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

# Introduction

To maintain stationary turbulence, 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.

$$\hat{f}_i$$ 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. 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.

$$\hat{\mathbf{b}}$$ is composed of six independent Uhlenbeck-Ornstein process.

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

# Solving Uhlenbeck-Ornstein process

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

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

in which $$W_t$$ denotes a Wiener process satisfying $dW_t \sim \mathcal{N} (0, \Delta t)$

Analytical solution to Langevin equation is given by

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

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 proces is

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

# Estimating Reynolds Number

## Given parameters

• $$\nu$$ : Fluid viscosity
• $$\beta$$ : constant ($$=0.8$$)
• $$\kappa_0$$ : 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

• $$\epsilon \propto N_f \epsilon^*$$ \begin{align} \epsilon &= \dfrac{4\epsilon^* T_e N_f}{T_L + T_e} \newline T_e &= \dfrac{\beta}{(N_f \epsilon^* \kappa^2_{0})^{1/3}} \end{align}
• $$\kappa^{-1}_{0}$$ : Integral length scales

## Computed parameters

• $$N_f$$ : The number of forced modes, $$\kappa < \kappa_f>$$, counted manually
• Predicted value of the energy dissipation, replace $$T_e$$ by $$\beta / (N_f \epsilon^* \kappa^2_{0})^{1/3}$$ $\epsilon^*_{T} \equiv \dfrac{4 \epsilon^* N_f}{1 + T_{L} N^{1/3}_{F}/\beta}$
• Predicted Kolmogorov microscale $\eta_T \equiv (\nu^3 / \epsilon_T)$

## Predicted $$Re$$

Using above parameters Taylor Reynolds number is estimated by

$Re \simeq \dfrac{8.5}{(\eta_T \kappa_0)^{5/6} N^{2/9}_{F}}$

# TODO

• More detail explanation of Uhlenbeck-Ornstein process
• Add code for computing $$Re$$