An empirical expression for on the axis of a slightly heated turbulent round jet

2019 ◽  
Vol 867 ◽  
pp. 392-413 ◽  
Author(s):  
J. Lemay ◽  
L. Djenidi ◽  
R. A. Antonia ◽  
A. Benaïssa
Keyword(s):  
Structure Function ◽  
Dissipation Rate ◽  
Velocity Structure ◽  
Power Laws ◽  
Second Order ◽  
Temperature Structure ◽  
Analytical Expression ◽  
Round Jet ◽  
Mean Temperature ◽  
The Mean

Self-preservation analyses of the equations for the mean temperature and the second-order temperature structure function on the axis of a slightly heated turbulent round jet are exploited in an attempt to develop an analytical expression for$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$, the mean dissipation rate of$\overline{\unicode[STIX]{x1D703}^{2}}/2$, where$\overline{\unicode[STIX]{x1D703}^{2}}$is the temperature variance. The analytical approach follows that of Thiessetet al.(J. Fluid Mech., vol. 748, 2014, R2) who developed an expression for$\unicode[STIX]{x1D716}_{k}$, the mean turbulent kinetic energy dissipation rate, using the transport equation for$\overline{(\unicode[STIX]{x1D6FF}u)^{2}}$, the second-order velocity structure function. Experimental data show that complete self-preservation for all scales of motion is very well satisfied along the jet axis for streamwise distances larger than approximately 30 times the nozzle diameter. This validation of the analytical results is of particular interest as it provides justification and confidence in the analytical derivation of power laws representing the streamwise evolution of different physical quantities along the axis, such as:$\unicode[STIX]{x1D702}$,$\unicode[STIX]{x1D706}$,$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$,$R_{U}$,$R_{\unicode[STIX]{x1D6E9}}$(all representing characteristic length scales), the mean temperature excess$\unicode[STIX]{x1D6E9}_{0}$, the mixed velocity–temperature moments$\overline{u\unicode[STIX]{x1D703}^{2}}$,$\overline{v\unicode[STIX]{x1D703}^{2}}$and$\overline{\unicode[STIX]{x1D703}^{2}}$and$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$. Simple models are proposed for$\overline{u\unicode[STIX]{x1D703}^{2}}$and$\overline{v\unicode[STIX]{x1D703}^{2}}$in order to derive an analytical expression for$A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}$, the prefactor of the power law describing the streamwise evolution of$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$. Further, expressions are also derived for the turbulent Péclet number and the thermal-to-mechanical time scale ratio. These expressions involve global parameters that are most likely to be influenced by the initial and/or boundary conditions and are therefore expected to be flow dependent.

scholarly journals Long-term Lidar Observations of the Gravity Wave Activity near the Mesopause at Arecibo

2018 ◽  
Author(s):  
Xianchang Yue ◽  
Jonathan S. Friedman ◽  
Qihou Zhou ◽  
Xiongbin Wu ◽  
Jens Lautenbach
Keyword(s):  
Potential Energy ◽  
Gravity Wave ◽  
Inversion Layer ◽  
Lower Thermosphere ◽  
Temperature Inversion ◽  
Temperature Structure ◽  
Mean Temperature ◽  
The Mean ◽  
Highly Correlated ◽  
Lidar Observations

Abstract. 11-years long K Doppler lidar observations of temperature profiles in the mesosphere and lower thermosphere (MLT) between 85 and 100 km, conducted at the Arecibo Observatory, Puerto Rico (18.35° N, 66.75° W), are used to estimate seasonal variations of the mean temperature, the squared Brunt-Väisälä frequency, and the gravity wave potential energy in a composite year. The following unique features are obtained: (1) The mean temperature structure shows similar characteristics as a prior report based on a smaller dataset: (2) The profiles of the squared Brunt-Väisälä frequency usually reach the maxima at or just below the temperature inversion layer when that layer is present. The first complete range-resolved climatology of potential energy of temperature fluctuations in the tropical MLT exhibits an altitude dependent combination of annual oscillation (AO) and semiannual oscillation (SAO). Between 88 to 96 km altitude, the amplitudes of AO and SAO are comparable, and their phases are almost the same and quite close to day of year (DOY) 100. Below 88 km, the SAO amplitude is significantly larger than AO and the AO phase shifts to DOY 200 and after. At 97 to 98 km altitude, the amplitudes of AO and SAO reach their minima, and both phases shift significantly. Above that, the AO amplitude becomes greater. The annual mean potential energy profile reaches the minimum at 91 to 92 km altitude. The altitude-dependent SAO of the potential energy is found to be highly correlated with the satellite observed mean zonal winds reported in the literature.

Three-component vorticity measurements in a turbulent grid flow

1998 ◽  
Vol 374 ◽  
pp. 29-57 ◽  
Author(s):  
R. A. ANTONIA ◽  
T. ZHOU ◽  
Y. ZHU
Keyword(s):  
Energy Dissipation ◽  
Dissipation Rate ◽  
Velocity Structure ◽  
Transverse Velocity ◽  
Longitudinal Velocity ◽  
Energy Dissipation Rate ◽  
Grid Turbulence ◽  
Local Isotropy ◽  
Velocity Increments ◽  
The Mean

All components of the fluctuating vorticity vector have been measured in decaying grid turbulence using a vorticity probe of relatively simple geometry (four X-probes, i.e. a total of eight hot wires). The data indicate that local isotropy is more closely satisfied than global isotropy, the r.m.s. vorticities being more nearly equal than the r.m.s. velocities. Two checks indicate that the performance of the probe is satisfactory. Firstly, the fully measured mean energy dissipation rate 〈ε〉 is in good agreement with the value inferred from the rate of decay of the mean turbulent energy 〈q2〉 in the quasi-homogeneous region; the isotropic mean energy dissipation rate 〈εiso〉 agrees closely with this value even though individual elements of 〈ε〉 indicate departures from isotropy. Secondly, the measured decay rate of the mean-square vorticity 〈ω2〉 is consistent with that of 〈q2〉 and in reasonable agreement with the isotropic form of the transport equation for 〈ω2〉. Although 〈ε〉≃〈εiso〉, there are discernible differences between the statistics of ε and εiso; in particular, εiso is poorly correlated with either ε or ω2. The behaviour of velocity increments has been examined over a narrow range of separations for which the third-order longitudinal velocity structure function is approximately linear. In this range, transverse velocity increments show larger departures than longitudinal increments from predictions of Kolmogorov (1941). The data indicate that this discrepancy is only partly associated with differences between statistics of locally averaged ε and ω2, the latter remaining more intermittent than the former across this range. It is more likely caused by a departure from isotropy due to the small value of Rλ, the Taylor microscale Reynolds number, in this experiment.

scholarly journals On the Relationship between the TKE Dissipation Rate and the Temperature Structure Function Parameter in the Convective Boundary Layer

2020 ◽  
Vol 77 (7) ◽  
pp. 2311-2326
Author(s):  
Hubert Luce ◽  
Lakshmi Kantha ◽  
Hiroyuki Hashiguchi ◽  
Abhiram Doddi ◽  
Dale Lawrence ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Structure Function ◽  
Convective Boundary Layer ◽  
Dissipation Rate ◽  
Function Parameter ◽  
Mixing Efficiency ◽  
Buoyancy Frequency ◽  
Temperature Structure ◽  
Convective Boundary

AbstractUnder stably stratified conditions, the dissipation rate ε of turbulence kinetic energy (TKE) is related to the structure function parameter for temperature , through the buoyancy frequency and the so-called mixing efficiency. A similar relationship does not exist for convective turbulence. In this paper, we propose an analytical expression relating ε and in the convective boundary layer (CBL), by taking into account the effects of nonlocal heat transport under convective conditions using the Deardorff countergradient model. Measurements using unmanned aerial vehicles (UAVs) equipped with high-frequency response sensors to measure velocity and temperature fluctuations obtained during the two field campaigns conducted at Shigaraki MU observatory in June 2016 and 2017 are used to test this relationship between ε and in the CBL. The selection of CBL cases for analysis was aided by auxiliary measurements from additional sensors (mainly radars), and these are described. Comparison with earlier results in the literature suggests that the proposed relationship works, if the countergradient term γD in the Deardorff model, which is proportional to the ratio of the variances of potential temperature θ and vertical velocity w, is evaluated from in situ (airplane and UAV) observational data, but fails if evaluated from large-eddy simulation (LES) results. This appears to be caused by the tendency of the variance of θ in the upper part of the CBL and at the bottom of the entrainment zone to be underestimated by LES relative to in situ measurements from UAVs and aircraft. We discuss this anomaly and explore reasons for it.

scholarly journals Exact second-order structure-function relationships

2002 ◽  
Vol 468 ◽  
pp. 317-326 ◽  
Author(s):  
REGINALD J. HILL
Keyword(s):  
Structure Function ◽  
Scaling Laws ◽  
Velocity Structure ◽  
Structure Functions ◽  
Energy Dissipation Rate ◽  
Second Order ◽  
Order Structure ◽  
Third Order ◽  
Navier Stokes Equation ◽  
Local Isotropy

Equations that follow from the Navier–Stokes equation and incompressibility but with no other approximations are ‘exact’. Exact equations relating second- and third- order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: (i) the trace of the structure functions, (ii) DNS that has periodic boundary conditions, and (iii) an average over a sphere in r-space. Case (iii) introduces the average over orientations of r into the structure-function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

scholarly journals Impact of ADCP motion on structure function estimates of turbulent kinetic energy dissipation rate

2021 ◽  
Author(s):  
Brian Daniel Scannell ◽  
Yueng-Djern Lenn ◽  
Tom P. Rippeth
Keyword(s):  
Structure Function ◽  
Dissipation Rate ◽  
Function Method ◽  
Energy Dissipation Rate ◽  
Flow Speed ◽  
Angular Range ◽  
Beam Velocity ◽  
Flow Speeds ◽  
The Mean ◽  
The Impact

Abstract. Turbulent mixing is a key process in the transport of heat, salt and nutrients in the marine environment, with fluxes commonly derived directly from estimates of the turbulent kinetic energy dissipation rate, ϵ. Time series of ϵ estimates are therefore useful in helping to identify and quantify key biogeochemical processes. Estimates of ϵ are typically derived using shear microstructure profilers, which provide high resolution vertical profiles, but require a surface vessel, incurring costs and limiting the duration of observations and the conditions under which they can be made. The velocity structure function method can be used to determine time series of ϵ estimates using along-beam velocity measurements from suitably configured acoustic Doppler current profilers (ADCP). Shear in the background current can bias such estimates, therefore standard practice is to deduct the mean or linear trend from the along-beam velocity over the period of an observation burst. This procedure is effective if the orientation of the ADCP to the current remains constant over the burst period. However, if the orientation of a tethered ADCP varies, a proportion of the velocity difference between bins is retained in the structure function and the resulting ϵ estimates will be biased. Long-term observations from a mooring with three inline ADCP show the heading oscillating with an angular range that depends on the flow speed; from large, slow oscillations at low flow speeds to smaller, higher frequency oscillations at higher flow speeds. The mean tilt was also determined by the flow speed, whilst the tilt oscillation range was primarily determined by surface wave height. Synthesised along-beam velocity data for an ADCP subject to sinusoidal oscillation in a sheared flow indicates that the retained proportion of the potential bias is primarily determined by the angular range of the oscillation, with the impact varying between beams depending on the mean heading relative to the flow. Since the heading is typically unconstrained in a tethered mooring, heading oscillation is likely to be the most significant influence on the retained bias for a given level of shear. Use of an instrument housing designed to reduce oscillation would mitigate the impact, whilst if the shear is linear over the observation depth range, the bias can be corrected using a modified structure function method designed to correct for bias due to surface waves.

Complete self-preservation on the axis of a turbulent round jet

2016 ◽  
Vol 790 ◽  
pp. 57-70 ◽  
Author(s):  
L. Djenidi ◽  
R. A. Antonia ◽  
N. Lefeuvre ◽  
J. Lemay
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Energy Budget ◽  
Dissipation Rate ◽  
Longitudinal Velocity ◽  
Order Structure ◽  
Mean Velocity ◽  
Round Jet ◽  
The Mean

Self-preservation (SP) solutions on the axis of a turbulent round jet are derived for the transport equation of the second-order structure function of the turbulent kinetic energy ($k$), which may be interpreted as a scale-by-scale (s.b.s.) energy budget. The analysis shows that the mean turbulent energy dissipation rate, $\overline{{\it\epsilon}}$, evolves like $x^{-4}$ ($x$ is the streamwise direction). It is important to stress that this derivation does not use the constancy of the non-dimensional dissipation rate parameter $C_{{\it\epsilon}}=\overline{{\it\epsilon}}u^{\prime 3}/L_{u}$ ($L_{u}$ and $u^{\prime }$ are the integral length scale and root mean square of the longitudinal velocity fluctuation respectively). We show, in fact, that the constancy of $C_{{\it\epsilon}}$ is simply a consequence of complete SP (i.e. SP at all scales of motion). The significance of the analysis relates to the fact that the SP requirements for the mean velocity and mean turbulent kinetic energy (i.e. $U\sim x^{-1}$ and $k\sim x^{-2}$ respectively) are derived without invoking the transport equations for $U$ and $k$. Experimental hot-wire data along the axis of a turbulent round jet show that, after a transient downstream distance which increases with Reynolds number, the turbulence statistics comply with complete SP. For example, the measured $\overline{{\it\epsilon}}$ agrees well with the SP prediction, i.e. $\overline{{\it\epsilon}}\sim x^{-4}$, while the Taylor microscale Reynolds number $Re_{{\it\lambda}}$ remains constant. The analytical expression for the prefactor $A_{{\it\epsilon}}$ for $\overline{{\it\epsilon}}\sim (x-x_{o})^{-4}$ (where $x_{o}$ is a virtual origin), first developed by Thiesset et al. (J. Fluid Mech., vol. 748, 2014, R2) and rederived here solely from the SP analysis of the s.b.s. energy budget, is validated and provides a relatively simple and accurate method for estimating $\overline{{\it\epsilon}}$ along the axis of a turbulent round jet.

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