Steady Flow Structures and Pressure Drops in Wavy-Walled Tubes

1987 ◽  
Vol 109 (3) ◽  
pp. 255-261 ◽  
Author(s):  
M. E. Ralph
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Creeping Flow ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Pressure Drops ◽  
Mean Pressure ◽  
Reynolds Number Flows

Solutions of the Navier-Stokes equations for steady axisymmetric flows in tubes with sinusoidal walls were obtained numerically, for Reynolds numbers (based on the tube radius and mean velocity at a constriction) up to 500, and for varying depth and wavelength of the wall perturbations. Results for the highest Reynolds numbers showed features suggestive of the boundary layer theory of Smith [23]. In the other Reynolds number limit, it has been found that creeping flow solutions can exhibit flow reversal if the perturbation depth is large enough. Experimentally measured pressure drops for a particular tube geometry were in agreement with computed predictions up to a Reynolds number of about 300, where transitional effects began to disturb the experiments. The dimensionless mean pressure gradient was found to decrease with increasing Reynolds number, although the rate of decrease was less rapid than in a straight-walled tube. Numerical results showed that the mean pressure gradient decreases as both the perturbation wavelength and depth increase, with the higher Reynolds number flows tending to be more influenced by the wavelength and the lower Reynolds number flows more affected by the depth.

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.

Capsular flow in pipelines

1972 ◽  
Vol 56 (1) ◽  
pp. 49-59 ◽  
Author(s):  
A. E. Vardy ◽  
M. I. G. Bloor ◽  
J. A. Fox
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Steady Motion ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Hydraulic Pressure ◽  
Central Difference ◽  
Navier Stokes Equations

The problem considered is that of the steady motion of a series of neutrally buoyant, flat-faced, rigid, cylindrical capsules along the axis of a pipeline under the influence of a hydraulic pressure gradient. The Navier-Stokes equations are non-dimensionalized and expressed in central-difference form. Numerical solutions are found by the method of relaxation for Reynolds numbers up to 20 000 and a close agreement is obtained with readings from a laboratory apparatus for Reynolds numbers up to 2200.The flow is examined in detail and the existence of toroidal vortices between successive capsules is demonstrated. Their shape is shown to be increasingly influenced by inertial forces as the Reynolds number increases, but the overall pressure gradient is not greatly dependent on the Reynolds number.

Flow Within a Pipe Annulus With Injection and Suction Through a Porous Wall With High Wall Reynolds Numbers

2011 ◽  
Vol 133 (1) ◽  
Author(s):  
Y. Moussy ◽  
A. D. Snider
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Porous Wall ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Singular Perturbation Method ◽  
Element Analysis ◽  
Navier Stokes Equations ◽  
Reynolds Number Flows ◽  
Number Case

An approximate solution to the Navier–Stokes equations was found for the case describing two-dimensional steady-state laminar flow over an array of porous pipes with high wall Reynolds number. The Navier–Stokes equations in cylindrical coordinates reduced to a fourth-order nonlinear differential equation, which was solved for high wall Reynolds number flows through the porous wall using a zeroth- and first-order singular perturbation method. Our analytic solution for the high wall Reynolds number case is consistent with solutions found from the low Reynolds number case and that found using finite element analysis.

scholarly journals Finite-amplitude steady waves in plane viscous shear flows

1985 ◽  
Vol 160 ◽  
pp. 281-295 ◽  
Author(s):  
F. A. Milinazzo ◽  
P. G. Saffman
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Navier Stokes ◽  
Unidirectional Flow ◽  
Viscous Shear ◽  
Finite Amplitude Waves

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

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.

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.

Direct simulations of low-Reynolds-number turbulent flow in a rotating channel

1993 ◽  
Vol 256 ◽  
pp. 163-197 ◽  
Author(s):  
Reidar Kristoffersen ◽  
Helge I. Andersson
Keyword(s):  
Reynolds Number ◽  
Turbulent Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Complete Suppression ◽  
Reynolds Stresses ◽  
Mean Velocity ◽  
Computational Mesh ◽  
Developed Pressure ◽  
Rotating Channel

Direct numerical simulations of fully developed pressure-driven turbulent flow in a rotating channel have been performed. The unsteady Navier–Stokes equations were written for flow in a constantly rotating frame of reference and solved numerically by means of a finite-difference technique on a 128 × 128 × 128 computational mesh. The Reynolds number, based on the bulk mean velocity Um and the channel half-width h, was about 2900, while the rotation number Ro = 2|Ω|h/Um varied from 0 to 0.5. Without system rotation, results of the simulation were in good agreement with the accurate reference simulation of Kim, Moin & Moser (1987) and available experimental data. The simulated flow fields subject to rotation revealed fascinating effects exerted by the Coriolis force on channel flow turbulence. With weak rotation (Ro = 0.01) the turbulence statistics across the channel varied only slightly compared with the nonrotating case, and opposite effects were observed near the pressure and suction sides of the channel. With increasing rotation the augmentation and damping of the turbulence along the pressure and suction sides, respectively, became more significant, resulting in highly asymmetric profiles of mean velocity and turbulent Reynolds stresses. In accordance with the experimental observations of Johnston, Halleen & Lezius (1972), the mean velocity profile exhibited an appreciable region with slope 2Ω. At Ro = 0.50 the Reynolds stresses vanished in the vicinity of the stabilized side, and the nearly complete suppression of the turbulent agitation was confirmed by marker particle trackings and two-point velocity correlations. Rotational-induced Taylor-Görtler-like counter-rotating streamwise vortices have been identified, and the simulations suggest that the vortices are shifted slightly towards the pressure side with increasing rotation rates, and the number of vortex pairs therefore tend to increase with Ro.

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.

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.

Structure of Turbulent Flows Over Forward Facing Steps With Adverse Pressure Gradient

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Hassan Iftekhar ◽  
Martin Agelin-Chaab
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Turbulent Flows ◽  
Large Scale ◽  
Adverse Pressure Gradient ◽  
Reynolds Numbers ◽  
Step Height ◽  
Reattachment Length ◽  
Mean Velocity ◽  
The Mean

This paper reports an experimental study on the effects of adverse pressure gradient (APG) and Reynolds number on turbulent flows over a forward facing step (FFS) by employing three APGs and three Reynolds numbers. A particle image velocimetry (PIV) technique was used to conduct velocity measurements at several locations downstream, and the flow statistics up to 68 step heights are reported. The step height was maintained at 6 mm, and the Reynolds numbers based on the step height and freestream mean velocity were 1600, 3200, and 4800. The mean reattachment length increases with the increase in Reynolds number without the APG whereas the mean reattachment length remains constant for increasing APG. The proper orthogonal decomposition (POD) results confirmed that higher Reynolds numbers caused the large-scale structures to be more defined and organized close to the step surface.

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