Analysis of Instantaneous Turbulent Velocity Vector and Temperature Profiles in Transitional Rough Channel Flow

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
Noor Afzal
Keyword(s):  
Velocity Vector ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Instantaneous Velocity ◽  
Navier Stokes ◽  
Order Effects ◽  
Mean Velocity ◽  
Instantaneous Temperature ◽  
The Mean ◽  
Instantaneous Velocity Vector

The instantaneous velocity vector and instantaneous temperature in a turbulent flow in a transitionally rough channel have been analyzed from unsteady Navier–Stokes equations and unsteady thermal energy equation for large Reynolds numbers. The inner and outer layers asymptotic expansions for the instantaneous velocity vector and instantaneous temperature have been matched in the overlap region by the Izakson–Millikan–Kolmogorov hypothesis. The higher order effects and implications of the intermediate (or meso) layer are analyzed for the instantaneous velocity vector and instantaneous temperature. Uniformly valid solutions for instantaneous velocity vector have been decomposed into the mean velocity vector, and fluctuations in velocity vector, as well as the instantaneous temperature, have been decomposed into mean temperature and fluctuations in temperature. It is shown in the present work that if the mean velocity vector in the work of Afzal (1976, “Millikan Argument at Moderately Large Reynolds Numbers,” Phys. Fluids, 16, pp. 600–602) is replaced by instantaneous velocity vector, we get the results of Lundgren (2007, “Asymptotic Analysis of the Constant Pressure Turbulent Boundary Layer,” Phys. Fluids, 19, pp. 055105) for instantaneous velocity vector. The comparison of the predictions for momentum and thermal mesolayers is supported by direct numerical simulation (DNS) and experimental data.

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.

Experimental and Numerical Investigation of Two-Dimensional Parallel Jets

2001 ◽  
Vol 123 (2) ◽  
pp. 401-406 ◽  
Author(s):  
Elgin A. Anderson ◽  
Robert E. Spall
Keyword(s):  
Good Accuracy ◽  
Stokes Equations ◽  
Numerical Models ◽  
Symmetry Plane ◽  
Turbulence Models ◽  
Turbulent Jets ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
The Mean

The flowfield of dual, parallel planar turbulent jets is investigated experimentally using an x-type hot-wire probe and numerically by solving the Reynolds-averaged Navier-Stokes equations. The performance of both differential Reynolds stress (RSM) and standard k-ε turbulence models is evaluated. Results show that the numerical models predict the merge and combined point characteristics to good accuracy. However, both turbulence models show a narrower width of the jet envelope than measured by experiment. The predicted profiles of the mean velocity along the symmetry plane agree well with the experimental results.

Instantaneous velocity and temperature measurements in oscillating diffusion flames

1974 ◽  
Vol 338 (1615) ◽  
pp. 479-501 ◽  
Keyword(s):  
Diffusion Flames ◽  
Frequency Doubling ◽  
Instantaneous Velocity ◽  
Temperature Measurements ◽  
Mean Velocity ◽  
Temperature Distributions ◽  
Mean Temperature ◽  
Instantaneous Temperature ◽  
The Mean ◽  
The Many

Measurements of instantaneous velocity, instantaneous temperature, and the corresponding mean and r. m. s. values, obtained in a range of diffusion flames, are presented. The velocity measurements were obtained with a laser anemometer and the temperature measurements with a thermocouple. The flames were formed by burning methane, town gas and hydrogen at the exit from burner tubes of external diameters 15.0, 9.2, 6.3 and 3.2 mm; the corresponding inside diameters were 13.0, 5.1, 5.1 and 2.5 mm. The mean velocity of the gas at exit from the tube ranged from 0.6 to 5.3 m/s. All flames exhibited discrete frequencies in the vicinity of 11 Hz. The instantaneous velocity and temperature signals were close to sinusoidal, except in the vicinity of the reaction zone where double frequencies and spiky signals were observed. The r. m. s. temperature distributions exhibited minima in the region of the maximum values of mean temperature; the r. m. s. velocity distributions were similar in form but the location of the minimum occurred downstream of the corresponding r. m. s. temperature minimum. The minimum in the r. m. s. velocity and temperature distributions were consistant with the observed frequency doubling and stemmed from the need for the frequency to increase to allow an increase in mean temperature in regions where the instantaneous temperature had attained its adiabatic flame value. A flame model is postulated and shown to represent the many observed features of the oscillating flames. It appears that the oscillations stem from aerodynamic instabilities associated with inflexion points in the local, instantaneous velocity distributions.

Similarity in non-rotating and rotating turbulent pipe flows

1999 ◽  
Vol 379 ◽  
pp. 1-22 ◽  
Author(s):  
MARTIN OBERLACK
Keyword(s):  
Reynolds Number ◽  
Scaling Laws ◽  
Stokes Equations ◽  
Scaling Law ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Pipe Flows ◽  
The Mean

The Lie group approach developed by Oberlack (1997) is used to derive new scaling laws for high-Reynolds-number turbulent pipe flows. The scaling laws, or, in the methodology of Lie groups, the invariant solutions, are based on the mean and fluctuation momentum equations. For their derivation no assumptions other than similarity of the Navier–Stokes equations have been introduced where the Reynolds decomposition into the mean and fluctuation quantities has been implemented. The set of solutions for the axial mean velocity includes a logarithmic scaling law, which is distinct from the usual law of the wall, and an algebraic scaling law. Furthermore, an algebraic scaling law for the azimuthal mean velocity is obtained. In all scaling laws the origin of the independent coordinate is located on the pipe axis, which is in contrast to the usual wall-based scaling laws. The present scaling laws show good agreement with both experimental and DNS data. As observed in experiments, it is shown that the axial mean velocity normalized with the mean bulk velocity um has a fixed point where the mean velocity equals the bulk velocity independent of the Reynolds number. An approximate location for the fixed point on the pipe radius is also given. All invariant solutions are consistent with all higher-order correlation equations. A large-Reynolds-number asymptotic expansion of the Navier–Stokes equations on the curved wall has been utilized to show that the near-wall scaling laws for at surfaces also apply to the near-wall regions of the turbulent pipe flow.

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.

Dissipation element analysis in turbulent channel flow

2012 ◽  
Vol 694 ◽  
pp. 332-351 ◽  
Author(s):  
Fettah Aldudak ◽  
Martin Oberlack
Keyword(s):  
Stokes Equations ◽  
Turbulent Channel Flow ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Length Scale ◽  
Navier Stokes Equations ◽  
Wall Distance ◽  
Extremal Points ◽  
The Mean ◽  
Log Normal

AbstractIn order to analyse the geometric structure of turbulent flow patterns and their statistics for various scalar fields we adopt the dissipation element (DE) approach and apply it to turbulent channel flow by employing direct numerical simulations (DNS) of the Navier–Stokes equations. Gradient trajectories starting from any point in a scalar field $\phi (x, y, z, t)$ in the directions of ascending and descending scalar gradients will always reach an extremum, i.e. a minimum or a maximum point, where $\boldsymbol{\nabla} \phi = 0$. The set of all points and trajectories belonging to the same pair of extremal points defines a dissipation element. Extending previous DE approaches, which were only applied to homogeneous turbulence, we here focus on exploring the influence of solid walls on the dissipation element distribution. Employing group-theoretical methods and known symmetries of Navier–Stokes equations, we observe for the core region of the flow, i.e. the region beyond the buffer layer, that the probability distribution function (p.d.f.) of the DE length exhibits an invariant functional form, in other words, self-similar behaviour with respect to the wall distance. This is further augmented by the scaling behaviour of the mean DE length scale which shows a linear scaling with the wall distance. The known proportionality of the mean DE length and the Taylor length scale is also revisited. Utilizing a geometric analogy we give the number of DE elements as a function of the wall distance. Further, it is observed that the DE p.d.f. is rather insensitive, i.e. invariant with respect both to the Reynolds number and the actual scalar $\phi $ which has been employed for the analysis. In fact, a very remarkable degree of isotropy is observed for the DE p.d.f. in regions of high shear. This is in stark contrast to classical Kolmogorov scaling laws which usually exhibit a strong dependence on quantities such as shear, anisotropy and Reynolds number. In addition, Kolmogorov’s scaling behaviour is in many cases only visible for very large Reynolds numbers. This is rather different in the present DE approach which applies also for low Reynolds numbers. Moreover, we show that the DE p.d.f. agrees very well with the log-normal distribution and derive a log-normal p.d.f. model taking into account the wall-normal dependence. Finally, the conditional mean scalar differences of the turbulent kinetic energy at the extremal points of DE are examined. We present a power law with scaling exponent of $2/ 3$ known from Kolmogorov’s hypothesis for the centre of the channel and a logarithmic law near the wall.

A Study of Unsteady Rotor–Stator Interactions

1989 ◽  
Vol 111 (4) ◽  
pp. 394-400 ◽  
Author(s):  
Reda R. Mankbadi
Keyword(s):  
Pressure Field ◽  
Stokes Equations ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Turbulence Energy ◽  
Mean Flow ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
The Mean ◽  
Field Decay

This work is concerned with simulations of rotor-generated unsteady response of the turbulent flow in a stator. The rotor’s effect is represented by moving cylinders of equivalent drag coefficient that produce passing wakes at the entrance of the stator. The unsteady incompressible Navier–Stokes equations are solved on a staggered grid and eddy viscosities are obtained using a k–ε model. The rotor-generated wakes were found to produce a pressure field at the stator’s entrance that increases in the direction of the wake traverse. At a streamwise distance equal to the distance between the stator blades, the pressure becomes uniform across the channel and the oscillations in the pressure field decay. Because of the initial asymmetry of the pressure field, the time-averaged mean velocity is no longer symmetric. This asymmetry of the mean flow continues along the passage even after the pressure has regained its symmetry. As a result of the passing of the rotor-generated wakes, large periodic oscillations are introduced into the mean velocity and turbulence energy. The time-averaged turbulence energy and the wall shear stress increases in the direction of the rotor traverse.

scholarly journals WATERWAVES CALCULATION BY NAVIER-STOKES EQUATIONS

1982 ◽  
Vol 1 (18) ◽  
pp. 53
Author(s):  
O. Daubert ◽  
A. Hauguel ◽  
J. Cahouet
Keyword(s):  
Stokes Equations ◽  
Vertical Plane ◽  
Viscosity Coefficient ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Simple Simulation ◽  
The Mean ◽  
Basic Equations ◽  
Eddy Viscosity Coefficient

N.S.L. program is a finite-difference code for two dimensionnal flows with a free surface in a vertical plane. Basic equations are Navier-Stokes Equations with a simple simulation of turbulent effects by an eddy viscosity coefficient related to the mixing length and the mean velocity gradient. Theses equations are solved in a variable domain in time. The main features of the numerical method are presented. Some comparisons with theoretical solutions give a good validation of the code both in linear and non linear cases. Other examples of application are given.

RANS Predictions in an Idealized Lower-Plenum Model

2006 ◽  
Author(s):  
Joshua D. Hodson ◽  
Robert E. Spall ◽  
Barton L. Smith
Keyword(s):  
Stokes Equations ◽  
Turbulence Models ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Pressure Losses ◽  
Pressure Loss Coefficient ◽  
The Mean ◽  
Recirculation Length

The two-dimensional, unsteady, Reynolds-averaged Navier-Stokes equations have been solved for the flow across a row of confined cylinders with a pitch-to-diameter ratio of 1.7, a configuration which was designed to model a next generation nuclear plant lower-plenum. Four different turbulence models were used: k–ε, k–ω, v2–f, and differential Reynolds-stress transport. Comparisons with available experimental data were made for pressure losses, recirculation lengths, and mean velocity profiles. The results indicate that all models did a reasonable job of predicting the pressure loss coefficient. However, in terms of the mean velocities and recirculation length, the determination of which model performed best is not clear.

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