Showing papers on "K-epsilon turbulence model published in 2004"

Effects of hydrophobic surface on skin-friction drag

Abstract: Effects of hydrophobic surface on skin-friction drag are investigated through direct numerical simulations of a turbulent channel flow. Hydrophobic surface is represented by a slip-boundary condition on the surface. When a slip-boundary condition is used in the streamwise direction, the skin-friction drag decreases and turbulence intensities and turbulence structures, near-wall streamwise vortices in particular, are significantly weakened. When a slip-boundary condition is used in the spanwise direction, on the other hand, the drag is increased. It is found that near-wall turbulence structures are modified differently, resulting in drag increase. It is also found that the slip length must be greater than a certain value in order to have a noticeable effect on turbulence. An important implication of the present finding is that drag reduction in turbulent boundary layers is unlikely with hydrophobic surface with its slip length on the order of a submicron scale.

A priori and a posteriori tests of inflow conditions for large-eddy simulation

Abstract: Comparisons of inflow conditions for large-eddy simulations of turbulent, wall-bounded flows are carried out. Consistent with previous investigations, it is found that the spectral content of the inflow velocity is important. Inflow conditions based on random-noise, or small-scale eddies only, dissipate quickly. Temporal and spatial filtering of a time series obtained from a separate calculation indicates that it is important to capture eddies of dimensions equal to or larger than the integral length scale of the flow. Three methods for generating inflow velocity fields are tested in a simulation of spatially developing turbulent channel flow. Synthetic turbulence generation methods that introduce realistic length scales are more suitable than uncorrelated random noise, but still require fairly long development lengths before realistic turbulence is established. A recycling method based on the use of turbulent data obtained from a separate calculation, in different flow conditions, was found to result in more rapid transition. A forcing method that includes a control loop also appears to be effective by generating turbulence with the correct Reynolds stresses and correlations within less than ten channel half heights.

Expanded analogy between Boltzmann kinetic theory of fluids and turbulence

Abstract: We demonstrate that the effects of turbulent fluctuations have a striking resemblance to those of microscale (thermal) fluctuations in laminar flows, even to higher order in the Knudsen number. This suggests that there may be a good basis for understanding turbulence in terms of Boltzmann kinetic theory. If so, turbulence may be better described in terms of ‘mixing times’ rather than the more classical ‘mixing lengths’. Comparisons are made to Reynolds-stress turbulence models.

Elastic turbulence in curvilinear flows of polymer solutions.

Abstract: Following our first report (A Groisman and V Steinberg 2000 Nature 405 53), we present an extended account of experimental observations of elasticity-induced turbulence in three different systems: a swirling flow between two plates, a Couette–Taylor (CT) flow between two cylinders, and a flow in a curvilinear channel (Dean flow). All three set-ups had a high ratio of the width of the region available for flow to the radius of curvature of the streamlines. The experiments were carried out with dilute solutions of high-molecular-weight polyacrylamide in concentrated sugar syrups. High polymer relaxation time and solution viscosity ensured prevalence of non-linear elastic effects over inertial non-linearity, and development of purely elastic instabilities at low Reynolds number (Re) in all three flows. Above the elastic instability threshold, flows in all three systems exhibit features of developed turbulence. They include: (i) randomly fluctuating fluid motion excited in a broad range of spatial and temporal scales and (ii) significant increase in the rates of momentum and mass transfer (compared with those expected for a steady flow with a smooth velocity profile). Phenomenology, driving mechanisms and parameter dependence of the elastic turbulence are compared with those of the conventional high-Re hydrodynamic turbulence in Newtonian fluids. Some similarities as well as multiple principal differences were found. In two out of three systems (swirling flow between two plates and flow in the curvilinear channel), power spectra of velocity fluctuations decayed rather quickly, following power laws with exponents of about −3.5. It suggests that, being random in time, the flow is rather smooth in space, in the sense that the main contribution to deformation and mixing (and, possibly, elastic energy) is coming from flow at the largest scale of the system. This situation, random in time and smooth in space, is analogous to flows at small scales (below the Kolmogorov dissipation scale) in high-Re turbulence.

On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows

Abstract: Numerical simulations of turbulent polymer solutions using the FENE-P model are used to characterize the action of polymers on turbulence in drag-reduced flows. The energetics of turbulence is investigated by correlating the work done by polymers on the flow with turbulent structures. Polymers are found to store and to release energy to the flow in a well-organized manner. The storage of energy occurs around near-wall vortices as has been anticipated for a long time. Quite unexpectedly, coherent release of energy is observed in the very near-wall region. Large fluctuations of polymer work are shown to re-energize decaying streamwise velocity fluctuations in high-speed streaks just above the viscous sublayer. These distinct behaviours are used to propose an autonomous regeneration cycle of polymer wall turbulence, in the spirit of Jimenez & Pinelli (1999).

Rayleigh Taylor turbulence: self-similar analysis and direct numerical simulations

Abstract: Direct numerical simulations and a self-similar analysis of the single-fluid Boussinesq Rayleigh-Taylor instability and transition to turbulence are used to investigate Rayleigh-Taylor turbulence. The Schmidt, Atwood and bulk Reynolds numbers are Sc = 1, A = 0.01, Re ≤ 3000. High-Reynolds-number moment self-similarity, consistent with the the energy cascade interpretation of dissipation, is used to analyse the DNS results

Turbulence spreading into the linearly stable zone and transport scaling

Abstract: We study the simplest problem of turbulence spreading corresponding to the spatio-temporal propagation of a patch of turbulence from a region where it is locally excited to a region of weaker excitation or even local damping. A single model equation for the local turbulence intensity, I(x, t), includes the effects of local linear growth and damping, spatially local nonlinear coupling to dissipation and spatial scattering of turbulence energy induced by nonlinear coupling. In the absence of dissipation, front propagation into the linearly stable zone occurs with the property of rapid progression at small t, followed by slower sub-diffusive progression at late times. The turbulence radial spreading into the linearly stable zone reduces the turbulent intensity in the linearly unstable zone and introduces an additional dependence on the ρ* ≡ ρi/a to the turbulent intensity and the transport scaling. These are in broad, semi-quantitative agreement with a number of global gyrokinetic simulation results with zonal flows and without zonal flows. Front propagation stops when the radial flux of fluctuation energy from the linearly unstable region is balanced by local dissipation in the linearly stable region.

A low-dimensional model for turbulent shear flows

Abstract: We analyse a low-dimensional model for turbulent shear flows. The model is based on Fourier modes and describes sinusoidal shear flow, in which the fluid between two free-slip walls experiences a sinusoidal body force. The model contains a total of nine modes, including modes describing the basic mean velocity profile and its modification, downstream vortices, streaks, and instabilities of streaks, with other modes being a consequence of the non- linear interactions. The transition to turbulence for the model is subcritical and intermittent, and the distributions of turbulent lifetimes are exponential, in agreement with observations in many shear flows.

Creating homogeneous and isotropic turbulence without a mean flow

Abstract: A novel method of creating homogeneous and isotropic turbulence with small mean flow has been developed. Eight synthetic jet actuators on the corners of a cubic chamber can create energetic turbulence with root-mean-square (rms) velocities as large as 0.87 m/s, corresponding to a Taylor microscale Reynolds number, Re λ , of 218. Stationary turbulence results show that the turbulence was isotropic, with the rms velocity ratio equal to 1.03, and also homogeneous within the region of interest. Natural decaying turbulence measurements confirmed the power-law decay of the turbulent kinetic energy, with the decay exponent n equal to 1.86 for an initial Re λ of 224.

Response of the wake of an isolated particle to an isotropic turbulent flow

Abstract: The interaction of an isolated spherical particle with an isotropic turbulent flow is considered using direct numerical simulations (DNS). The particle Reynolds number is varied from about 50 to 600 and the particle diameter is varied from about 1.5 to 10 times the Kolmogorov scale. The Reynolds number based on the Taylor microscale of the free-stream turbulent field considered here is 164. The DNS technique employed here is the first of its kind to address particle–turbulence interaction and it resolves the smallest scales in the free-stream turbulent flow and the complex vortical structures in the particle wake. The primary objective of this paper is to present new results on the effect of the free-stream turbulence on the particle wake and vortex shedding, and the modulation of free-stream turbulence in the particle wake. The parameters of the present simulations are comparable to those of the experimental study by Wu & Faeth (1994), and agreement between the present computational results and the experimental measurement is demonstrated.The effect of free-stream turbulence on the mean and instantaneous wake structure is studied. The time-averaged mean wake in a turbulent ambient flow shows a lower velocity deficit and a flatter profile. However, in agreement with the experimental results of Wu & Faeth the mean wake in a turbulent flow behaves like a self-preserving laminar wake. At Reynolds numbers below about 210 the effect of free-stream turbulence is to introduce wake oscillations. For Reynolds numbers in the range 210 to 280, free-stream turbulence is observed to promote early onset of vortex shedding. The nature of the shed vortices is somewhat different from that in a uniform flow. Increasing the free-stream turbulence intensity suppresses the process of vortex shedding, and only marginally increases the wake oscillation. The modulation of free-stream turbulence in the wake is studied in terms of the distribution of kinetic energy and RMS velocity fluctuation. The free-stream energy lost in the wake is recovered faster in a turbulent ambient flow than in a uniform ambient flow. The energy of the velocity fluctuation is enhanced in the wake at low free-stream intensities, and is damped or marginally increased at higher intensities. The fluctuation energy is not equi-partitioned among the streamwise and cross-stream components. The RMS streamwise fluctuation is always enhanced, whereas the RMS cross-stream fluctuation is enhanced only at low free-stream intensities, and damped at higher intensities.

Direct numerical simulations of isotropic compressible turbulence: Influence of compressibility on dynamics and structures

Abstract: In the present paper the statistical properties of compressible isotropic turbulence are analyzed by means of direct numerical simulations. The scope of the work is to evaluate the influence of compressibility on the time evolution of mean turbulence properties and to quantify the statistical properties of turbulent structures, their dynamics and similarities with the incompressible case. Simulations have been carried out at various turbulent Mach numbers and compressibility ratios by using a conservative hybrid scheme that relies on an optimized weighted essentially nonoscillatory approach for the convective terms and compact differencing for the viscous contributions. In order to identify similarities with incompressible turbulence we have also carried out an analysis in the plane of the second (Q*) and third (R*) invariants of the anisotropic part of the deformation rate tensor. The simulations show that the joint probability density function (Q*,R*) has a universal structure, as found in incompressibl...

Turbulence Characteristics and Interaction between Particles and Fluid in Particle-Laden Open Channel Flows

Abstract: Mechanism of sediment transport is composed of complicated interactions between turbulent flow, particle motion, and bed configurations. Of particular significance is the interaction between turbulence and particle motion, although turbulence measurements of particle-laden two phase flow have been a problem for a long time, especially in the near-wall region. In this study, simultaneous measurements of both the particles and fluid (water) were conducted in particle-laden two phase open channel flows by means of a discriminator particle-tracking velocimetry. The mean velocity and turbulence characteristics for fluid and particles each were examined in comparison with those in clear-water (particle-free) flow, together with previous existing data measured by laser Doppler anemometer and phase Doppler anemometer. The relative velocity and the turbulence modulation, which are the most important topics in two phase-flow approach, were revealed by varying the particle diameter and specific density. The fluid-sweeps are more contributory to the motion of particles than the fluid ejections in the near-wall region. In turn, the particle-sweeps transport the high momentum to the carrier fluid and enhance the turbulence intensities of fluid.

In-tube cooling heat transfer of supercritical carbon dioxide. Part 2. Comparison of numerical calculation with different turbulence models

Abstract: Major difficulty in the numerical calculation of heat transfers of supercritical carbon dioxide is the proper selection of the turbulence model. Because the thermophysical properties significantly depend on temperature and pressure, conventional turbulence models proposed for constant-property conditions might not be valid for supercritical pressure conditions, and therefore need to be analyzed carefully. Here, four turbulence models were applied to both heating and cooling conditions of supercritical carbon dioxide, and the simulation results of heat transfer coefficient were then compared with experimental data. The JL model (low Reynolds number k – e model by Jones and Launder) showed the best agreement with the experimental data. The three other models (a mixing length model by Bellmore and Reid, and two other low Reynolds number k – e models, respectively, by Launder and Sharma and by Myong and Kasagi) should be re-examined because they use a dimensionless distance from the wall y + . The turbulent Prandtl number did not significantly affect the calculation results of heat transfer coefficient.

Turbulence Structure of Hydraulic Jumps of Low Froude Numbers

Abstract: Turbulence characteristics of hydraulic jumps with Froude numbers of 2.0, 2.5, and 3.32 are presented. A Micro Acoustic Doppler velocimeter was used to obtain measurements of the velocities, turbulence intensities, Reynolds stresses, and power spectra. The maximum turbulence intensities and Reynolds stress at any section were found to decrease rapidly from the toe of the jump towards downstream within the jump and then gradually level off in the transition region from the end of the jump to the friction dominated open channel flow downstream. The maximum turbulence kinetic energy at each section decreases linearly with the longitudinal distance within the jump and gradually levels off in the transition region. The Reynolds stress and turbulence intensities within the jump show some degree of similarity. The dissipative eddy size was estimated to vary from 0.04 mm within the jump to 0.15 mm at the end of the transition region. The dominant frequency is in the range from 0 to 4 Hz for both horizontal and vertical velocity components.

Numerical Modelling of the Turbulent Flow Developing Within and Over a 3-D Building Array, Part I: A High-Resolution Reynolds-Averaged Navier Stokes Approach

Abstract: A study of the neutrally-stratified flow within and over an array of three-dimensional buildings (cubes) was undertaken using simple Reynolds-averaged Navier—Stokes (RANS) flow models. These models consist of a general solution of the ensemble-averaged, steady-state, three-dimensional Navier—Stokes equations, where the k-e turbulence model (k is turbulence kinetic energy and e is viscous dissipation rate) has been used to close the system of equations. Two turbulence closure models were tested, namely, the standard and Kato—Launder k-e models. The latter model is a modified k-e model designed specifically to overcome the stagnation point anomaly in flows past a bluff body where the standard k-e model overpredicts the production of turbulence kinetic energy near the stagnation point. Results of a detailed comparison between a wind-tunnel experiment and the RANS flow model predictions are presented. More specifically, vertical profiles of the predicted mean streamwise velocity, mean vertical velocity, and turbulence kinetic energy at a number of streamwise locations that extend from the impingement zone upstream of the array, through the array interior, to the exit region downstream of the array are presented and compared to those measured in the wind-tunnel experiment. Generally, the numerical predictions show good agreement for the mean flow velocities. The turbulence kinetic energy was underestimated by the two different closure models. After validation, the results of the high-resolution RANS flow model predictions were used to diagnose the dispersive stress, within and above the building array. The importance of dispersive stresses, which arise from point-to-point variations in the mean flow field, relative to the spatially-averaged Reynolds stresses are assessed for the building array.

Stochastic Models of Quasigeostrophic Turbulence

Abstract: Atmospheric and oceanic eddies are believed to be manifestations of quasigeostrophic turbulence — turbulence that occurs in rapidly rotating, vertically stratified fluid systems. The heat, momentum, and water transport by these eddies constitute a significant component of the climate balance, without which climate change cannot be understood. A major, unsolved problem is whether the turbulent eddy fluxes can be parameterized in terms of the large-scale, background flow. In the past, stochastic models have been used quite extensively to investigate quasigeostrophic turbulence in the case in which the eddy statistics are isotropic and homogeneous. Unfortunately, these models ignore the background shear which is absolutely essential to maintaining the eddies in the presence of dissipation. Recent attempts to extend stochastic models to shear flows have shown significant skill in predicting the structure of the eddy fluxes in arbitrary, three-dimensionally varying flows. This paper provides an accessible introduction to these models. The topics reviewed include quasigeostrophic turbulence and two-dimensional turbulence, non-modal andoptimal perturbations, mathematical theory of stochastic models, stochastic model simulations with realistic background states, and recent closure theories. A list of unsolved problems concludes this review.

Advances in wave turbulence: rapidly rotating flows

Abstract: At asymptotically high rotation rates, rotating turbulence can be described as a field of interacting dispersive waves by the general theory of weak wave turbulence. However, rotating turbulence has some complicating features, including the anisotropy of the wave dispersion relation and the vanishing of the wave frequency on a non-vanishing set of 'slow' modes. These features prevent straightforward application of existing theories and lead to some interesting properties, including the transfer of energy towards the slow modes. This transfer competes with, and might even replace, the transfer to small scales envisioned in standard turbulence theories.In this paper, anisotropic spectra for rotating turbulence are proposed based on weak turbulence theory; some evidence for their existence is given based on numerical calculations of the wave turbulence equations. Previous arguments based on the properties of resonant wave interactions suggest that the slow modes decouple from the others. Here, an extended wave turbulence theory with non-resonant interactions is proposed in which all modes are coupled; these interactions are possible only because of the anisotropy of the dispersion relation. Finally, the vanishing of the wave frequency on the slow modes implies that these modes cannot be described by weak turbulence theory. A more comprehensive approach to rotating turbulence is proposed to overcome this limitation.

Reynolds stress modeling of vegetated open-channel flows

Abstract: The Reynolds stress model is applied to open-channel flows with vegetation. For the computation of pressure-strain term, the Speziale, Sarkar, and Gatski's model is employed. Mellor and Herring's model and Rotta's model are used for diffusion and dissipation rate of Reynolds stress, respectively. Flow structures of open-channels under two vegetative conditions are simulated, namely submerged and emergent plants. Plain open-channel flows are also simulated for comparisons. Computed profiles are compared with the results from the κ-e model and the algebraic stress model as well as measured data available in the literature. For the plain open-channel flow and the open-channel flow with emergent vegetation, the Reynolds stress model is observed to simulate the non-isotropic nature of the flows better than the algebraic stress model and the κ-e model. For the open-channel flow with submerged vegetation, it is found that the Reynolds stress model predicts the mean flow and turbulence quantities best compared wi...

The Unsteady Development of a Turbulent Wake Through a Downstream Low-Pressure Turbine Blade Passage

Abstract: This paper presents two-dimensional LDA measurements of the convection of a wake through a low-pressure (LP) turbine cascade. Previous studies have shown the wake convection to be kinematic but have not provided details of the turbulent field. The spatial resolution of these measurements has facilitated the calculation of the production of turbulent kinetic energy and this has revealed a mechanism for turbulence production as the wake convects through the bladerow. The measured ensemble-averaged velocity field confirmed the previously reported kinematics of wake convection while the measurements of the turbulence quantities showed the wake fluid to be characterised by elevated levels of turbulent kinetic energy (TKE) and to have an anisotropic structure. Based on the measured mean and turbulence quantities, the production of turbulent kinetic energy was calculated. This highlighted a TKE production mechanism that resulted in increased levels of turbulence over the rear suction surface where boundary layer transition occurs. The turbulence production mechanism within the bladerow was also observed to produce more nearly isotropic turbulence. Production occurs when the principal stresses within the wake are aligned with the mean strains. This coincides with the maximum distortion of the wake within the blade passage and provides a mechanism for the production of turbulence outside of the boundary layer.Copyright © 2004 by ASME

Magnetic turbulence in the plasma sheet

Abstract: [1] Small-scale magnetic turbulence observed by the Cluster spacecraft in the plasma sheet is investigated by means of a wavelet estimator suitable for detecting distinct scaling characteristics even in noisy measurements. The spectral estimators used for this purpose are affected by a frequency-dependent bias. The variances of the wavelet coefficients, however, match the power-law shaped spectra, which makes the wavelet estimator essentially unbiased. These scaling characteristics of the magnetic field data appear to be essentially nonsteady and intermittent. The scaling properties of bursty bulk flow (BBF) and non-BBF associated magnetic fluctuations are analyzed with the aim of understanding processes of energy transfer between scales. Small-scale (∼0.08-0.3 s) magnetic fluctuations having the same scaling index α ∼ 2.6 as the large-scale (∼0.7-5 s) magnetic fluctuations occur during BBF-associated periods. During non-BBF associated periods the energy transfer to small scales is absent, and the large-scale scaling index a ∼ 1.7 is closer to Kraichnan or Iroshnikov-Kraichnan scalings. The anisotropy characteristics of magnetic fluctuations show both scale-dependent and scale-independent behavior. The former can be partly explained in terms of the Goldreich-Sridhar model of MHD turbulence, which leads to the picture of Alfvenic turbulence parallel and of eddy turbulence perpendicular to the mean magnetic field direction. Nonetheless, other physical mechanisms, such as transverse magnetic structures, velocity shears, or boundary effects can contribute to the anisotropy characteristics of plasma sheet turbulence. The scale-independent features are related to anisotropy characteristics which occur during a period of magnetic reconnection and fast tailward flow.

Two-equation Turbulence Modeling of Pulsatile Flow in a Stenosed Tube

Abstract: The study of pulsatile flow in stenosed vessels is of particular importance because of its significance in relation to blood flow in human pathophysiology. To date, however, there have been few comprehensive publications detailing systematic numerical simulations of turbulent pulsatile flow through stenotic tubes evaluated against comparable experiments. In this paper, two-equation turbulence modeling has been explored for sinusoidally pulsatile flow in 75% and 90% area reduction stenosed vessels, which undergoes a transition from laminar to turbulent flow as well as relaminarization. Wilcox's standard k-omega model and a transitional variant of the same model are employed for the numerical simulations. Steady flow through the stenosed tubes was considered first to establish the grid resolution and the correct inlet conditions on the basis of comprehensive comparisons of the detailed velocity and turbulence fields to experimental data. Inlet conditions based on Womersley flow were imposed at the inlet for all pulsatile cases and the results were compared to experimental data from the literature. In general, the transitional version of the k-omega model is shown to give a better overall representation of both steady and pulsatile flow. The standard model consistently over predicts turbulence at and downstream of the stenosis, which leads to premature recovery of the flow. While the transitional model often under-predicts the magnitude of the turbulence, the trends are well-described and the velocity field is superior to that predicted using the standard model. On the basis of this study, there appears to be some promise for simulating physiological pulsatile flows using a relatively simple two-equation turbulence model.

Turbulent diffusivity in the free atmosphere inferred from MST radar measurements: a review

Abstract: . The actual impact on vertical transport of small-scale turbulence in the free atmosphere is still a debated issue. Numerous estimates of an eddy diffusivity exist, clearly showing a lack of consensus. MST radars were, and continue to be, very useful for studying atmospheric turbulence, as radar measurements allow one to estimate the dissipation rates of energy (kinetic and potential) associated with turbulent events. The two commonly used methods for estimating the dissipation rates, from the backscattered power and from the Doppler width, are discussed. The inference methods of a local diffusivity (local meaning here "within" the turbulent patch) by using the dissipation rates are reviewed, with some of the uncertainty causes being stressed. Climatological results of turbulence diffusivity inferred from radar measurements are reviewed and compared. As revealed by high resolution MST radar measurements, atmospheric turbulence is intermittent in space and time. Recent theoretical works suggest that the effective diffusivity of such a patchy turbulence is related to statistical parameters describing the morphology of turbulent events: filling factor, lifetime and height of the patches. It thus appears that a statistical description of the turbulent patches' characteristics is required in order to evaluate and parameterize the actual impact of small-scale turbulence on transport of energy and materials. Clearly, MST radars could be an essential tool in that matter.

Separation of magnetic field lines in two-component turbulence

Abstract: The problem of the separation of random magnetic field lines in collisionless astrophysical plasmas is closely related to the problem of the magnetic field line random walk and is highly relevant to the transport of charged particles in turbulent plasmas. In order to generalize treatments based on quasi-linear theory, here we examine the separation of nearby magnetic field lines by employing a nonperturbative technique based on the Corrsin independence hypothesis. Specifically, we consider the case of two-component turbulence in which the magnetic field fluctuations are a mixture of one-dimensional (slab) and two-dimensional ingredients, as a concrete example of anisotropic turbulence that provides a useful description of turbulence in the solar wind. We find that random field trajectories can separate in general through three regimes of the behavior of the running diffusion coefficient: slow diffusive separation, an intermediate regime of superdiffusion, and fast diffusive separation at large distances. These features are associated with the gradual, exponential divergence of field lines within islands of two-dimensional turbulence, followed by diffusive separation at long distances. The types of behavior are determined not by the Kubo number but rather a related ratio that takes the turbulence anisotropy into account. These results are confirmed by computer simulations. We discuss implications for space observations of energetic charged particles, including ‘‘dropouts’’ of solar energetic particles.

Can high-order moments be meaningfully estimated from experimental turbulence measurements?

Abstract: Although high-order moments are widely used in the study of fully developed turbulence, their statistical properties remain poorly known. It is well known that beyond a given order, moment estimates based on finite samples cannot be trusted. We provide an empirical criterion for determining that order and illustrate it using a long record of boundary layer turbulence. The results show that even with modest levels of intermittency, structure functions in the inertial range of turbulence cannot be meaningfully assessed for orders as low as 5 or 6.