Equilibrium similarity solution of the turbulent transport equation along the centreline of a round jet

2015 ◽  
Vol 772 ◽  
pp. 740-755 ◽  
Author(s):  
H. Sadeghi ◽  
P. Lavoie ◽  
A. Pollard
Keyword(s):  
Experimental Data ◽  
Kinetic Energy ◽  
Structure Function ◽  
Transport Equation ◽  
Power Law ◽  
Turbulent Jet ◽  
Order Structure ◽  
Current Analysis ◽  
Power Law Decay ◽  
The Mean

A novel similarity-based form is derived of the transport equation for the second-order velocity structure function of$\langle ({\it\delta}q)^{2}\rangle$along the centreline of a round turbulent jet using an equilibrium similarity analysis. This self-similar equation has the advantage of requiring less extensive measurements to calculate the inhomogeneous (decay and production) terms of the transport equation. It is suggested that the normalised third-order structure function can be uniquely determined when the normalised second-order structure function, the power-law exponent of$\langle q^{2}\rangle$and the decay rate constants of$\langle u^{2}\rangle$and$\langle v^{2}\rangle$are available. In addition, the current analysis demonstrates that the assumption of similarity, combined with an inverse relation between the mean velocity$U$and the streamwise distance$x-x_{0}$from the virtual origin (i.e. $U\propto (x-x_{0})^{-1}$), is sufficient to predict a power-law decay for the turbulence kinetic energy ($\langle q^{2}\rangle \propto (x-x_{0})^{m}$), rather than requiring a power-law decay ($m=-2$) as an additionalad hocassumption. On the basis of the current analysis, it is suggested that the mean kinetic energy dissipation rate,$\langle {\it\epsilon}\rangle$, varies as$(x-x_{0})^{m-2}$. These theoretical results are tested against new experimental data obtained along the centreline of a round turbulent jet as well as previously published data on round jets for$11\,000\leqslant \mathit{Re}_{D}\leqslant 184\,000$over the range$10\leqslant x/D\leqslant 90$. For the present experiments,$\langle q^{2}\rangle$exhibits power-law behaviour with$m=-1.83$. The validity of this solution is confirmed using other experimental data where$\langle q^{2}\rangle$follows a power law with$-1.89\leqslant m\leqslant -1.78$. The present similarity form of the transport equation for$\langle ({\it\delta}q)^{2}\rangle$is also shown to be closely satisfied by the experimental data.

Transport equation for the mean turbulent energy dissipation rate in low- grid turbulence

2014 ◽  
Vol 747 ◽  
pp. 288-315 ◽  
Author(s):  
L. Djenidi ◽  
R. A. Antonia
Keyword(s):  
Reynolds Number ◽  
Energy Dissipation ◽  
Transport Equation ◽  
Power Law ◽  
Dissipation Rate ◽  
Low Reynolds Number ◽  
Energy Dissipation Rate ◽  
Grid Turbulence ◽  
Power Law Decay ◽  
The Mean

AbstractA direct numerical simulation (DNS) based on the lattice Boltzmann method (LBM) is carried out in low-Reynolds-number grid turbulence to analyse the mean turbulent kinetic energy dissipation rate, $\overline{\epsilon }$, and its transport equation during decay. All the components of $\overline{\epsilon }$ and its transport equation terms are computed, providing for the first time the opportunity to assess the contribution of each term to the decay. The results indicate that although small departures from isotropy are observed in the components of $\overline{\epsilon }$ and its destruction term, there is sufficient compensation among the components for these two quantities to satisfy isotropy to a close approximation. A short distance downstream of the grid, the transport equation of $\overline{\epsilon }$ simplifies to its high-Reynolds-number homogeneous and isotropic form. The decay rate of $\overline{\epsilon }$ is governed by the imbalance between the production due to vortex stretching and the destruction caused by the action of viscosity, the latter becoming larger than the former as the distance from the grid increases. This imbalance, which is not constant during the decay as argued by Batchelor & Townsend (Proc. R. Soc. Lond. A, vol. 190, 1947, pp. 534–550), varies according to a power law of $x$, the distance downstream of the grid. The non-constancy implies a lack of dynamical similarity in the mechanisms controlling the transport of $\overline{\epsilon }$. This is consistent with the fact that the power-law-decay ($\overline{q^2} \sim x^n$) exponent $n$ is not equal to $-$1. It is actually close to $-$1.6, a value in keeping with the relatively low Reynolds number of the simulation. These results highlight the importance of the imbalance in establishing the value of $n$. The $\overline{\epsilon }$-transport equation is also analysed in relation to the power-law decay. The results show that the power-law exponent $n$ is controlled by the imbalance between production and destruction. Further, a relatively straightforward analysis provides information on the behaviour of $n$ during the entire decay process and an interesting theoretical result, which is yet to be confirmed, when $R_{\lambda } \rightarrow 0 $, namely, the destruction coefficient $G$ is constant and its value must lie between $15/7$ and $30/7$. These two limits encompass the predictions for the final period of decay by Batchelor & Townsend (1947) and Saffman (J. Fluid Mech., vol. 27, 1967, pp. 581–593).

A vortex-street model of the flow in the similarity region of a two-dimensional free turbulent jet

1982 ◽  
Vol 123 ◽  
pp. 523-535 ◽  
Author(s):  
J. W. Oler ◽  
V. W. Goldschmidt
Keyword(s):  
Experimental Data ◽  
Reynolds Stress ◽  
Turbulent Jet ◽  
Vortex Street ◽  
Two Dimensional ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Core ◽  
The Mean ◽  
Similarity Relationships

The mean-velocity profiles and entrainment rates in the similarity region of a two-dimensional jet are generated by a simple superposition of Rankine vortices arranged to represent a vortex street. The spacings between the vortex centres, their two-dimensional offsets from the centreline, as well as the core radii and circulation strengths, are all governed by similarity relationships and based upon experimental data.Major details of the mean flow field such as the axial and lateral mean-velocity components and the magnitude of the Reynolds stress are properly determined by the model. The sign of the Reynolds stress is, however, not properly predicted.

scholarly journals A note on acoustic turbulence

2019 ◽  
Vol 874 ◽  
Author(s):  
Erik Lindborg
Keyword(s):  
Structure Function ◽  
Entropy Production ◽  
Acoustic Field ◽  
Three Dimensional ◽  
Weak Shock ◽  
Separation Distance ◽  
Ideal Gas ◽  
Order Structure ◽  
Specific Heats ◽  
The Mean

We consider a three-dimensional acoustic field of an ideal gas in which all entropy production is confined to weak shocks and show that similar scaling relations hold for such a field as for forced Burgers turbulence, where the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$ and the $p$th-order structure function scales as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, $\unicode[STIX]{x1D716}$ being the mean energy dissipation per unit mass, $d$ the mean distance between the shocks and $r$ the separation distance. However, for the acoustic field, $\unicode[STIX]{x1D716}$ should be replaced by $\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712}$, where $\unicode[STIX]{x1D712}$ is associated with entropy production due to heat conduction. In particular, the third-order longitudinal structure function scales as $\langle \unicode[STIX]{x1D6FF}u_{r}^{3}\rangle =-C(\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712})r$, where $C$ takes the value $12/5(\unicode[STIX]{x1D6FE}+1)$ in the weak shock limit, $\unicode[STIX]{x1D6FE}=c_{p}/c_{v}$ being the ratio between the specific heats at constant pressure and constant volume.

scholarly journals Shallow water wave turbulence

2019 ◽  
Vol 874 ◽  
pp. 1169-1196 ◽  
Author(s):  
Pierre Augier ◽  
Ashwin Vishnu Mohanan ◽  
Erik Lindborg
Keyword(s):  
Reynolds Number ◽  
Structure Function ◽  
Shallow Water ◽  
Water Wave ◽  
Structure Functions ◽  
Wave Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Shallow Water Wave ◽  
The Mean

The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the ‘four-fifths law’ of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$, where $\unicode[STIX]{x1D716}$ is the mean dissipation rate and $d$ the mean distance between the shocks, and that the $p$th-order displacement and velocity structure functions scale as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, where $r$ is the separation. Then we carry out a series of forced simulations with resolutions up to $7680^{2}$, varying the Froude number, $F_{f}=(\unicode[STIX]{x1D716}L_{f})^{1/3}/c$, where $L_{f}$ is the forcing length scale and $c$ is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as $E\sim \sqrt{\unicode[STIX]{x1D716}L_{f}c}$, and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson–Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as $d\sim F_{f}^{1/2}L_{f}$. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, $F_{f}\rightarrow 0$, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given $F_{f}$, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.

STUDY OF POWER-LAW CORRECTIONS TO SCALING IN A RELATIVISTIC QUARK MODEL

1990 ◽  
Vol 05 (13) ◽  
pp. 2591-2604
Author(s):  
L. S. CELENZA ◽  
A. PANTZIRIS ◽  
C. M. SHAKIN ◽  
HUI-WEN WANG
Keyword(s):  
Experimental Data ◽  
Structure Function ◽  
Quark Model ◽  
Power Law ◽  
Correction Term ◽  
Analytic Form ◽  
Recent Analysis ◽  
Relativistic Quark Model ◽  
Corrections To Scaling ◽  
Correction To Scaling

We study the approach to scaling in a relativistic quark model which we have used previously to fit the experimental data for the structure function [Formula: see text] (for all x) and for [Formula: see text] (for x > 0.4). We present values for these structure functions calculated in our model and also present an analytic form which provides a good fit to the numerical results. Our model supports one form of the power-law corrections suggested in the literature. [We reproduce the sign and magnitude of the parameter which appears in a phenomenological form used in a recent analysis made by the BEBC Collaboration. Our results are also consistent with a recent BCDMS Collaboration analysis which finds no need for "higher-twist" terms for Q2 > 20 (Gev) 2.] We also discuss certain complications which may arise when one attempts to study "higher-twist" behavior at both small and large x with a single sign for the phenomenological power-law correction term. (In particular, we predict a change in sign of the power-law correction to scaling between x = 0.65 and x = 0.7.) In addition, we calculate R(x, Q2) and find a good fit to the experimental data using the same parameters which were used in our fits to [Formula: see text] and [Formula: see text].

FRACTAL GROWTH OF BACTERIAL COLONIES

Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 869-877 ◽  
Author(s):  
FEREYDOON FAMILY
Keyword(s):  
Experimental Data ◽  
Escherichia Coli ◽  
Bacillus Subtilis ◽  
Power Law ◽  
Bacterial Growth ◽  
Phenomenological Approach ◽  
Nutrient Concentrations ◽  
Statistical Fluctuations ◽  
The Mean ◽  
Different Strains

The dynamics of the growth of three different strains of bacteria, ATCC 25589, Bacillus subtilis and Escherichia coli, was studied under different conditions of low as well as rich nutrient concentrations. We find that within the statistical fluctuations in the experimental data, the mean radius of the bacterial colonies grow with a power of time and the exponent characterizing this power law growth has an anomalous value. We present and discuss a simple phenomenological approach for explaining the existence of anomalous power-law exponents in bacterial growth. This approach may be useful in determining the key mechanisms which control the growth and morphology of bacterial colonies.

Studying the Number of Lineages Through Monte Carlo Simulations of Biological Ageing

1998 ◽  
Vol 09 (06) ◽  
pp. 809-813 ◽  
Author(s):  
S. Moss de Oliveira ◽  
G. A. de Medeiros ◽  
P. M. C. de Oliveira ◽  
D. Stauffer
Keyword(s):  
Power Law ◽  
Biological Diversity ◽  
String Model ◽  
Biological Ageing ◽  
Power Law Decay ◽  
Family Names ◽  
Size Number ◽  
The Family ◽  
The Mean ◽  
Number Of Individuals

We studied different versions of the Penna bit-string model for biological ageing and found that, after many generations, the number of lineages N (maternal family names) always decays to one as a power-law N∝t-z with an exponent z roughly equal to one. Measuring the mean correlation between the ancestor genome and those of the actual population we obtained the result that it goes to zero much earlier before the number of families goes to one, the population keeping thus its biological diversity. Considering maternal and paternal family names (doubled names) we also finished with only one pair of common ancestors. Computing the number of families of a given size as a function of the size (number of individuals the family has had during its whole existence) again a power-law decay is obtained.

scholarly journals First-Order Structure Function Analysis of Statistical Scale Invariance in the AIRS-Observed Water Vapor Field

2012 ◽  
Vol 25 (16) ◽  
pp. 5538-5555 ◽  
Author(s):  
Kyle G. Pressel ◽  
William D. Collins
Keyword(s):  
Water Vapor ◽  
Structure Function ◽  
Power Law ◽  
Scale Invariance ◽  
Structure Functions ◽  
Scale Dependence ◽  
Global Maximum ◽  
Order Structure ◽  
Scaling Exponents ◽  
First Order

Abstract The power-law scale dependence, or scaling, of first-order structure functions of the tropospheric water vapor field between 58°S and 58°N is investigated using observations from the Atmospheric Infrared Sounder (AIRS). Power-law scale dependence of the first-order structure function would indicate that the water vapor field exhibits statistical scale invariance. Directional and directionally independent first-order structure functions are computed to assess the directional dependence of derived first-order structure function scaling exponents (H) for a range of scales from 50 to 500 km. In comparison to other methods of assessing statistical scale invariance, the methodology used here requires minimal assumptions regarding the homogeneity of the spatial distribution of data within regions of analysis. Additionally, the methodology facilitates the evaluation of anisotropy and quantifies the extent to which the structure functions exhibit scale invariance. The spatial and seasonal dependence of the computed scaling exponents are explored. Minimum scaling exponents at all levels are shown to occur proximate to the equator, while the global maximum is shown to occur in the middle troposphere near the tropical–subtropical margin of the winter hemisphere. From a detailed analysis of AIRS maritime scaling exponents, it is concluded that the AIRS observations suggest the existence of two scaling regimes in the extratropics. One of these regimes characterizes the statistical scale invariance the free troposphere with H approximately = 0.55 and a second that characterizes the statistical scale invariance of the boundary layer with H approximately = ⅓.

The Calculation of Properties of the Flow over a Backward Facing Step With the k-ε Model of Turbulence

1984 ◽  
Vol 8 (3) ◽  
pp. 165-170
Author(s):  
L.P. Hackman ◽  
A.B. Strong ◽  
G.D. Raithby
Keyword(s):  
Experimental Data ◽  
Kinetic Energy ◽  
Turbulent Flow ◽  
Skin Friction ◽  
Turbulent Kinetic Energy ◽  
Reasonable Agreement ◽  
Shear Stresses ◽  
Backward Facing Step ◽  
Mean Velocity ◽  
The Mean

This paper reports predictions of the mean velocity, the turbulent kinetic energy and the pressure and skin friction coefficients for turbulent flow over a backward facing step based on the standard k – ε closure for the turbulence shear stresses. In previous publications, errors due to the numerical algorithm as distinct from the turbulence model have been carefully assessed using different numerical schemes and finite volume geometries and it is argued that the current results are numerically accurate. Thus one can now assess the accuracy of the k – ε model of turbulence independently of numerical error. The results predicted herein were found to be in reasonable agreement with relevant experimental data.

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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