Solving Navier Stokes equation using Spectral method (1)

Posted on Mon 17 July 2017 in Blog

Introduction

The spectral method is solving certain differential equation by some "basis function", typically sinusoids with Fourier method With the Navier-Stokes equation, it can remove presssure term in N-S equation and solve viscous term analytically. This is huge because solving Poisson equation (pressure term) usually takes very long time compared to other terms. Moreover, I can expect more accuracy than simple numericla discretization. However, I can apply spectral method only to periodic domain, but it doesn't matter when solving isotropic turbulence.

Governing Equtation

Original Navier-Stokes equation in convection form is

\begin{align} \dfrac{\partial u_i}{\partial t} &= -\dfrac{\nabla p}{\rho} - (u \cdot \nabla) u + \nu \nabla^2 u \\ \nabla \cdot u &= 0 \end{align}

Using following vector identical solution,

\begin{align} \dfrac{1}{2} \nabla (A \cdot A) = (A \cdot \nabla) A + A \times (\nabla \times A) \end{align}

I can convert original equation to rotational form.

\begin{align} \dfrac{\partial u_i}{\partial t} &= -\dfrac{\partial P}{\partial x_i} + H_i + \nu \nabla^2 u \\ \dfrac{\partial u_i}{\partial x_i} &= 0 \end{align}

where

\begin{align} P &= \dfrac{p}{\rho} + \dfrac{1}{2} u_j u_j \\ H_i &= \epsilon_{i,j,k} u_j \omega_k = u \times (\nabla \times u) \end{align}

Removing pressure term

If I take divergence from Navier-Stokes equation in rotational form,

\begin{align} \nabla^2 P = \dfrac{\partial H_j}{\partial x_j} \end{align}

By expanding both equation (N-S equation & Poisson equation) to Fourier space, I got following equation

\begin{align} \dfrac{d \hat{u}_i }{d t} &= -i \kappa_i \hat{P} + \hat{H}_i - \nu \kappa^2 \hat{u}_i \\\ -\kappa^2 \hat{P} &= i \kappa_j \hat{H}_j \end{align}

Combining two equation and then..

\begin{align} \dfrac{d \hat{u}_i }{d t} &= -i \kappa_i \left( -i \dfrac{\kappa_j}{\kappa^2} \hat{H}_j \right ) + \hat{H}_i - \nu \kappa^2 \hat{u}_i \\\ \dfrac{d \hat{u}_i }{d t} &= -\dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i - \nu \kappa^2 \hat{u}_i \end{align}

where $\kappa$ is a wavenumber. Now I have new N-S equation without pressure term

\begin{align} \dfrac{d \hat{u}_i }{d t} &= -\dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i - \nu \kappa^2 \hat{u}_i \end{align}

Treating viscous term analytically

To treat a viscous terms analytically, I just multiply following formula to N-S equation without pressure form

\begin{equation} f(t) = e^{\nu \kappa^2 t} \end{equation}

Then the equation changed like this

\begin{align} \left[ \dfrac{d \hat{u}}{dt} + \nu \kappa^2 \hat{u}_j \right] \times f(t) &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] \times f(t) \\\ f(t) \dfrac{d \hat{u}}{dt} + (\nu \kappa^2 f(t))\hat{u}_j &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] f(t) \\\ f(t) \dfrac{d \hat{u}}{dt} + (\nu \kappa^2 e^{\nu \kappa^2 t})\hat{u}_j &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] f(t) \\\ f(t) \dfrac{d \hat{u}}{dt} + (\dfrac{d e^{\nu \kappa^2 t}}{dt})\hat{u}_j &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] f(t) \\\ \dfrac{d \hat{u}_i f(t)}{dt} &= \left[ - \dfrac{\kappa_i \kappa_j}{\kappa^2} \hat{H}_j + \hat{H}_i \right ] f(t) \end{align}

this can be more simpler by introducing new term $\widehat{NL}$

\begin{equation} \dfrac{d \hat{u}_i e^{\nu \kappa^2 t}}{dt} = \widehat{NL} e^{\nu \kappa^2 t} \end{equation}

Time Discretization by RK3 method

For low-storage RK3 method (2-register, 3-stage, 3rd order), the coefficients are following table

order $a_n$ $b_n$ $c_n$
1st 8/15 0 0
2nd 5/12 -17/60 8/15
3rd 3/4 -5/12 2/3

If I assume to solve following equation,

\begin{equation} \dfrac{\partial Q}{\partial t} = R(Q) \end{equation}

The low-storage RK3 method applied to the equation using above coefficients.

\begin{align} Q^1 &= Q^n + \Delta t \left( \dfrac{8}{15} R^n \right) \\\ Q^2 &= Q^1 + \Delta t \left( \dfrac{5}{12} R^n - \dfrac{17}{60} R^1\right) \\\ Q^{n+1} &= Q^2 + \Delta t \left( \dfrac{3}{4} R^n - \dfrac{5}{12} R^2\right) \end{align}

Before Navier-Stokes equation, I can apply low-storage RK3 method to reaction-diffusion equation

\begin{align} \dfrac{\partial \psi}{\partial t} &= G + L \psi \\\ \psi^{n+1} &= \psi^{n} + a_n \Delta t G^n + b_n \Delta t G^{n-1} + (a_n + b_n)\Delta t\left(\dfrac{L \psi^{n+1} + L \psi^n}{2} \right) \end{align}

Then the N-S equation should be..

\begin{align} \dfrac{d \hat{u} e^{\nu \kappa^2 t}}{dt} &= \widehat{NL} e^{\nu \kappa^2 t} \\\ \dfrac{\hat{u}^{n+1}_i e^{\nu \kappa^2 (t+a_n \Delta t + b_n \Delta t)} - \hat{u}^{n}_i e^{\nu \kappa^2 t}}{\Delta t} &= a_n\widehat{NL}^n e^{\nu \kappa^2 t} + b_n\widehat{NL}^{n - 1} e^{\nu \kappa^2 (t - a_{n-1} \Delta t - b_{n-1} \Delta t)} \\\ \hat{u}^{n+1}_i e^{\nu \kappa^2 (a_n \Delta t + b_n \Delta t)} - \hat{u}^{n}_i &= a_n \Delta t \widehat{NL}^n + b_n \Delta t \widehat{NL}^{n - 1} e^{\nu \kappa^2 (- a_{n-1} \Delta t - b_{n-1} \Delta t)} \\\ \hat{u}^{n+1}_i &= \left[a_n \Delta t \widehat{NL}^n + \hat{u}^{n}_i \right ] e^{-\nu \kappa^2 (a_n + b_n) \Delta t} + b_n \Delta t \widehat{NL}^{n - 1} e^{\nu \kappa^2 -(a_n + b_n + a_{n-1}+ b_{n-1}) \Delta t} \\\ \end{align}

Reference

RK3 method

  • Lundbladh, Anders, et al. "An efficient spectral method for simulation of incompressible flow over a flat plate." Trita-mek. Tech. Rep 11 (1999).
  • Wray, A. A. "Minimal storage time advancement schemes for spectral methods." NASA Ames Research Center, California, Report No. MS 202 (1990).
  • Yu, Sheng-Tao. "Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations." (1992).