User:Car.cin/sandbox/stochastic models backup

Title: Stochastic models for turbulent flows

The stochastic models for turbulent flow are a family of mathematical models that make use of random process or stochastic variable to take into account the effects of turbulence. They are used both for theoretical analysis of fluid dynamics and for numerical simulations, i.e. in computational fluid dynamics.

=Introduction=

Despite it is difficult to give a precise definition of turbulence, there is a general consensus on some characteristics exhibited by turbulent flows. In his milestone textbook, John L. Lumley underlines that:

"One characteristic is the irregularity, or randomness, of all turbulent flows. This makes a deterministic approach to turbulent problems impossible; instead, one relies on statistical models."

- J.L. Lumley

Indeed, one of the firsts and more fruitful approach to the problem of turbulent flow modelling is a statistical one: the Reynolds-Averaged Navier-Stokes (RANS) simulation methodology. In such a methodology, the equations of motion are solved for the averaged-in-time velocity field and the contribution due to velocity fluctuation (the Reynolds stress) is modelled through additional physical assumptions. During the XX century, other techniques have been developed in the framework of deterministic approach, i.e. proposing models based on deterministic partial differential equations. Instead, a relative modest improvements in the stochastic modelling of turbulence are reported in literature.

The major stochastic models for turbulent flow are reported. Essentially, they rely to two basic points of view: The earlier models (like Langevin models) are based on the representation of the velocity field as a random variable, and introduce directly the randomness in the governing equations of motion. A more recent model (MULU) starts from the representation of the fluid-particle trajectory as a random process and uses the stochastic calculus to derive consistent governing equations for the fluid velocity. Thus, stochasticity is introduced indirectly.

= Langevin models =

The Langevin models for turbulent flows rely to the analogy between the motion of a fluid-particle in turbulent flows and the Brownian motion. The basic idea behind this approach is that the fluid-particle velocity (in the Lagrangian coordinates) is described by a Langevin equation, which is employed to determined the probability density function (PDF) of turbulent velocity in Eulerian coordinates. A first approach in this sense was introduced by Chung and was subsequently applied to different flows also with chemical reactions. In the homogeneous isotropic turbulence regime, it is sufficient to consider one component of the fluid-particle velocity that is described by the following Langevin stochastic partial differential equation:

where $$X$$ is the Lagrangian coordinate, $$T_L$$ is the Lagrangian integral time scale, $$d W_t$$ is an isotropic Wiener process and $$\sigma$$ is a constant related to the standard deviation of the Weiner process. The first term on the right-hand side is the drift velocity, while the second term accounts for the random velocity contribution. In this framework, $$U_t$$ is a stochastic process that takes the name of Ornstein-Uhlenbeck process and its probability density function is a solution of the Fokker-Planck equation. It can be proven that $$U_t$$ is: Moreover, it has continuous simple path nowhere differentiable. Pope shows that equation ($$) can be express in terms of turbulence kinetic energy $$k$$ and turbulence kinetic energy dissipation $$\epsilon$$. It reads:
 * a statistically stationary process,
 * a Gaussian process,
 * a Markov process.

where $$C_0$$ is the Kolmogorov constant.

Generalised Langevin model for inhomogeneous turbulence
A model for inhomogeneous turbulent flow was proposed by Pope, referred to as the Generalised Langevin Model (GLM). The starting point for the Langevin models is to express the velocity as a random process; the GLM adopts a different approach: it is obtained by decomposing the Navier-Stokes equations in a mean and fluctuating parts and by modelling the contribution by fluctuation as a random process. Particularly, such a random process is expressed as an isotropic white noise, following the analogy with the general Langevin model. The final governing equation for fluid flow (in Lagrangian coordinates) is:

where the index $$i=1,2,3$$ indicates the vector component (it can be placed indifferently at the top or the bottom), the brackets $$ \left \langle \cdot \right \rangle $$ denote the averaged in some sense, $$G_{ij}$$ is a second order tensor function of local mean quantities, $$\nu$$ is the dynamic viscosity and $$p$$ is the pressure. In this case $$ W_t^i$$ is a vector-value Wiener process. This model is proven to be consistent with the Kolmogorov microscales theory in the inertial sub-layer and with the Reynolds stress equation model for turbulent flows.

The anisotropic linear term $$G_{ij}$$ accounts for turbulent effects and has to be modelled. A simple expression of such tensor is given in Pope ; it reads:

Haworth and Pope (1986) suggested a more general formulation of the anisotropic tensor in terms of Reynolds stresses, velocity gradient, and turbulence time scale. Also Durbin and Speziale (1994) proposed an alternative functional expression for such a tensor.

=Two-point closure in Fourier space=

The class of stochastic models that goes under the name of two-point closure models are developed in the Fourier space. They are so called because they involve the correlation of velocity in two space points. The Fourier transform is applied to the Navier-Stokes equations, that are expressed as:

where $$u_j$$ is the velocity, the hat indicates the Fourier transform, $$k,p,q$$ are the wavelength, $$P_{jn}$$ is the projector tensor and the Einstein notation is adopted. In the second line of equation ($$), the coupling operator $$M_{imn}$$ is introduced to simplify the notation. Equation ($$) is used to find an expression of the spectral tensor $$\hat{U}_{nm}$$ that represents the velocity correlation between two wave-vectors. It is defined as the Fourier transform of the second order velocity correlation tensor, which reads:

where the angular brackets denote an average operation. Equation ($$) is evaluated into two points $$\hat{u}(k)$$ and $$\hat{u}(k')$$, and the resulting formulae are multiply. Performing the average and exploiting the properties of the spectral tensor, one obtains :

form which the spectral tensor is recovered. Formula ($$) is not closed because the triple-correlation $$ \langle \hat{u}_m \hat{u}_s \hat{u}_r \rangle $$ needs to be specified. In order to do that, an additional equation for triple-correlation is obtained in a similar manner:

Equation ($$) allows to solve the triple-correlation but requires to specify the fourth-correlation. It can be easily inferred that this procedure can be indefinitely iterated, giving rise to a hierarchy of not closed equations for correlations. Different closure techniques have been prosed, see for example the textbook of Lesieur (2008) , the most fruitful among them being the eddy-damped quasi-normal Markovian model.

Eddy-damped quasi-normal markovian (EDQNM) model
The EDQNM model is the result of several empirical approximations that lead to a simple closure equation. It was proposed by Orszag (1977) (see also the work of Leslie (1973) ). The steps to derive the model are summarised as follow:


 * Gaussian random function hypothesis: the probability density function of velocity is assumed to be a Gaussian random function. Then, the following relations hold for the third-correlation and the fourth-correlation term (respectively):


 * Whit these formulae the hierarchy of equations are closed. Empirically, the velocity probability density function is found to not follow the normal distribution, and in general it is not realistic to describe velocity as a Gaussian random function. Moreover, the first equations in ($$) implies that the right-hand-side in equation ($$) is zero, which means that there is no more non-linear terms. For these reasons, this model is too simplistic and not adequate.


 * Quasi-normal approximation: the previous hypothesis is make weaker by adopting the second relation in ($$) for the fourth-correlation term, but not the first (i.e. no constrain is imposed on the third-correlation term). Then, an evolution equation for the energy spectrum $$E_k = \langle |u_j u_j| \rangle$$ is recovered. Unfortunately, also this model is not suitable for the fluid dynamics description since it can lead to negative values of energy, which is non-physical.


 * Direct interaction approximation: it was proposed by Kraichnan (1971) . He suggested that the negative energy value are due to the excessively long memory time, proportional to $$\mu_k^{-1} = (\nu k^2)^{-1}$$, which is longer than the eddy tour-over time. Hence, a turbulent damping time $$ \tilde{\mu}_k^{-1}$$ is introduced to scale correctly the memory time with respect to the flow turbulent content. The main problems with this model are that it is not consistent with the Kolmogorov energy cascade theory and that it is not invariant under random Galilean transformations.


 * Eddy-damped quasi-normal Markovian approximation: consists in providing an empirical expression of $$ \tilde{\mu}_k$$ that satisfy the Kolmogorov theory and the invariant properties. The model is affected by a certain degree of arbitrariness, given by the particular expression chosen for $$ \tilde{\mu}_k$$. An additional simplification done is the Markovianization, which consists in ignoring any memory effect in the system.

The final equation for the energy spectrum in the EDQNM reads:

where

is the relaxation time by molecular viscosity and non-linear transfer, $$\mu_{kpq}= \mu_k+\mu_p+\mu_q$$, and $$a_{kpq}$$ is a positive coefficient.

Expression ($$) is equivalent to write the Langevin equation:

for the stochastic velocity $$\hat{u}_j$$, where $$\hat{\gamma}_j$$ is a white noise stochastic variable and $$\alpha_j$$ is a deterministic damping rate. Notice that $$\hat{u}_j$$ is not the same as $${u}_j$$, but they have the same variance.

Others closure models were developed with the attempt to eliminate the dependency on free parameters, among the others the Test Field Model and the Lagrangian History Direct Interaction Approximation. The disadvantage in these cases is the increase of mathematical complexity of the model.

= Kraichnan model =

An other stochastic model in Fourier space is due to Kraichnan (1961) , and was further developed by Herring and Kraichnan (1972). It exhibits some similarities with the EDQNM model, but it is developed starting form a different base: the Navier-Stokes equations are replaced by a set of arbitrary equations which have the same structural properties but allows a solution to the closure problem. Such equations have the form:

that represents $$N$$ equations for the $$N$$ velocity random fields $$ \hat{u}^\alpha $$. Here $$\Phi_{\alpha \beta \gamma} (k,p,q)$$ are $$N^3$$ Gaussian random functions in time, that are: symmetric with respect the permutation of $$\alpha \beta \gamma$$; with zero mean; with the same statistcal properties.

Notice that the EDQNM model can be recover from this model chosing a suitable expression of  $$\Phi_{\alpha \beta \gamma}$$. The Kraichnan model allows to gain significant insight on turbulence flows, both on mathematical and physical side.

= Model under location uncertainty (MULU) =

A different family of stochastic methods for fluid dynamics is introduced (theoretically) by the early works of Mikulevicius and Rozovskii (2001) and Mikulevicius and Rozovskii (2004). This methodology is named model under location uncertainty (MULU) after the work of Mémin (2014), that applies it to geophysical fluid dynamics applications. The approach is radically different with respect to the previous ones: the stochasticity is not introduced in the velocity equation, but in the expression of fluid-particle trajectory. The basic assumption is that, in Lagrangian coordinates, a fluid-particle displacement in a spatial domain $$ \Omega$$ is described by a stochastic partial differential equation of the type:

where $$ X_t^i$$ is the (random) trajectory followed by a fluid-particle initially located at $$ x_0$$, $$ w_i $$ is a processus of finite-variation (differentiable in time) that represents the drift velocity, and the second term of the right-hand side is a martingale modelling the random contribution to the motion. This martingale is formed by a cylindrical Wiener process not differentiable intime, and a symmetric diffusion tensor $$ \sigma_{ik}$$ differentiable in time and space. Notice that the random displacement:

is a stochastic process uncorrelated in time a correlated in space. From formula ($$) the expression of velocity field (in Eulerian coordinates) is recovered:

where $$w_i$$ is the expected value of velocity, while the second therm on the right-hand-side is a noise, defined as the weak derivative of random displacement ($$).

Starting from this assumption, Mikulevicius and Rozovskii (2004) re-derives the governing equations of motion in a Lagrangian framework. The key point in such a procedure is to use the Itō-Wentzell formula for computing the time derivative of $$U_i(X_t)$$, that is a composition of two random processes. Mémin (2014) derived the same set of equations in an Eulerian freamwork. In order to do that, the Itō-Wentzell formula is used to find a stochastic version of the Reynolds transport theorem, then the conservation of mass and momentum are imposed following the classical procedure.

The MULU for an incompressible and Newtonian fluid reads:

{{NumBlk|:| $$ \left\{ \begin{aligned} & \frac{\partial w_i}{\partial t} + \frac{\partial (w_j w_i) }{\partial x_j} =  - \frac{\partial p}{\partial x_i}  + \nu \frac{\partial^2 w_i }{\partial x_j \partial x_j} + \frac{\nu}{3} \frac{\partial}{\partial x_i} \frac{\partial w_\ell}{\partial x_\ell} + \frac{1}{2} \frac{\partial^2 \left( a_{sk} w_i \right)}{\partial x_s \partial x_k} \\ & \frac{\partial }{\partial x_i} \left( w_i - \frac{1}{2} \frac{\partial }{\partial x_j} a_{ij} \right) = 0 \\ & \frac{1}{\rho} \frac{\partial}{\partial x_i} d \xi_t = \nu \frac{\partial^2}{\partial x_j \partial x_j} d \eta_t^i - d \eta_t^j \frac{\partial  w_i}{\partial x_j} \\ & \frac{\partial}{\partial x_i} d \eta_t^i = 0 \end{aligned} \right. $$ |$$}}

where $$\xi_t$$ is the random term of the pressure variable (that is also a random process) and the variance variance tensor $$a_{ij}$$ is defined as:

Therefore, the system ($$) is a close set of deterministic and stochastic partial differential equations in the unknowns $$w_i$$ and $$\sigma_{ij}$$.

Analogous equations are studied theoretically by Flandoli (2011) , while the MULU is applied to numerical simulations of geophysical flows in a series of papers by Resseguier et al. (2017) . Holm (2015) derived in a different way an equivalent system of stochastic equations for fluid dynamics. The MULU is the basis of the pseudo-stochastic simulation methodology for numerical simulation of turbulent flow, that can be considered a generalisation of the large eddy simulation technique with eddy-viscosity models.

= Notes =

=References=