(This content is originally written by ‪Kyongmin Yeo’s manual)

Introduction

The small scale statistics of turbulence are important research topic.

Small-scale behavior in turbulent flows tends to be characterized by statistical homoegenity, isotropy, and universality. Because of this universality we can hope to
understand small-scale behavior by studying the simplest turbulent flows, i.e. homoegeneous, isotropic turbulence.

(Eswaran and Pope 1988)

To maintain statistically stationary turbulence, adding force to low wavenumber (large scale) velocity components artificially. Therfore, external force term is added to Navier-Stokes equation

\[\dfrac{d \hat{u}_i}{dt} = - i \kappa_{i} \hat{P} + \hat{H}_i - \nu \kappa^2 \hat{u}_i + \hat{f}_i\]

where \(\hat{f}_i\) is a external forcing term.

The forcing \(\hat{f}_i\) is applied to circle of low wavenumber band. \(\hat{f}_i\) is defined as the projection of a vector \(\hat{\mathbf{b}}\) onto the plane normal to the wavenumber vector \(\mathbf{\kappa}\) to ensure divergence-free condition.

\[\hat{f}_{i} = \hat{b}_{i} - \dfrac{\kappa_i}{\kappa^2} \kappa_j \hat{b}_{j}\]

So, how we define vector \(\hat{\mathbf{b}}\)? Eswaran & Pope suggested stochastic forcing (Eswaran and Pope 1988). They define 3D complex vector \(\hat{\mathbf{b}}\) which is non-zero in the range \(0 < \kappa < \kappa_f \), in which \(\kappa_f\) is the maximum forcing wavenumber. This can be interpreted as forcing to sphere in wavenumber space.

They used Uhlenbeck-Ornstein process to generate \(\hat{\mathbf{b}} = \hat{b} (\kappa, t) \) with following properties, the average and the correlation.

\[\begin{align} \langle \hat{b} (\kappa, t) \rangle &= 0 \\ \langle \hat{b} (\kappa, t) \hat{b}^* (\kappa, t + s) \rangle &= 2\sigma^2 \delta_{ij} \exp{(-s/T_L)} \end{align}\]

where an asterisk dentoes a complex conjugate, angle bracket is the ensemble average, \( \delta_{ij} \) is the Kronecker delta. \( \sigma^2 \) and \( T_L \) are the variance and time-scale of UO process. Obviously, if \( T_L \) increases with fixed \(\sigma \), the correlation will converge to zero. This is by no means the desired result, so \( \epsilon^* \equiv \sigma^2 T_L \) is fixed.

The three-dimensional vector \(\hat{\mathbf{b}}\) is composed of six independent Uhlenbeck-Ornstein process.

\[\hat{\mathbf{b}} = \begin{bmatrix} UO1 \\ UO3 \\ UO5 \\ \end{bmatrix} + i \begin{bmatrix} UO2 \\ UO4 \\ UO6 \\ \end{bmatrix}\]

Solving Uhlenbeck-Ornstein process

Each stochastic process, \(UO1 \) ~ \( UO6\), is chosen so as to satisfy the Langevin equation with a time scale \(T^f_L\) and stadnard deviation \(\sigma_f\).

In (Wojnowicz 2012), UO process are defined as

\[dx_t = \dfrac{(\mu - x_t)}{\tau} dt + \sqrt{\dfrac{2\nu}{\tau}} dW_t\]

After applying zero mean property of forcing term and adjusting parameters makes above eqaution to

\[dUO = - \dfrac{UO}{T^f_L} \Delta t + \left( \dfrac{2\sigma^2_f}{T^f_L} \right)^{1/2} dW_t\]

in which \(W_t\) denotes a Wiener process satisfying

\[dW_t \sim \mathcal{N} (0, \Delta t)\]

The analytical solution of the Langevin equation is given by following equation, which describes the Browninan motion of particle.

\[\begin{align} x(t) &= x_0 + \int_0^t v(s) ds \\ v(t) &= e^{-t/T^f_L} v_0 + \dfrac{1}{m} \int^t_0 e^{-(t-s)/T^f_L} dW(s) \end{align}\]

where \(x_0 = x(0)\) and \(v_0 = v(0) \). The forcing \(\hat{f}_i\) term can be viewed as forcing acclereration, then \(UO \) can be denoted to \(v(t)\).

\[UO(t) = UO(0) e^{-t/T^f_L} + e^{-t/T^f_L} \int^t_{0} e^{s/T^f_L} (2\sigma^2_f/T^f_L)^{1/2}dW_s\]

Above solution can be solved discretely by applying Itô integral. With RK3 method, UO process discretized solution is

\[UO^{n+1} = e^{-(a_n+b_n)\Delta t / T^f_L}\left[ UO^{n} + e^{s/T^f_L} (2\sigma^2_f/T^f_L)^{1/2}dW_s dW^n \right]\]

in which discretized Wiener process is

\[dW^n \sim \mathcal{N} (0, (a_n + b_n) \Delta t)\]

This is the extension of Euler-Maruyama method.

Estimating Reynolds Number

Input parameters

The input parameters are \( \kappa_0 \) (the lowest wavenumber), \( \kappa_\textrm{max} \) (the highest wavenumber), \( K_F \) (the maximum wavenumber of the forced modes), \( \nu \) (the kinematic viscosity), \( T_L \) (the forcing time scale, time scale in UO process), and \( \epsilon^* = \sigma^2 T_L \).

The nondimensional parameters are \(\kappa_{\textrm{max}} / \kappa_0 \), \( K_{F} / \kappa_0\),

\[\begin{align} Re^* &\equiv \epsilon^* \kappa_0^{-4/3} / \nu \\ T^*_L &\equiv T_L {\epsilon^{*}}^{1/3} \kappa_0^{2/3} \end{align}\]

Given parameters

  • \( \nu \) : Fluid viscosity
  • \( \beta \) : constant (\( \beta=0.8 \))
  • \( \kappa_0 \) : smallest wavenumber
  • \( \kappa_f \) : maximum forcing wavenumber
  • \( T_L \) : Forcing time scale
  • \( \epsilon^* \equiv \sigma^2 T_L \) where \( \sigma \) is a forcing amplitude, usually just given by constant

Assumptions

  • \( \epsilon \propto N_f \epsilon^* \)
  • \( T_e \approx \dfrac{\beta}{(N_f \epsilon^* \kappa^{2}_0)^{1/3}}\) (posteriori assumption)
  • \( \kappa^{-1}_{0}\) : Integral length scales

Computed parameters

  • \( N_f \) : The number of forced modes, \( \kappa < \kappa_f \), counted manually
  • Predicted energy dissipation,

    \[\begin{align} T_e &= \dfrac{\beta}{(N_f \epsilon^* \kappa^2_{0})^{1/3}} \\ T^*_L &\equiv T_L {\epsilon^{*}}^{1/3} \kappa_0^{2/3} \\ \epsilon^*_{T} &= \epsilon \\ &\equiv \dfrac{4\epsilon^* T_e N_f}{T_L + T_e} \\ &= \dfrac{4 \epsilon^* N_f}{1 + T^*_{L} N^{1/3}_{F}/\beta} \end{align}\]
  • Predicted Kolmogorov microscale \(\eta_{T} \equiv (\nu^3 / \epsilon^*_T)\)

Predicted \( Re \)

Using above parameters Taylor Reynolds number is estimated by

\[Re \simeq \dfrac{8.5}{(\eta_{T} \kappa_0)^{5/6} N^{2/9}_{F}}\]

Reference

  1. Eswaran, V., and S. B. Pope. 1988. “An Examination of Forcing in Direct Numerical Simulations of Turbulence.” Computers and Fluids. https://doi.org/10.1016/0045-7930(88)90013-8.
  2. Wojnowicz, Michael Thomas. 2012. “The Ornstein-Uhlenbeck Process In Neural Decision-Making: Mathematical Foundations And Simulations Suggesting The Adaptiveness Of Robustly Integrating Stochastic Neural Evidence.” Phdthesis. https://digital.lib.washington.edu:443/researchworks/handle/1773/21760.