Towards a stochastic multi-point description of turbulence

It has been found in previous work that the multi-scale statistics of homogeneous isotropic turbulence can be described by a stochastic ‘cascade’ process of the velocity increment from scale to scale, which is governed by a Fokker–Planck equation. We show in this paper how this description can be extended to obtain the complete multi-point statistics of the velocity field. We extend the stochastic cascade description by conditioning on the velocity value itself and find that the corresponding process is also governed by a Fokker–Planck equation, which contains as a leading term a simple additional velocity-dependent coefficient in the drift function. Taking into account the velocity dependence of the Fokker–Planck equation, the multi-point statistics in the inertial range can be expressed by the two-scale statistics of velocity increments, which are equivalent to the three-point statistics of the velocity field. Thus, we propose a stochastic three-point closure for the velocity field of homogeneous isotropic turbulence.

Nawroth and Peinke [23] noted that this multi-scale description can also be used for the generation of synthetic time series with the same statistical multi-scale properties as, for example, a given turbulent velocity time series, or for the prediction of financial time series [24]. They also noted that it is necessary to take into account the statistics of the velocity itself in order to obtain stationary synthetic data [23]. In this work, we want to go beyond this approach and show precisely how the joint N -scale PDF p[ξ(r 1 ), ξ(r 2 ), . . . , ξ(r N )] is related to the joint (N + 1)-point PDF p[U (x), U (x + r 1 ), . . . , U (x + r N )]. As a consequence of the Markov properties of the velocity increments, the multi-point statistics can be obtained based on the knowledge of the velocity field at three points. Thus, we propose a stochastic three-point closure for homogeneous isotropic turbulence.
This paper is organized as follows. In section 2, we examine the relationship between multi-scale and multi-point statistics. We derive a velocity-dependent Fokker-Planck equation for the velocity increments, which gives access to the multi-point statistics. In section 3, we present experimental evidence for the validity of such a stochastic description, and examine the empirical velocity dependence of the Fokker-Planck equation. Section 4 concludes the paper.

Multi-scale and multi-point statistics
In the following, we use the notation u := U −Ū for the velocity fluctuation, where U is the velocity andŪ is the mean flow velocity. We also use the short-hand notations u i := u(x i ) and u i+ j := u(x i + r j ), and for the velocity increment ξ at scale r j we write The starting point of this paper is the finding that the N -scale statistics of velocity increments can be expressed by two-scale conditional probability densities p(ξ j |ξ j+1 ). More precisely, in several papers, cf [18]- [22], it has been shown by experimental evidence that the stochastic process of the velocity increment ξ in scale r has the Markov property where the process direction has been chosen from large to small scales, by sorting the scale variables r j as r 1 < r 2 < · · · < r N . This equation is usually studied for N = 2, as an adequate simplification for finite data sets [14], We note that (2) and (3) hold only for step sizes r := r j+1 − r j that are larger than the socalled Einstein-Markov coherence length l EM , which is of the order of magnitude of the Taylor microscale λ [16,25]. If (2) holds, the N -scale joint PDF of the velocity increments can be expressed by a product of conditional PDFs, As the next step we consider the (N + 1)-point statistics that can be expressed by 4 as can be easily seen, since the increments ξ j of this equation are calculated from the velocity values that appear on the left-hand side 3 . Let us assume that the Markov property of the interscale process is conserved when the process is conditioned on the velocity, i.e. let us assume that which can be simplified to a sufficient condition for finite data, Equation (6) implies the following factorization of the multi-point joint PDF: The evolution of the conditional PDFs of (8) in scale r j can be expressed by a Kramers-Moyal expansion [26], where r k > r j , with the Kramers-Moyal coefficients Note that in contrast to the usual notation, we multiplied both sides of (9) by r j , and we have a negative sign on the left-hand side due to the process direction from large to small scales. If the Kramers-Moyal coefficient of fourth order, D (4) , vanishes, the expansion truncates after the second term and becomes a Fokker-Planck equation [26], In the same way, a velocity-independent Fokker-Planck equation can be derived from (4), It has been shown for homogeneous isotropic turbulence that the coefficientD (4) (ξ j , r j ) can in fact be neglected [18,20]. Therefore, the Fokker-Planck equation (12) is an adequate description of the interscale process of the velocity increments, giving access to the N -scale joint PDF of (4). In order to step from the description of multi-scale to multi-point statistics, we have to examine the validity of (7), check whether or not the coefficient D (4) (ξ j , r j , u i ) can be neglected and study the empirical dependence of the Fokker-Planck equation (11) on the velocity u i . This will be done in section 3.

Experimental results
We analyze hot-wire measurement data from three different flow types at different Reynolds numbers. The first flow type is the wake of a cylinder with diameter D = 2 cm in a wind tunnel at downstream distance x = 100D, for Taylor microscale Reynolds numbers R λ over the range 86-338 [27]. We use the data at R λ = 338 for the most detailed analysis. These data are characterized by a dissipation scale η = 0.10 mm, Taylor microscale λ = 3.7 mm and integral scale L = 119 mm. The second flow type is an axisymmetric air free jet at 145 nozzle-diameters distance from the nozzle, at R λ = 190, with η = 0.25 mm, λ = 6.6 mm and L = 67 mm. These data are described in detail in [18]. Finally, we also analyze data from the flow behind a so-called space-filling fractal square grid, described in detail in [28], measured at different positions in the decay region of the turbulence in the wind tunnel, for different flow speeds, at Reynolds numbers R λ between 175 and 740. The cylinder wake and free jet data sets have a total length of 12.5 × 10 6 values each, and the fractal grid data sets have a length of 3 × 10 6 values each. We analyze the streamwise velocity component of the cylinder wake and fractal grid data, which has been measured with cross-wire probes. The free jet data have been measured with a single wire.
We examine the dependence of the conditional PDFs of the velocity increments on the velocity itself by comparing both sides of the equation Note that if equation (13) holds, the Fokker-Planck equations (11) and (12) will be identical.
In figure 1, we compare the contour plots of both conditional distributions of (13) for the free jet at scales r 1 = 2λ ≈ L/4 and r 2 = 2r 1 for two different values of u i . The distributions agree almost perfectly for u i = 0, while for a larger velocity, u i = 2σ u (where σ u = u 2 ), the distribution p(ξ 1 |ξ 2 , u i ) is slightly shifted downward with respect to the other distribution p(ξ 1 |ξ 2 ). According to the errors of p(ξ 1 |ξ 2 , u i ) shown in this figure, the shift is statistically significant 4 . Such a shift of the conditional PDF for velocities u i = 0 can be observed at any scale r 1 for all the examined flows. As we will see below, this shift will lead to a velocitydependent term in the drift function of the Fokker-Planck equation.
In section 2, we argued that the multi-point statistics might be obtained from the velocitydependent Fokker-Planck equation (11) of the interscale process. A necessary condition for this description to hold is the validity of (7), which states that the Markov property of the interscale process is conserved under the additional condition of u i . In order to verify the validity of (7), we apply the (Mann-Whitney-) Wilcoxon test [18,29,30], which tests whether or not two samples of different sizes have the same statistical distribution. Since the two distributions of (7) are necessarily of different sizes, the Wilcoxon test is an appropriate method for estimating its validity. The test is described in detail in the appendix. Figure 2 shows the results of the Wilcoxon test for (7) for different values of u i and ξ j+2 . In the present implementation of the Wilcoxon test, a statistical test value Q * is computed, which must be close to 1 for   (7) for different values of u i and ξ j+2 , as indicated in the legend (ξ j+2 = 0 is short for ξ j+2 = 0 ± σ u /8, and ξ j+2 = (−)σ u is short for ξ j+2 = (−)σ u ± σ u /6). Cylinder wake at R λ = 338. 7 acceptance of the hypothesis expressed by (7). The values Q * in figure 2 are in fact identical to those for 1, except for some inevitable scattering, for large enough scale distances r > l EM ≈ 0.6λ. Thus, the Markov property of the interscale process is conserved under the additional condition of u i , and equations (7) and (8) do apply.
The drift and diffusion functions of (12) for homogeneous isotropic turbulence are approximately linear and second-order functions in ξ , as was found previously in [18,19]: Here and for the remaining part of the paper, we skip the indices in ξ j , r j and u i for simplicity. The velocity increments ξ are given in units of their standard deviation in the limit r → ∞, σ ∞ , which is identical to √ 2σ u ≡ 2 u 2 [18]. This normalization allows a comparison of the Kramers-Moyal coefficients of different flows. As pointed out in [18], it is known that a diffusion function that is constant in ξ has Gaussian solutions, whereas the additional ξdependent terms present in (14b) are responsible for intermittency effects and anomalous scaling of the structure functions.
In the following, we examine the velocity dependence of the drift and diffusion functions.
The most significant difference to (14a) and (14b) is the presence of the additional velocitydependent term d 10 in the drift function. The coefficient d 10 is the leading velocity-dependent term, which can also be seen from the shift of the conditional PDFs in figures 1(c) and (d).
We presented the drift functions in figure 3 at a relatively large scale r = 20λ, where the vertical shift for u = 0 is relatively strong, and the diffusion functions at a relatively small scale r = 3λ, where the quadratic shape of the diffusion is quite pronounced (they become flatter with increasing scale). The same observations-a shift of the drift and no significant change in the diffusion for different values of u-can be made at any scale. Possible further dependences on u, like the weak 'bending' observed in the drift in figure 3(a) for very large values of |ξ |, are within estimation errors and are therefore not statistically significant for the examined data. They are considered in the following as possible higher-order effects.
The r -dependence of d 10 u = 0 ± σ u /6 u = −σ u ± σ u /6 u = σ u ± σ u /6  (1) is shown for r j = 20λ and D (2) and D (4) for r = 3λ. Cylinder wake at R λ = 338.  in u for the free jet in figure 4(b). In all cases, d 10 can be approximated by a second-order polynomial in r , The dependence of the coefficients d 101 and d 102 on the velocity u is shown in figure 5. We find that the parameters d 101 and d 102 depend on the flow geometry, as well as on the Reynolds number. In the cases of the cylinder wake and fractal grid, the term d 101 is almost linear in u for low Reynolds numbers, and becomes more S-shaped for higher Reynolds numbers (figures 5(a) and (c)); the term d 102 is approximately linear in u, with Reynolds-number-dependent slopes in the case of the cylinder wake (figures 5(d) and (f)). For the free jet, neither d 101 nor d 102 is linear in u, and d 101 is also strongly asymmetric (figures 5(b) and (e)). Figure 6 shows the coefficients d 11 , d 20 , d 21

Conclusions
It was found in [14] that the stochastic cascade process of velocity increments from scale to scale can be described by a Fokker-Planck equation. This description gives access to the joint multi-scale PDF p[ξ(r 1 ), ξ(r 2 ), . . . , ξ(r N )] of the velocity increments ξ at the scales r j . The structure functions S n (r ) ≡ ξ(r ) n can also be obtained from this description (see also [15], [17]- [22]). In this paper, we showed how this method can be extended in order to obtain the joint multipoint PDF p[U (x 1 ), U (x 2 ), . . . , U (x N )] of the velocity U at the points x i . This description is more complete than the multi-scale description, since it takes into account the velocity dependence of the small-scale statistics. The multi-point statistics can be obtained from a Fokker-Planck equation for the conditional PDF p[ξ(r 1 )|ξ(r 2 ), u], where u is the fluctuating velocity. The Fokker-Planck equation follows from the Markov property of the underlying stochastic process and from the experimental observation that the fourth-order Kramers-Moyal coefficient can be neglected.
The Fokker-Planck equation for the multi-point statistics differs from the Fokker-Planck equation for the multi-scale statistics mainly by the presence of a simple additional term in the drift function. This term, d 10 (r, u), represents a vertical, velocity-dependent shift of the drift function. It implies that if, for example, ξ(r 2 ) = 0 and u 0, then ξ(r 1 ) at the scale r 1 < r 2 is likely to be negative. This is reasonable because ξ(r 2 ) and ξ(r 1 ) are to some extent independent and, loosely speaking, the increments have the tendency to drive the velocity signal back to zero, since it is stationary at large scales r L. The shift of the drift function corresponds to the shift of the conditional PDF p[ξ(r 1 )|ξ(r 2 ), u], observed in figure 1.
It was found in previous work that the coefficients of the Fokker-Planck equation for the interscale process are not universal, but depend on the Reynolds number [19] and/or flow geometry [22]. We have now found a similar result for the velocity-dependent coefficient d 10 , which depends on the flow type and, at least in the case of the cylinder wake, also on the Reynolds number. On the basis of the examined data sets with lengths of up to 10 7 , we cannot make definite statements on the velocity dependence of the other coefficients.
The coefficients of the Fokker-Planck equation can be estimated directly from the measured data. With the knowledge of these coefficients, the N -point statistics of the velocity field are given by the three-point statistics p[u(x + r 1 )|u(x + r 2 ), u(x)], which are equivalent to p[ξ(r 1 )|ξ(r 2 ), u(x)]. Thus, a stochastic three-point closure for the turbulent velocity is given.
The analysis presented in this paper is restricted to a single velocity component, but in principle it can be extended to a velocity vector with three components. The drift and diffusion functions would then become tensors D (1) i (ξ , r, u) and D (2) i j (ξ , r, u), which contain coupling terms between the different velocity (increment) components. Siefert and Peinke [21] investigated the Fokker-Planck equation for two components of the velocity increment ξ , and found that the drift function decouples, while the diffusion function contains non-vanishing coupling terms between longitudinal and transversal velocity increments. These coupling terms were found to have simple functional forms, and thus the analysis can be easily extended to more than one velocity component. However, such an analysis requires more data-Siefert and Peinke [21] needed data of length 10 8 to perform the two-dimensional analysis without conditioning on the velocity-and is therefore left to future studies. With the knowledge of (or appropriate assumptions about) the coupling coefficients, it would be possible to extend the method of time-series generation proposed in [23,24]-with the additional conditioning on the velocity proposed in the present paper-to the generation of synthetic turbulent velocity signals with three components. which is the absolute value of a standard normal distributed variable, has a mean value of √ 2/π. In the present implementation of the test, the quantity is calculated for a fixed value of ξ j+2 for a total of 100 bins for ξ j+1 , which span the complete range of ξ j+1 . The mean value Q * is calculated by taking the average over the 100 values of Q * . Under the null hypothesis of the Markov properties, the expectation value of Q * is 1.