On distributions of velocity random fields in turbulent flows

The purpose of the present paper is to derive a partial differential equation (PDE) for the single-time single-point probability density function (PDF) of the velocity field of a turbulent flow. The PDF PDE is a highly non-linear parabolic-transport equation, which depends on two conditional statistical numerics of important physical significance. The PDF PDE is a general form of the classical Reynolds mean flow equation, and is a precise formulation of the PDF transport equation. The PDF PDE provides us with a new method for modelling turbulence. An explicit example is constructed, though the example is seemingly artificial, but it demonstrates the PDF method based on the new PDF PDE.


INTRODUCTION
The research on statistical properties of turbulent flows can be traced back to the semi-empirical theories of turbulence in the 1920s and 1930s, and seminal advances in the area include Prandtl [11], von Kármán [15] and Taylor [13,14]. The goal of statistical fluid mechanics is to provide good descriptions and computational tools for understanding the distributions of the velocity random fields of turbulent fluid flows. Unlike some other unsolved problems in theoretical physics, the equations of motion for fluid dynamics, even for turbulent flows, have been known for over a century. These equations are highly non-linear and non-local partial differential equations (PDEs), and it is difficult to extract information about the evolution of fluid flows in a deterministic manner. Thus, the velocity field of a turbulent flow is better to be considered as a random field arising from either the random initial data or a random external force, or both. To understand the statistics of turbulent flows, it is desired to know, if it is possible, the evolution of some distributional characteristics of fluid flows. The distribution of a random field, such as the velocity field, is rather complicated, and determining the distribution of turbulent flows is a challenging task even when the initial distribution is known. Hopf [2] (see also [8]) has derived a functional differential equation for the law of the velocity random field, but his equation involves functional derivatives.
In the past decades, the probability density function (PDF) method, based on the transport equation, a formal adjoint equation of the Navier-Stokes equation, has been developed into a useful tool for modelling turbulent flows. This method focuses on evaluating the one-point onetime PDF ( ; , ) of the velocity field ( , ), or equivalently, the centred field ( , ) − ( , ). The exact transport equation for the PDF, which involves the mean of the pressure term as well as the conditional expectation of the pressure term, has been derived by Pope and can be found in [9,10] for details. However, it is still not an easy task to apply the PDF transport equation for the purpose of modelling turbulent flows. Applications of the PDF method have been mainly restricted to the generalised Langevin model, where the time-dependent velocity ( ) of a particle at position ( ) is assumed to satisfy a stochastic differential equation (SDE).
The main contribution of this paper is the derivation of the PDF PDE for the law of the velocity at any location and any instance, which is much more explicit than the formal PDF transport equation. This PDE is a generalisation of Reynolds mean flow equations, which can be closed by introducing the Reynolds stress tensor field. Having six dimensions in space ( , ) and one dimension in time , our PDF PDE can be regarded as a parabolic-transport equation which has a parabolic operator 1 2 + ⋅ ∇ − Δ in and transport operator 1 2 + ∇ ⋅ in . The PDF PDE for a velocity field is a second-order PDE which is, in general, not parabolic due to the appearance of a mixed derivative term . Even when this mixed derivative does not appear in the PDF PDE (which is the case for some turbulent flows and will be explained in the next section) and the parabolic part in the PDF PDE involves variable only, the PDF PDE is still highly degenerate. This feature of the PDF PDE distinguishes itself from the prevalent parabolic PDEs and other types of PDE theories in the literature.
The PDF PDE that we have obtained relies on two conditional structure functions, which are the conditional average increment These conditional structure functions describe the interactions of the velocity random field at different positions, and hence, they are natural to appear in the PDF PDE. The fact that the distribution of velocity random fields is characterised by conditional first and second moments is an interesting feature revealed in this paper. These statistical characteristics have the capacity to determine the PDF PDE. Moreover, these local statistical characteristics can localise many concepts, such as homogeneity, isotropy, and so on, introduced firstly by Taylor and Kolmogorov [5,6,14], allowing us to generalise such concepts to their weak versions. We outline the main structure of this paper as follows. In Section 2, we introduce definitions related to random fields. The evolution equation for the velocity random field distribution of turbulence over time will be derived, assuming that the random field is regular. The PDE will be applied to various types of flows in Section 3, including both viscid and inviscid cases. We also obtain a stochastic representation formula for the solution of the PDF PDE, together with the constraint that ensures the solution is indeed a PDF for all time ⩾ 0 and position ∈ ℝ 3 . These theoretical results are important when we apply PDF PDE for modelling turbulent flows. Section 4 is thus devoted to an example of modelling the PDF of turbulence, which demonstrates the change of distribution at a fixed position over time. Our paper will be closed with a few remarks in the last section.
Conventions on notations. The following set of conventions is employed throughout the paper. Firstly, Einstein's convention on summation over repeated indices through their ranges is assumed unless otherwise specified. If is a vector or a vector field (usually in the space of dimension three) dependent on some parameters, then its components are labelled with upperscript indices so that ∶= ( ) = ( 1 , 2 , 3 ). The same convention applies to coordinates too. The derivative operators ∇ and Δ are labelled with subscripts to indicate the variable to which the operator is applied, such as Δ ∶= and ∇ ⋅ ∶= . Finally, the velocity vector field will be denoted by = ( ).

PDF EQUATION OF VELOCITY FIELDS
In this section, we aim to introduce some fundamental concepts on random fields and to derive the evolution equation for the velocity ( , ), where ∈ ℝ 3 , ⩾ 0. Note that ( , ) takes values in ℝ 3 .

Random fields and their statistical characteristics
Given a random field { ( , )} , on a probability space (Ω,  , ℙ), ( , ) is, by definition, an ℝ 3valued random variable for every ∈ ℝ 3 and ⩾ 0. The law or the distribution of ( , ) for fixed and is a probability measure on the Borel -algebra of ℝ 3 . The distribution of the random field consists of, by definition, all possible finite-dimensional marginal joint distributions of where ∈ ℝ 3 , ⩾ 0 and any positive integer . For example, by saying that the random field { ( , ) ∶ ∈ ℝ 3 , ⩾ 0} is Gaussian, we refer to the fact that any finite-dimensional marginal joint distribution is a Gaussian distribution, which, in particular, implies that the marginal distribution of ( , ) for any ( , ) has a normal distribution. We remark that the converse argument is not true in general.
The most important statistical numerics for understanding a random field is the correlation function of two random variables, which plays the dominant role in the study of turbulence [1,8]. In this paper, we, however, emphasise the use of a few statistical characteristics based on the conditional distribution. Let us introduce these statistical numerics, which we believe are of the most importance. and ( , , , ) = 0 for all , and . The conditional covariance function ( , , , ) can be treated as the conditional Reynold stress. These statistical characteristics have explicit representations in terms of the two-point joint distribution. For our purpose, it is convenient to assume that the distribution of ( , ) for every ( , ) has a PDF with respect to Lebesgue measure on ℝ 3 , denoted by ( ; , ), in the sense that for fixed and , Similarly, the joint distribution of ( , ) and ( , ) at two distinct points ≠ has a joint PDF, denoted by 2 ( 1 , 2 ; , , ). It follows that the conditional law of ( , ) given that ( , ) = possesses the following conditional PDF: As a result, we are allowed to represent the conditional average increment function (2.1) and covariance function (2.2) as an integral relevant to the conditional density, namely The use of the conditioning techniques is, in fact, the main reason for advocating the foundation of statistical fluid mechanics based on the probability theory rather than on an average procedure, which was first explicitly proposed by Kolmogorov [5,6]. The concept of homogeneity and isotropy can be defined for random fields indexed by a space variable ∈ ℝ 3 , which has been introduced into the study of turbulence by G. I. Taylor. The concept of local homogeneous and local isotropic flows was introduced by Kolmogorov for formulating the K41 theory (and its improved version K61 theory). According to Kolmogorov [5,6], a random field { ( , )} , is locally homogeneous if for any , ∈ ℝ 3 , the conditional distribution of ( , ) − ( , ) given ( , ) = depends on − and , and further, it is locally isotropic if the conditional distribution depends only on | − | and . By using the conditional average and the conditional covariance functions, it is possible to generalise these terminologies to their weak versions. We are now in a position to state technical assumptions on the random field. Unlike Kolmogorov's definition of isotropic flows and homogeneous flows, our concept of weak isotropy has no direct relationship to weak homogeneity. Nevertheless, a regular locally homogeneous turbulent flow in the sense of Kolmogorov satisfies the condition that depends only on , since the conditional average increment of such a flow must obey ( , , , ) = g ( − , ) for some function g and Moreover, if we further assume that the flow is locally isotropic, the conditional average increment function satisfies ( , , , ) = g (| − |, ) and is well defined if and only if = 0. Therefore, this turbulent flow is both weakly homogeneous and weakly isotropic in our sense.
Apart from extending Kolmogorov's definitions of homogeneity and isotropy, the significance of introducing these terminologies is they will eliminate the mixed terms in the PDF PDE, which will be thoroughly explained in Section 3.

The evolution equation for the velocity distribution
In this subsection, we derive the main theoretical result, which provides the theoretical foundation for modelling PDFs of turbulent flows based on two statistical characteristics. We consider an incompressible turbulent flow, inviscid or viscous, with kinetic viscosity constant , which is positive for viscous fluid, or reads as zero for inviscid fluid. The turbulent flow is described by its velocity = ( 1 , 2 , 3 ), the pressure , which are random fields, and the three-dimensional Navier-Stokes equations where = 1, 2, 3, and ⩾ 0 is the viscosity constant, together with the constraint The initial condition 0 is also treated as a random field on ℝ 3 and each sample path ∈ Ω corresponds to a deterministic function ( , ; ), which serves as a solution to Equation (2.7) with initial data 0 ( ; ). We will discuss an ideal case, for the purpose of understanding the local properties of turbulent flows, where the region occupied by the fluid is the entire space ℝ 3 . Moreover, without further qualifications, the dynamical variables such as ( , ) and ( , ) decay to zero sufficiently fast as | | tends to infinity. In addition, to avoid technical difficulties, but not in any way implying that these issues are not important, we will assume that the dynamical variables ( , ) and ( , ) are sufficiently smooth functions of ( , ).
Due to the divergence-free condition (2.8), the pressure term satisfies the following Poisson equation: Therefore, according to the Green formula, we have the integral representation which implies, in particular, that the distribution of is completely determined by the distribution of the velocity random field. We assume that { ( , )} , is a regular random field. Since ( , ) is divergence-free as in Equation (2.8), we have for all , , as well as the following integral condition for PDF of ( , ): which will appear as a natural constraint for the PDF PDE we will derive. Recall that the Reynolds equation (see [12]) is obtained by taking the average in (2.7). More explicitly, The conventional treatment for the non-linear term on the left-hand side is to write is the Reynolds stress. The PDF equation can be obtained by carrying out this computation for the average [ ( )] where ∶ ℝ 3 → ℝ is set as a smooth function with a compact support instead of choosing ( ) = for each velocity component in the case of Reynold stress. We are now in a position to establish the most important work in this paper.
where 0 ( ; ) is the PDF of the 0 ( ) in the random field { 0 ( )} and Here, Proof. Let ∈ ∞ (ℝ 3 ) be a test function, which is a smooth function taking values in ℝ with a compact support. For simplicity, we denote ( ) ∶= ( ) ∈ ∞ (ℝ 3 ). Applying to the average [ ( )] followed by exchanging the integral and the derivative operator, we have On the other hand, we have ] . (2.14) Utilising the Navier-Stokes equations (2.7), we get Substituting this into (2.14), we obtain that where the first two terms are equivalent to Subsequently, the remaining 1 , 2 are of the form ] .
The evaluation of 1 and 2 requires invoking the joint distribution at two points together with taking limits. Here, we depart from the approach by expressing this term via the PDF, which allows us to perform similar computations for a general case. Assuming that we are able to take the average under the limit, that is, the dominated convergence theorem can be applied, we are able to write the space derivatives ( ) in 1 in terms of the following limits: where (1) = (1, 0, 0), and so on. Taking expectation first, we obtain that Using the conditional probability notation that we introduced, we may write this as ) ( , + ℎ ( ) , , ).
The last equality is a result of applying (2.4), which converts integrals into conditional average increments . As a consequence of the regularity condition on the random field, we make use of (2.5) to deduce and therefore, We perform integration by parts to derive Next, we handle 2 . Applying the representation (2.9), we arrive at Writing the derivative in terms of then using the joint PDF, integrating and taking the limit as ℎ → 0 lead us to where the integral 2 has the following integral form: Putting all terms together, we deduce that for all such ∈ ∞ (ℝ 3 ). Therefore, we must have (2.11). □ We finish this section by adding several comments. The PDF PDE (2.11) may be written as ( where, for simplicity, we introduce the following vector field: ) is a mixed type of parabolic and transport PDEs. The parabolic operator is independent of fluid flows, which is a significant feature. Compared with the traditional PDF transport equation (cf. [10]), the PDF PDE (2.11) is more explicit, and is expressed in terms of conditional statistics of the underlying turbulent flow. Both PDF equations involve conditional statistics, so they are not closed. The traditional PDF transport equation contains the conditional expectations of the viscosity term and the fluctuating pressure, while the PDF PDE (2.11) requires the conditional mean and the conditional covariance in the same spirit of Reynolds' mean flow equation. This feature makes the modelling of turbulent flows based on the PDF PDE more flexible. For example, one may close the PDF PDE by importing data on conditional statistics from some Gaussian random fields to derive turbulence models, a research line we will explore in separate papers.
Although the PDF PDE (2.15) appears linear in the PDF ( ; , ), it is much more complicated than it looks. In particular, the coefficients , and are functionals of the conditional average and covariance functions, which are, in general, not determined by the PDF ( ; , ) alone. Therefore, the PDF PDE (2.15) is not a closed PDE. The significance of the PDF PDE lies in the fact that if the statistical numerics and are considered as given, which will be the case for modelling turbulent flows, then the PDF PDE is a PDE of second-order, though the mixed type of parabolic and transport in general.
Nevertheless, the PDE (2.11) is a challenging obstacle even if , , are all considered as given. The function can be understood as the mean velocity gradient at ( , ) conditioned on the velocity vector at ( , ), which brings the mixed derivatives , while the corresponding diffusion matrix ( ) 1⩽ , ⩽6 collecting the second-order terms is of the form if we consider ( , ) as a whole. The matrix ( ) , can be asymmetric, and it is not necessarily non-negative definite even if it is symmetric. It poses a challenging mathematical problem developing a theory of this kind of mixed-type PDEs to facilitate the modelling of turbulent flows based on the PDF PDE.

APPLICATION TO TURBULENT FLOWS
As we have seen, our PDE (2.11) does not fit into any existing categories of PDE theories. However, functionals , , will be derived if the conditional statistics can be obtained or estimated through practical experiments. Therefore, tracking the PDF of the turbulent flow is equivalent to measuring or modelling the conditional mean and conditional variance, followed by solving the PDF PDE (2.11) using some numerical methods. This brings a new approach to the modelling of turbulent flows. In this part, we establish some mathematical tools for the purpose of modelling turbulence based on the PDF PDE.
For convenience, let us introduce the following technical assumptions on a function .

Weakly homogeneous and weakly isotropic flows
When the viscous incompressible flow is weakly homogeneous, the mixed-derivative term disappears and the PDF PDE is simplified to ( where = − , and and are given by Equations (2.12) and (2.13), respectively. By the definition of , a weakly homogeneous flow has the property that the velocity gradient conditioned on the velocity vector at the same location is a centred random vector. The weak homogeneity allows us to state the representation formula, which provides a useful tool when we model weakly homogeneous turbulent flows.

) The defined by Equation (3.2) is the unique vanishing viscosity limit solution, that is, the solution obtained via the method of vanishing viscosity, to the PDF PDE (3.1).
Proof. If is Lipschitz, the previous system of SDEs for ( , ) has a unique solution ( , ) and is given via the exponential function. Notice that ( , , ) depends on ( , ) as well. Let Consider the following parabolic PDE: ( ; , 0) = 0, ( ; ).
It admits a unique classical smooth solution by classical PDE theory. Moreover, the solution possesses the representation if we make use of (1). Applying Burkholder-Davis-Gundy and Grönwall inequalities (or following routine arguments in [4]), we have uniformly on compact sets. Therefore, Equation (3.2) is a viscosity solution by Proposition 5.8 in [16], whereas the uniqueness follows from [3]. □ The PDF PDE (3.1) boils down to a degenerate parabolic PDE after the weak homogeneity has been applied. The stochastic representation (3.2) therefore offers a route to numerically solving the PDE. The PDE (3.1) has six dimensions in space and one dimension in time, which is a challenging task for classical finite difference methods due to the size of grid in space. Instead, we can simulate the solution based on the Monte-Carlo method directly. We are now in a position to verify that the solution of our PDF PDE (3.1) is indeed a PDF. That is, it must carry two properties, including positivity and the mass preservation property. It turns out that under some technical assumptions, the mass preservation property is equivalent to the divergence-free condition.   for all ( ; ) ⩾ 0. Therefore, Applying Lebesgue's dominated convergence theorem, we get Recall that 0 ( ; ) has at most polynomial growth in ( , ), that is, 0 ( ; ) ⩽ (| | + | | ) for some , ⩾ 1. Denoting (0, ) as the ball centred at the origin with radius > 0, the moment bound can be obtained by where is chosen large enough such that | | ( ; , ) ⩽ ′ for all | | > and some constant ′ > 0.
We introduce a sufficient condition for the mass-preservation property when satisfies the constraints in the following lemma. Proof. We apply ⋅ ∇ on both sides of the PDE (3.1) followed by integrating with respect to , resulting as well as Eventually, the right-hand side can be written in terms of provided | ( ; , ) ( , , ) | → 0 as | | → ∞. This has been guaranteed by the growth condition on . □ If we further assume that the incompressible viscous turbulent fluid flow is both weakly homogeneous and weakly isotropic, the PDF PDE is a parabolic-transport equation Therefore, if 0 ( ; ) ⩾ 0 for every , then so is ( ; , ) for any ( , ). If 0 ( ; ) is a PDF for all , then ( ; , ) is again a PDF for all ( , ) if only if the following constraint holds: Proof. By definition, and are deterministic processes, while is a three-dimensional Gaussian process such that for all ∈ B(ℝ 3 ) and ∈ [0, ], The remaining part is a consequence of Lemma 3.3. □

Inviscid flows
The modelling of inviscid incompressible flows is significantly simplified compared to modelling viscid flow, because the PDF PDE can be solved without imposing weak homogeneity or weak isotropy conditions. As = 0, the velocity ( , ) fulfils the Euler equations, while the PDF PDE becomes the following transport equation: ( ; , 0) = 0 ( ; ).

MODELLING THE PDF: A CONCRETE EXAMPLE
On the one hand, from the modelling point of view, only those solutions to the PDF PDE which satisfy the natural mass conservation condition (3.3) can be used as models of distributions of turbulent velocity fields. On the other hand, from the viewpoint of PDEs, the mass conservation property of solutions to the PDF PDE imposes a strong constraint on its solutions. As a matter of fact, most solutions of PDF PDE with given , , do not satisfy this constraint. In general, solutions to the PDF PDE do not have an explicit expression, although our stochastic representations established in the previous sections may be helpful in dealing with the mass conservation.
Consequently, it brings the next level of difficulty to verify the constraint (3.3). It turns out that the natural condition that ∫ ℝ 3 ( ; , ) d = 1 for a solution ( ; , ) to the PDF PDE, which is equivalent to the mass conservation, is a very strict constraint for whatever the coefficients , and which may be modelled or measured. At least our experience demonstrates that the PDF solutions to the PDF PDE are rare, and indicates that the PDF solution ( ; , ) is not so sensitive for the choices of , and , although we are unable to prove this claim in the present paper. Therefore, the PDF PDE together with the mass conservation constraint is very rigid, and hence, is good for modelling the PDF of turbulent flows. The authors hope to see further exploration in this direction in the future.
In this section, we study an explicit example of the PDF PDE, where the mixed-derivative term vanishes in the PDE, that is, the turbulent flow is weakly homogeneous. The example seems artificial, but as we have explained above, we believe that this example has relevance to real turbulent flows.

Space homogeneous density with perturbation
For simplicity, the viscosity parameter in this example is set to be = 1. As the solution to the PDE (3.1) is solely determined by the initial data 0 ( ; ) as well as the function ( , , ), we consider the simplest scenario = 0. Meanwhile, instead of setting up a common distribution such as Gaussian or exponential distribution for the initial data, we introduce the following nonnegative function: for every ∈ ℝ 3 , let 0 ( ; ) = ( ) + ( ) ( ), (4.1) where ( ) is a PDF given by corresponding to the PDF of a centred Gaussian vector with independent components. Here, satisfies ∫ ℝ 3 ( ) d = 0, and is chosen to be the product of reciprocals which vanishes at | | → ∞. We select these functions so that the positivity and mass-preserving properties are fulfilled, and the initial velocity is a random field whose marginal density at is given by (4.1). By the stochastic representation formula, the PDF of the random field at ( , ) has the form ( ; , ) = ( ) + ( ) We focus on the PDF of the random field { ( , )} , at = (0, 0, 0) and plot the graph of against ( ; , ) at 3 = 0.3 for different time . At = 1 2 , ( ; , ) is discontinuous on the boundary of as in Figure 4.1. If we compare the density of ( ; , 1 2 ) and ( ; , 0) by evaluating ( ; , 1 2 ) − ( ; , 0), we can see the change of density in the region . Meanwhile, the discontinuity becomes less apparent on the plot when we increase the time to = 40. From Figures 4.2 and 4.3, the PDF of ( , ) is close to the density of a Gaussian random variable with density function , even if the discontinuity still exists. This is due to vanishes at infinity and [ ( )] → 0 as → ∞. However, the impact of does not disappear from the velocity field. There is a strong discontinuity near ( 1 4 , 1 4 , 2 7 ) on ( ; , ) when = (12, 12, 12) and = 40. The PDF at = (12, 12, 12) is asymmetric and has a different evolution than the density at the origin, which demonstrates that the impact shifts from the origin to somewhere far away as time changes.

Motivation for the construction
The mass-preservation property of the PDE (3.1) corresponds to the divergence-free condition (3.3), which is difficult to verify explicitly even when we have the stochastic representation. Apart from describing the motivation for choosing , and , we will demonstrate that our solution to the PDF PDE (4.2) satisfies the divergence-free constraint. Assuming = 0 does not imply that the turbulence flow associated to solution ( ; , ) is weakly isotropic. For example, we force the conditional average increment to satisfy The PDF (4.2) at ( , ) is discontinuous in the variable , but it is still a strong solution to the PDE, because the disappears in the PDE (3.4) in this circumstance.

CONCLUDING REMARKS
This paper derives a new PDE which describes the evolution of one-time one-point PDF of the velocity random field of a turbulent flow. The PDF PDE, which is highly non-linear and is determined by two conditional statistics of a turbulent flow, should be a useful tool in modelling distributions of turbulence velocity fields. The modelling of viscous turbulence in various environments by solving the PDF PDE (2.15) numerically with measured data or based on a priori determination of , and should be beneficial in understanding turbulent flows. To implement good models of PDFs for turbulent flows, we need to numerically calculate solutions of the PDF PDE, with fed data which determine the functions , and . The solution has to satisfy the natural constraint, that the mass must be preserved throughout the evolution of the PDF. The conservation of the total mass of the solution is an important topic itself and is worthy of further study. Finally, we would like to point out that the coefficients , and defined in Equation (2.13), which determine the statistics of the turbulence at one-time one-point, must have significant physical meaning in turbulence. These coefficients, which are considered as turbulent flow parameters, should play their roles in further research.

J O U R N A L I N F O R M AT I O N
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