout the wall turbulence literature. In addition, it is shown that a necessary condition for geometrically self-similar resolvent modes is the presence of a logarithmic turbulent mean velocity. Under the practical assumption that the mean velocity consists of a logarithmic region, we identify the scalings that constitute hierarchies of self-similar modes that are parameterized by the critical wall-normal location where the speed of the mode equals the local turbulent mean velocity. For the rank-1 model subject to broadband forcing, the integrated streamwise energy density takes a universal form which is consistent with the dominant near-wall turbulent motions. When the shape of the forcing is optimized to enforce matching with results from direct numerical simulations at low turbulent Reynolds numbers, further similarity appears. Representation of these weight functions using similarity laws enables prediction of the Reynolds number and wall-normal variations of the streamwise energy intensity at high Reynolds numbers (${Re}_{\tau } \approx 1{0}^{3} {\unicode{x2013}} 1{0}^{10} $). Results from this low-rank model of the Navier–Stokes equations compare favourably with experimental results in the literature.

Flow in Rectangular Cavities With Two Vertical Oscillating Walls

2008 ◽  
Author(s):  
Guillermo E. Ovando ◽  
Alberto Beltran ◽  
Sandy L. Ovando
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
The Finite Element Method ◽  
Navier Stokes Equations ◽  
Cell Structures ◽  
Lower Reynolds Number ◽  
Constant Displacement ◽  
Vertical Walls

Fluid dynamics in a two-dimensional rectangular cavity with vertical oscillatory walls out of phase was studied numerically. The Navier-Stokes equations were solved using the finite element method. We analyzed the behaviour of the velocity fields, the vorticity fields and we also obtained the streaklines of the fluid at the bottom left corner of the domain for one and two cycles, which is associated with the mixing of the fluid. The analysis was carried out for three different Reynolds numbers of 50, 500 and 1000 with constant displacement amplitude of the moving boundaries of 0.2. For this range of parameters the flow is characterized by two kind of symmetries. We found that for lower Reynolds number there is a good local mixing given by cell structures and the smooth behavior of the fluid inside the cavity; however for higher Reynolds number these structures disappear due to the fluid near the vertical walls impinges against the corner of the cavity, then this fluid is dispersed through the whole cavity during the cycle, increasing the global mixing of the fluid.

Numerical and asymptotic methods for certain viscous free-surface flows

1992 ◽  
Vol 340 (1656) ◽  
pp. 1-45 ◽  
Keyword(s):  
Reynolds Number ◽  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Free Surface Flows ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Navier Stokes Equations ◽  
Flow Rates

This paper concerns the two-dimensional motion of a viscous liquid down a perturbed inclined plane under the influence of gravity, and the main goal is the prediction of the surface height as the fluid flows over the perturbations. The specific perturbations chosen for the present study were two humps stretching laterally across an otherwise uniform plane, with the flow being confined in the lateral direction by the walls of a channel. Theoretical predictions of the flow have been obtained by finite-element approximations to the Navier-Stokes equations and also by a variety of lubrication approximations. The predictions from the various models are compared with experimental measurements of the free-surface profiles. The principal aim of this study is the establishment and assessment of certain numerical and asymptotic models for the description of a class of free-surface flows, exemplified by the particular case of flow over a perturbed inclined plane. The laboratory experiments were made over a range of flow rates such that the Reynolds number, based on the volume flux per unit width and the kinematical viscosity of the fluid, ranged between 0.369 and 36.6. It was found that, at the smaller Reynolds numbers, a standard lubrication approximation provided a very good representation of the experimental measurements but, as the flow rate was increased, the standard model did not capture several important features of the flow. On the other hand, a lubrication approximation allowing for surface tension and inertial effects expanded the range of applicability of the basic theory by almost an order of magnitude, up to Reynolds numbers approaching 10. At larger flow rates, numerical solutions to the full equations of motion provided a description of the experimental results to within about 4% , up to a Reynolds number of 25, beyond which we were unable to obtain numerical solutions. It is not known why numerical solutions were not possible at larger flow rates, but it is possible that there is a bifurcation of the Navier-Stokes equations to a branch of unsteady motions near a Reynolds number of 25.

scholarly journals The effort of increasing Reynolds number in POD-Galerkin Reduced Order Methods: from laminar to turbulent flows

2018 ◽  
Author(s):  
Shafqat Ali ◽  
Saddam Hijazi ◽  
Sokratia Georgaka ◽  
Francesco Ballarin ◽  
Giovanni Stabile ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Finite Element ◽  
Turbulent Flows ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Volume Method ◽  
Reduced Order Methods

We present different strategies to be able to increase Reynolds number in Reduced Order Methods (ROMs), from laminar to turbulent flows, in the context of the incompressible parametrised Navier-Stokes equations. The proposed methodologies are based on different full order discretisation techniques: the finite element method and the finite volume method. For what concerns finite element full order discretisations which in this work aim to be used from low to moderate Reynolds numbers the

scholarly journals A Numerical Investigation of Co-flow Jet Effects on Airfoil Aerodynamic Characteristics

2021 ◽  
Author(s):  
Shima Yazdani ◽  
Erfan Salimipour ◽  
Ayoob Salimipour
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Active Flow Control ◽  
Lift Coefficient ◽  
Reynolds Numbers ◽  
Aerodynamic Characteristics ◽  
Dynamic Stall ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Momentum Coefficient

Abstract The present paper numerically investigates the performance of a Co-Flow Jet (CFJ) on the static and dynamic stall control of the NACA 0024 airfoil at Reynolds number 1.5 × 105. The two-dimensional Reynolds-averaged Navier-Stokes equations are solved using the SST k-ω turbulence model. The results show that the lift coefficients at the low angles of attack (up to α = 15̊) are significantly increased at Cµ = 0.06, however for the higher momentum coefficients, it is not seen an improvement in the aerodynamic characteristics. Also, the dynamic stall for a range of α between 0̊ and 20̊ at the mentioned Reynolds number and with the reduced frequency of 0.15 for two CFJ cases with Cµ = 0.05 and 0.07 are investigated. For the case with Cµ = 0.07, the lift coefficient curve did not present a noticeable stall feature compared to Cµ = 0.05. The effect of this active flow control by increasing the Reynolds numbers from 0.5 × 105 to 3 × 105 is also investigated. At all studied Reynolds numbers, the lift coefficient enhances as the momentum coefficient increases where its best performance is obtained at the angle of attack α = 15̊.

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