Pressure fluctuations in isotropic turbulence

1951 ◽  
Vol 47 (2) ◽  
pp. 359-374 ◽  
Author(s):  
G. K. Batchelor
Keyword(s):  
Pressure Gradient ◽  
Isotropic Turbulence ◽  
Alternative Hypothesis ◽  
Pressure Fluctuations ◽  
Reynolds Numbers ◽  
Fourth Moment ◽  
Mean Square ◽  
Important Special Case ◽  
Odd Order ◽  
The Mean

ABSTRACTThis paper considers the correlation P(r) between the fluctuating pressures at two different points distance r apart in a field of homogeneous isotropic turbulence. P(r) can be expressed in terms of the fourth moment of the velocity fluctuation, which is evaluated with the aid of the hypothesis that fourth moments are related to second moments in the same way as for a normal joint distribution of the velocities at any two points. The experimental evidence relevant to this hypothesis (which cannot be exactly true since it gives zero odd-order moments) is examined. The alternative hypothesis made by Heisenberg, that the Fourier coefficients of the velocity distribution are statistically independent, has identical consequences for the fourth moments of the velocity, although it does not lead to such convenient results.The pressure correlation is worked out in detail for the important special case of very large Reynolds numbers of turbulence; the mean-square pressure fluctuation is found to be . The mean-square pressure gradient is evaluated, from the available data concerning the doublevelocity correlation, for the cases of very small and very large Reynolds numbers, and a simple interpolation between these results is suggested for the general case. Finally, the relation between the mean-square pressure gradient and rate of diffusion of marked fluid particles from a fixed source is established without the neglect of the viscosity effect, and the available observations of diffusion are used to obtain estimates of which are compared with the theoretical values.

scholarly journals Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence

1978 ◽  
Vol 88 (4) ◽  
pp. 685-709 ◽  
Author(s):  
U. Schumann ◽  
G. S. Patterson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Numerical Study ◽  
Pressure Fluctuations ◽  
Reynolds Numbers ◽  
Companion Paper ◽  
Real Space ◽  
Pressure Gradients ◽  
Velocity Statistics ◽  
Decaying Turbulence

The spectral method of Orszag & Patterson has been extended to calculate the static pressure fluctuations in incompressible homogeneous decaying turbulence at Reynolds numbers Reλ [lsim ] 35. In real space 323 points are treated. Several cases starting from different isotropic initial conditions have been studied. Some departure from isotropy exists owing to the small number of modes at small wavenumbers. Root-mean-square pressure fluctuations, pressure gradients and integral length scales have been evaluated. The results agree rather well with predictions based on velocity statistics and on the assumption of normality. The normality assumption has been tested extensively for the simulated fields and found to be approximately valid as far as fourth-order velocity correlations are concerned. In addition, a model for the dissipation tensor has been proposed. The application of the present method to the study of the return of axisymmetric turbulence to isotropy is described in the companion paper.

scholarly journals On some mean value results involving djζ(1/2 + it)dj

2003 ◽  
Vol 127 (28) ◽  
pp. 17-29
Author(s):  
Aleksandar Ivic
Keyword(s):  
Mean Value ◽  
Fourth Moment ◽  
Mean Square ◽  
Error Terms ◽  
The Mean

Several problems involving E(T) and E2(T), the error terms in the mean square and mean fourth moment formula for |?(1/2 + it)|, are discussed. In particular it is proved that ?0T? E(t)E2(T)dt?T7/4(logT)7/2loglogT. .

The scale effect in compressible turbulence

1969 ◽  
Vol 65 (2) ◽  
pp. 557-566 ◽  
Author(s):  
D. G. Crighton
Keyword(s):  
Mach Number ◽  
Acoustic Pressure ◽  
Pressure Fluctuations ◽  
Compressible Turbulence ◽  
Fourth Power ◽  
Mean Square ◽  
Viscous Effects ◽  
Turbulent Region ◽  
The Mean ◽  
Diffusive Effects

AbstractThe pressure fluctuations in a compressible fluid in statistically steady isotropic motion are studied. These depend crucially on the scale of the turbulent region. If the region is small enough, the mean square acoustic pressure varies as the linear scale of the region, and as the fourth power of the turbulence Mach number. In the limit of infinite scale, however, diffusive effects limit the otherwise infinite pressure fluctuations as first shown by Lighthill(6). The mean square acoustic pressure then varies as the Mach number and as the turbulence Reynolds number. The pressure fluctuations above a sheet composed of statistically uniformly fluctuating sources are also examined. Here the mean square pressure diverges, initially, in proportion to the logarithm of the scale of the sheet, until viscous effects again become significant in limiting the pressure.

Moderate Reynolds number flows through periodic and random arrays of aligned cylinders

1997 ◽  
Vol 349 ◽  
pp. 31-66 ◽  
Author(s):  
DONALD L. KOCH ◽  
ANTHONY J. C. LADD
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Volume Fraction ◽  
Unit Length ◽  
Reynolds Numbers ◽  
Moderate Reynolds Number ◽  
Random Arrays ◽  
Flow Through ◽  
The Mean ◽  
Square Array

The effects of fluid inertia on the pressure drop required to drive fluid flow through periodic and random arrays of aligned cylinders is investigated. Numerical simulations using a lattice-Boltzmann formulation are performed for Reynolds numbers up to about 180.The magnitude of the drag per unit length on cylinders in a square array at moderate Reynolds number is strongly dependent on the orientation of the drag (or pressure gradient) with respect to the axes of the array; this contrasts with Stokes flow through a square array, which is characterized by an isotropic permeability. Transitions to time-oscillatory and chaotically varying flows are observed at critical Reynolds numbers that depend on the orientation of the pressure gradient and the volume fraction.In the limit Re[Lt ]1, the mean drag per unit length, F, in both periodic and random arrays, is given by F/(μU) =k1+k2Re2, where μ is the fluid viscosity, U is the mean velocity in the bed, and k1 and k2 are functions of the solid volume fraction ϕ. Theoretical analyses based on point-particle and lubrication approximations are used to determine these coefficients in the limits of small and large concentration, respectively.In random arrays, the drag makes a transition from a quadratic to a linear Re-dependence at Reynolds numbers of between 2 and 5. Thus, the empirical Ergun formula, F/(μU) =c1+c2Re, is applicable for Re>5. We determine the constants c1 and c2 over a wide range of ϕ. The relative importance of inertia becomes smaller as the volume fraction approaches close packing, because the largest contribution to the dissipation in this limit comes from the viscous lubrication flow in the small gaps between the cylinders.

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.

scholarly journals Turbulent boundary layers and channels at moderate Reynolds numbers

2010 ◽  
Vol 657 ◽  
pp. 335-360 ◽  
Author(s):  
JAVIER JIMÉNEZ ◽  
SERGIO HOYAS ◽  
MARK P. SIMENS ◽  
YOSHINORI MIZUNO
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Turbulent Flows ◽  
Pressure Fluctuations ◽  
Reynolds Numbers ◽  
Streamwise Velocity ◽  
Rotational Flow ◽  
Mean Velocity ◽  
The Mean ◽  
External Flows

The behaviour of the velocity and pressure fluctuations in the outer layers of wall-bounded turbulent flows is analysed by comparing a new simulation of the zero-pressure-gradient boundary layer with older simulations of channels. The 99 % boundary-layer thickness is used as a reasonable analogue of the channel half-width, but the two flows are found to be too different for the analogy to be complete. In agreement with previous results, it is found that the fluctuations of the transverse velocities and of the pressure are stronger in the boundary layer, and this is traced to the pressure fluctuations induced in the outer intermittent layer by the differences between the potential and rotational flow regions. The same effect is also shown to be responsible for the stronger wake component of the mean velocity profile in external flows, whose increased energy production is the ultimate reason for the stronger fluctuations. Contrary to some previous results by our group, and by others, the streamwise velocity fluctuations are also found to be higher in boundary layers, although the effect is weaker. Within the limitations of the non-parallel nature of the boundary layer, the wall-parallel scales of all the fluctuations are similar in both the flows, suggesting that the scale-selection mechanism resides just below the intermittent region, y/δ = 0.3–0.5. This is also the location of the largest differences in the intensities, although the limited Reynolds number of the boundary-layer simulation (Reθ ≈ 2000) prevents firm conclusions on the scaling of this location. The statistics of the new boundary layer are available from http://torroja.dmt.upm.es/ftp/blayers/.

Near-Wall Measurements in Turbulent Boundary Layers Using Laser Doppler Anemometry

2002 ◽  
Author(s):  
T. Gunnar Johansson ◽  
Luciano Castillo
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Pressure Gradient ◽  
Laser Doppler Anemometry ◽  
Reynolds Numbers ◽  
Mean Velocity ◽  
The Mean ◽  
Wall Shear ◽  
Near Wall

Near wall measurements have been performed in a zero pressure gradient turbulent boundary layer at low to moderate local Reynolds numbers using Laser-Doppler Anemometry in order to investigate how accurately the wall shear stress can be determined. Also, scaling problems are particularly difficult at low Reynolds numbers since they involve simultaneous influences of both inner and outer scales and this is most clearly observed in the near-wall region. In order to fully describe the zero pressure gradient turbulent boundary layer at low to moderate local Reynolds numbers it is necessary to accurately measure a number of quantities. These include the mean velocity and Reynolds stresses, and their spatial derivatives all the way down to the wall (y+∼1). Integral parameters that need to be measured are the wall shear stress and boundary layer thickness, particularly the momentum thickness. Problems with the measurement of field properties get worse close to a wall, and they get worse for increasing local Reynolds number. Three different approaches to measure the wall shear stress were examined. It was found that small measurement errors in the mean velocity close to the wall significantly reduced the accuracy in determining the wall shear stress by measuring the velocity gradient at the wall. The constant stress layer was found to be affected by the advection terms. However, it was found that taking the small pressure gradient into account and improving on the spatial resolution in the outer part of the boundary layer made the momentum integral method reliable.

scholarly journals On the strength of the nonlinearity in isotropic turbulence

2013 ◽  
Vol 733 ◽  
pp. 158-170 ◽  
Author(s):  
W. J. T. Bos ◽  
R. Rubinstein
Keyword(s):  
Nonlinear Term ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Direct Interaction ◽  
Inertial Range ◽  
Gaussian Field ◽  
Mean Square ◽  
Closed Expression ◽  
Gaussian Estimate ◽  
The Mean

AbstractTurbulence governed by the Navier–Stokes equations shows a tendency to evolve towards a state in which the nonlinearity is diminished. In fully developed turbulence, this tendency can be measured by comparing the variance of the nonlinear term to the variance of the same quantity measured in a Gaussian field with the same energy distribution. In order to study this phenomenon at high Reynolds numbers, a version of the direct interaction approximation is used to obtain a closed expression for the statistical average of the mean-square nonlinearity. The wavenumber spectrum of the mean-square nonlinear term is evaluated and its scaling in the inertial range is investigated as a function of the Reynolds number. Its scaling is dominated by the sweeping by the energetic scales, but this sweeping is weaker than predicted by a random sweeping estimate. At inertial range scales, the depletion of nonlinearity as a function of the wavenumber is observed to be constant. At large scales it is observed that the mean-square nonlinearity is larger than its Gaussian estimate, which is shown to be related to the non-Gaussianity of the Reynolds-stress fluctuations at these scales.

Enstrophy transport in swirl combustion

2019 ◽  
Vol 876 ◽  
pp. 715-732 ◽  
Author(s):  
Askar Kazbekov ◽  
Keishi Kumashiro ◽  
Adam M. Steinberg
Keyword(s):  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
The Other ◽  
Homogeneous Isotropic Turbulence ◽  
Relative Significance ◽  
Swirl Combustion ◽  
Image Velocimetry ◽  
Baroclinic Torque ◽  
Planar Laser ◽  
The Mean

The contributions of vortex stretching, dilatation, baroclinic torque and viscous diffusion to Reynolds-averaged enstrophy transport in turbulent swirl flames were experimentally measured using tomographic particle image velocimetry and $\text{CH}_{2}\text{O}$ planar laser induced fluorescence at jet Reynolds numbers of 26 000–51 000. The mean baroclinic torque was determined by subtracting the other terms in the enstrophy transport equation from the mean Lagrangian derivative. Enstrophy production from baroclinic torque was found to be significant relative to the other transport terms across all conditions studies. This result contrasts with direct numerical simulations of flames in homogeneous isotropic turbulence, which show a decreasing relative significance of baroclinic torque with increasing turbulence intensity (e.g. Bobbitt, Lapointe & Blanquart, Phys. Fluids, vol. 28 (1), 2016, 015101). Hence, the significance of baroclinic enstrophy production in flames is not determined entirely by the local turbulence and flame properties, but also depends on the configuration-specific pressure field.

The effect of velocity sensitivity on temperature derivative statistics in isotropic turbulence

1971 ◽  
Vol 48 (4) ◽  
pp. 763-769 ◽  
Author(s):  
J. C. Wyngaard
Keyword(s):  
Temperature Gradient ◽  
Temperature Sensor ◽  
Isotropic Turbulence ◽  
Mean Square ◽  
Temperature Derivative ◽  
Spectral Form ◽  
Temperature Spectrum ◽  
Velocity Sensitivity ◽  
The Mean ◽  
Third Moment

The velocity sensitivity of a resistance-wire temperature sensor is expressed in terms of sensor parameters, and the resulting errors in temperature derivative moments in isotropic turbulence are evaluated. It is shown that velocity sensitivity of a degree completely negligible for most purposes causes severe contamination of the measured third moment. The contamination terms are shown to be production rates of the mean square temperature gradient and vorticity, respectively, and therefore create positive values of measured derivative skewness. The dominant contamination term is related to the temperature spectrum through the balance equation for the mean-square temperature gradient, and calculations based on an assumed spectral form show that under typical conditions the measured skewness is large. This mechanism could provide an alternative to anisotropy as an explanation of the positive skewnesses recently measured in the atmosphere.

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