Generalised higher-order Kolmogorov scales

2016 ◽  
Vol 794 ◽  
pp. 233-251 ◽  
Author(s):  
Jonas Boschung ◽  
Fabian Hennig ◽  
Michael Gauding ◽  
Heinz Pitsch ◽  
Norbert Peters
Keyword(s):  
Velocity Gradient ◽  
Longitudinal Velocity ◽  
Reynolds Numbers ◽  
Structure Functions ◽  
Local Isotropy ◽  
Even Order ◽  
The Mean ◽  
Grid Points ◽  
Exact Relations ◽  
Full Probability

Kolmogorov introduced dissipative scales based on the mean dissipation $\langle {\it\varepsilon}\rangle$ and the viscosity ${\it\nu}$, namely the Kolmogorov length ${\it\eta}=({\it\nu}^{3}/\langle {\it\varepsilon}\rangle )^{1/4}$ and the velocity $u_{{\it\eta}}=({\it\nu}\langle {\it\varepsilon}\rangle )^{1/4}$. However, the existence of smaller scales has been discussed in the literature based on phenomenological intermittency models. Here, we introduce exact dissipative scales for the even-order longitudinal structure functions. The derivation is based on exact relations between even-order moments of the longitudinal velocity gradient $(\partial u_{1}/\partial x_{1})^{2m}$ and the dissipation $\langle {\it\varepsilon}^{m}\rangle$. We then find a new length scale ${\it\eta}_{C,m}=({\it\nu}^{3}/\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$ and $u_{C,m}=({\it\nu}\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$, i.e. the dissipative scales depend rather on the moments of the dissipation $\langle {\it\varepsilon}^{m/2}\rangle$ and thus the full probability density function (p.d.f.) $P({\it\varepsilon})$ instead of powers of the mean $\langle {\it\varepsilon}\rangle ^{m/2}$. The results presented here are exact for longitudinal even-ordered structure functions under the assumptions of (local) isotropy, (local) homogeneity and incompressibility, and we find them to hold empirically also for the mixed and transverse as well as odd orders. We use direct numerical simulations (DNS) with Reynolds numbers from $Re_{{\it\lambda}}=88$ up to $Re_{{\it\lambda}}=754$ to compare the different scalings. We find that indeed $P({\it\varepsilon})$ or, more precisely, the scaling of $\langle {\it\varepsilon}^{m/2}\rangle /\langle {\it\varepsilon}\rangle ^{m/2}$ as a function of the Reynolds number is a key parameter, as it determines the ratio ${\it\eta}_{C,m}/{\it\eta}$ as well as the scaling of the moments of the velocity gradient p.d.f. As ${\it\eta}_{C,m}$ is smaller than ${\it\eta}$, this leads to a modification of the estimate of grid points required for DNS.

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.

Universality of the energy-containing structures in wall-bounded turbulence

2017 ◽  
Vol 823 ◽  
pp. 498-510 ◽  
Author(s):  
Charitha M. de Silva ◽  
Dominik Krug ◽  
Detlef Lohse ◽  
Ivan Marusic
Keyword(s):  
Turbulent Flows ◽  
Friction Velocity ◽  
Velocity Structure ◽  
Longitudinal Velocity ◽  
Reynolds Numbers ◽  
Structure Functions ◽  
Self Similarity ◽  
Logarithmic Region ◽  
High Reynolds ◽  
Wall Bounded Turbulence

The scaling behaviour of the longitudinal velocity structure functions $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$ (where $2p$ represents the order) is studied for various wall-bounded turbulent flows. It has been known that for very large Reynolds numbers within the logarithmic region, the structure functions can be described by $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}/U_{\unicode[STIX]{x1D70F}}^{2}\approx D_{p}\ln (r/z)+E_{p}$ (where $r$ is the longitudinal distance, $z$ the distance from the wall, $U_{\unicode[STIX]{x1D70F}}$ the friction velocity and $D_{p}$, $E_{p}$ are constants) in accordance with Townsend’s attached eddy hypothesis. Here we show that the ratios $D_{p}/D_{1}$ extracted from plots between structure functions – in the spirit of the extended self-similarity hypothesis – have further reaching universality for the energy containing range of scales. Specifically, we confirm that this description is universal across wall-bounded flows with different flow geometries, and also for both the longitudinal and transversal structure functions, where previously the scaling has been either difficult to discern or differences have been reported when examining the direct representation of $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$. In addition, we present evidence of this universality at much lower Reynolds numbers, which opens up avenues to examine structure functions that are not readily available from high Reynolds number databases.

Transport equations for the normalized nth-order moments of velocity derivatives in grid turbulence

2021 ◽  
Vol 930 ◽  
Author(s):  
S.L. Tang ◽  
R.A. Antonia ◽  
L. Djenidi
Keyword(s):  
Large Scale ◽  
Transverse Velocity ◽  
Longitudinal Velocity ◽  
Transport Equations ◽  
Small Scale ◽  
Grid Turbulence ◽  
Mean Velocity ◽  
Local Isotropy ◽  
Advection Term ◽  
The Mean

Transport equations for the normalized moments of the longitudinal velocity derivative ${F_{n + 1}}$ (here, $n$ is $1, 2, 3\ldots$ ) are derived from the Navier–Stokes (N–S) equations for shearless grid turbulence. The effect of the (large-scale) streamwise advection of ${F_{n + 1}}$ by the mean velocity on the normalized moments of the velocity derivatives can be expressed as $C_1 {F_{n + 1}}/Re_\lambda$ , where $C_1$ is a constant and $Re_\lambda$ is the Taylor microscale Reynolds number. Transport equations for the normalized odd moments of the transverse velocity derivatives ${F_{y,n + 1}}$ (here, $n$ is 2, 4, 6), which should be zero if local isotropy is satisfied, are also derived and discussed in sheared and shearless grid turbulence. The effect of the (large-scale) streamwise advection term on the normalized moments of the velocity derivatives can also be expressed in the form $C_2 {F_{y,n + 1}}/Re_\lambda$ , where $C_2$ is a constant. Finally, the contribution of the mean shear in the transport equation for ${F_{n + 1}}$ can be modelled as $15 B/Re_\lambda$ , where $B$ ( $=S^*{S_{s,n + 1}}$ ) is the product of the non-dimensional shear parameter $S^*$ and the normalized mixed longitudinal-transverse velocity derivatives ${{S_{s,n + 1}}}$ ; if local isotropy is satisfied, $S_{s,n + 1}$ should be zero. These results indicate that if ${F_{n + 1}}$ , ${F_{y,n + 1}}$ and $B$ do not increase as rapidly as $Re_\lambda$ , then the effect of the large-scale structures on small-scale turbulence will disappear when $Re_\lambda$ becomes sufficiently large.

Modified Self-Scaling Relation for the Inertial and Low Energy Containing Scales of Decaying Turbulence

1998 ◽  
Vol 12 (04) ◽  
pp. 405-431
Author(s):  
M. Hnatich ◽  
D. Horváth
Keyword(s):  
Reynolds Numbers ◽  
Structure Functions ◽  
Scaling Relation ◽  
Self Similarity ◽  
Extended Self ◽  
High Reynolds Numbers ◽  
The Mean ◽  
New Form ◽  
Decaying Turbulence ◽  
High Reynolds

The limits of a new form of scaling, named Extended Self Similarity (ESS) originally suggested [R. Benzi et al., Phys. Rev.E48 (1993), 29] for the inertial, dissipation and transition scales are discussed. A modification of the ESS concept is put forward using the model of decaying turbulence at high Reynolds numbers [L. Ts. Adzhemyan et al., Czech. J. Phys.45 (1995), 517]. In this model the statistical description is simplified by the hypotheses of homogeneity, isotropy, incompressibility and self-similarity, for the power law stage of decay the presence of a single scaling length — Karman scale — is assumed within the energy containing range. The second and third structure functions of the velocity field [S2(r) and S3(r)] have been calculated using the well-known connections between the mean energy spectrum and S2(r), and between mean spectral transfer and third structure function S3(r). Both structure functions have been investigated in the inertial and low enery containing ranges, then expressed in the form involving the leading Kolmogorov's K41 asymptotics [S2(r)∝ r2/3, S3(r)∝ r] and its asymptotical corrections. These corrections allow to determine corrections to the original ESS form [Formula: see text] (for K41) and to find out the modified variant of the ESS.

On the deficiency of even-order structure functions as inertial-range diagnostics

2008 ◽  
Vol 602 ◽  
pp. 287-302 ◽  
Author(s):  
P. A. DAVIDSON ◽  
P.-Å. KROGSTAD
Keyword(s):  
Structure Function ◽  
Isotropic Turbulence ◽  
Three Dimensional ◽  
Physical Space ◽  
Reynolds Numbers ◽  
Structure Functions ◽  
Inertial Range ◽  
Second Order ◽  
Order Structure ◽  
Even Order

In the limit of vanishing viscosity, ν→0, Kolmogorov's two-thirds, 〈(Δυ)2〉~ε2/3r2/3, and five-thirds, E~ε2/3k−5/3, laws are formally equivalent. (Here 〈(Δυ)2〉 is the second-order structure function, ε the dissipation rate, r the separation in physical space, E the three-dimensional energy spectrum, and k the wavenumber.) However, for the Reynolds numbers encountered in terrestrial experiments, or numerical simulations, it is invariably easier to observe the five-thirds law. We ask why this should be. To this end, we create artificial fields of isotropic turbulence composed of a random sea of Gaussian eddies whose size and energy distribution can be controlled. We choose the energy of eddies of scale, s, to vary as s2/3, in accordance with Kolmogorov's 1941 law, and vary the range of scales, γ=smax/smin, in any one realization from γ=25 to γ=800. This is equivalent to varying the Reynolds number in an experiment from Rλ=60 to Rλ=600. We find that, while there is some evidence of a five-thirds law for γ>50; (Rλ>100), the two-thirds law only starts to become apparent when γ approaches 200 (Rλ~240). The reason for this discrepancy is that the second-order structure function is a poor filter, mixing information about energy and enstrophy, and from scales larger and smaller than r. In particular, in the inertial range, 〈(Δυ)2〉 takes the form of a mixed power law, a1 + a2r2 + a3r2/3, where a2r2 tracks the variation in enstrophy and a3r2/3 the variation in energy. These findings are shown to be consistent with experimental data where the ‘pollution’ of the r2/3 law by the enstrophy contribution, a2r2, is clearly evident. We show that higher-order structure functions (of even order) suffer from a similar deficiency.

Pressure structure functions and spectra for locally isotropic turbulence

1995 ◽  
Vol 296 ◽  
pp. 247-269 ◽  
Author(s):  
Reginald J. Hill ◽  
James M. Wilczak
Keyword(s):  
Pressure Gradient ◽  
Structure Function ◽  
Fourth Order ◽  
Velocity Structure ◽  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Structure Functions ◽  
Inertial Range ◽  
Previous Theory ◽  
Local Isotropy

Beginning with the known relationship between the pressure structure function and the fourth-order two-point correlation of velocity derivatives, we obtain a new theory relating the pressure structure function and spectrum to fourth-order velocity structure functions. This new theory is valid for all Reynolds numbers and for all spatial separations and wavenumbers. We do not use the joint Gaussian assumption that was used in previous theory. The only assumptions are local homogeneity, local isotropy, incompressibility, and use of the Navier–Stokes equation. Specific formulae are given for the mean-squared pressure gradient, the correlation of pressure gradients, the viscous range of the pressure structure function, and the pressure variance. Of course, pressure variance is a descriptor of the energy-containing range. Therefore, for any Reynolds number, the formula for pressure variance requires the more restrictive assumption of isotropy. For the case of large Reynolds numbers, formulae are given for the inertial range of the pressure structure function and spectrum and of the pressure-gradient correlation; these are valid on the basis of local isotropy, as are the formulae for mean-squared pressure gradient and the viscous range of the pressure structure function. Using the experimentally verified extension to fourth-order velocity structure functions of Kolmogorov's theory, we obtain r4/3 and k−7/3 laws for the inertial range of the pressure structure function and spectrum. The modifications of these power laws to account for the effects of turbulence intermittency are also given. New universal constants are defined; these require experimental evaluation. The pressure structure function is sensitive to slight departures from local isotropy, implying stringent conditions on experimental data, but applicability of the previous theory is likewise constrained. The results are also sensitive to compressibility.

scholarly journals Aerodynamics of smooth and rough square-section prisms at incidence in very high Reynolds-number cross-flows

2021 ◽  
Vol 62 (3) ◽  
Author(s):  
Nils Paul van Hinsberg
Keyword(s):  
Reynolds Number ◽  
Cross Sections ◽  
Reynolds Numbers ◽  
Circular Cylinders ◽  
Surface Boundary ◽  
Experimental Proof ◽  
Surface Boundary Layer ◽  
Pitch Moment ◽  
The Mean ◽  
High Reynolds

Abstract The aerodynamics of smooth and slightly rough prisms with square cross-sections and sharp edges is investigated through wind tunnel experiments. Mean and fluctuating forces, the mean pitch moment, Strouhal numbers, the mean surface pressures and the mean wake profiles in the mid-span cross-section of the prism are recorded simultaneously for Reynolds numbers between 1$$\times$$ × 10$$^{5}$$ 5 $$\le$$ ≤ Re$$_{D}$$ D $$\le$$ ≤ 1$$\times$$ × 10$$^{7}$$ 7 . For the smooth prism with $$k_s$$ k s /D = 4$$\times$$ × 10$$^{-5}$$ - 5 , tests were performed at three angles of incidence, i.e. $$\alpha$$ α = 0$$^{\circ }$$ ∘ , −22.5$$^{\circ }$$ ∘ and −45$$^{\circ }$$ ∘ , whereas only both “symmetric” angles were studied for its slightly rough counterpart with $$k_s$$ k s /D = 1$$\times$$ × 10$$^{-3}$$ - 3 . First-time experimental proof is given that, within the accuracy of the data, no significant variation with Reynolds number occurs for all mean and fluctuating aerodynamic coefficients of smooth square prisms up to Reynolds numbers as high as $$\mathcal {O}$$ O (10$$^{7}$$ 7 ). This Reynolds-number independent behaviour applies to the Strouhal number and the wake profile as well. In contrast to what is known from square prisms with rounded edges and circular cylinders, an increase in surface roughness height by a factor 25 on the current sharp-edged square prism does not lead to any notable effects on the surface boundary layer and thus on the prism’s aerodynamics. For both prisms, distinct changes in the aerostatics between the various angles of incidence are seen to take place though. Graphic abstract

scholarly journals Plunging Airfoil: Reynolds Number and Angle of Attack Effects

Aerospace ◽  
2021 ◽  
Vol 8 (8) ◽  
pp. 216
Author(s):  
Emanuel A. R. Camacho ◽  
Fernando M. S. P. Neves ◽  
André R. R. Silva ◽  
Jorge M. M. Barata
Keyword(s):  
Reynolds Number ◽  
High Performance ◽  
Angle Of Attack ◽  
Systems Design ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Low Reynolds Numbers ◽  
Operational Conditions ◽  
Naca0012 Airfoil ◽  
The Mean

Natural flight has consistently been the wellspring of many creative minds, yet recreating the propulsive systems of natural flyers is quite hard and challenging. Regarding propulsive systems design, biomimetics offers a wide variety of solutions that can be applied at low Reynolds numbers, achieving high performance and maneuverability systems. The main goal of the current work is to computationally investigate the thrust-power intricacies while operating at different Reynolds numbers, reduced frequencies, nondimensional amplitudes, and mean angles of attack of the oscillatory motion of a NACA0012 airfoil. Simulations are performed utilizing a RANS (Reynolds Averaged Navier-Stokes) approach for a Reynolds number between 8.5×103 and 3.4×104, reduced frequencies within 1 and 5, and Strouhal numbers from 0.1 to 0.4. The influence of the mean angle-of-attack is also studied in the range of 0∘ to 10∘. The outcomes show ideal operational conditions for the diverse Reynolds numbers, and results regarding thrust-power correlations and the influence of the mean angle-of-attack on the aerodynamic coefficients and the propulsive efficiency are widely explored.

Axisymmetric internal waves generated by a travelling oscillating body

1969 ◽  
Vol 35 (2) ◽  
pp. 219-224 ◽  
Author(s):  
T. N. Stevenson
Keyword(s):  
Internal Waves ◽  
Salt Solution ◽  
Reynolds Numbers ◽  
Mean Velocity ◽  
Dispersive Waves ◽  
The Mean ◽  
Oscillating Sphere

Experiments are presented in which axisymmetric internal waves are generated by an oscillating sphere moving vertically in a stably stratified salt solution. The Reynolds numbers for the sphere based on the diameter and the mean velocity are between 10 and 200. Lighthill's theory for dispersive waves is used to calculate the phase configuration of the internal waves. The agreement between experiment and theory is reasonably good.

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