External Forcing of Homogeneous Isotropic Turbulence
(This content is originally written by Kyongmin Yeo’s manual)
Introduction
The small scale statistics of turbulence are important research topic.
Smallscale behavior in turbulent flows tends to be characterized by statistical homoegenity, isotropy, and universality. Because of this universality we can hope to
(Eswaran and Pope 1988)
understand smallscale behavior by studying the simplest turbulent flows, i.e. homoegeneous, isotropic turbulence.
To maintain statistically stationary turbulence, adding force to low wavenumber (large scale) velocity components artificially. Therfore, external force term is added to NavierStokes 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 divergencefree 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 nonzero 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 UhlenbeckOrnstein 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 timescale 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 threedimensional vector \(\hat{\mathbf{b}}\) is composed of six independent UhlenbeckOrnstein process.
\[\hat{\mathbf{b}} = \begin{bmatrix} UO1 \\ UO3 \\ UO5 \\ \end{bmatrix} + i \begin{bmatrix} UO2 \\ UO4 \\ UO6 \\ \end{bmatrix}\]Solving UhlenbeckOrnstein 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.
\[\begin{align} x(t) &= x_0 + \int_0^t v(s) ds \\ v(t) &= e^{t/T^f_L} v_0 + \dfrac{1}{m} \int^t_0 e^{(ts)/T^f_L} dW(s) \end{align}\]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 EulerMaruyama 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
 Eswaran, V., and S. B. Pope. 1988. “An Examination of Forcing in Direct Numerical Simulations of Turbulence.” Computers and Fluids. https://doi.org/10.1016/00457930(88)900138.
 Wojnowicz, Michael Thomas. 2012. “The OrnsteinUhlenbeck Process In Neural DecisionMaking: Mathematical Foundations And Simulations Suggesting The Adaptiveness Of Robustly Integrating Stochastic Neural Evidence.” Phdthesis. https://digital.lib.washington.edu:443/researchworks/handle/1773/21760.