Appunti Simulation and modelling of turbulent flows

Space discretization

Wavenumber and wavelength

Wavenumber: k_N = ^2πk/_Δ is a measure of the spatial frequency of a wave.

Wavelength: λ_n = ^2π/_kn

Fourier space and convolution

In Fourier space, a convolution integral becomes a simple multiplication:
(f * g) = ∫ f(u)g(x-u)du ⟷ 1/_λ(f̂ * ĝ)_k = f̂_k ĝ_k

Parseval's theorem

Parseval's theorem: ∫ |f(x)|² dx = ∑_-∞^∞ |f̂_k|² = ∑_-∞^∞ f̂_k(f̂_k)*

Derivative property

The Fourier coefficients of g = df/dx is:
dF/dx = d/dx∑_k=-∞^∞f̂_ke^ikx = ∑_k=-∞^∞jkf̂_ke^ikxĝ_k ⟷ (df/dx) = jkf̂_k

The Fourier coefficients of g = d²f/dx² is:
d²f/dx² = ∑_k=-∞^∞(-k²)f̂_ke^ikx ⟷ ĝ_k = (d²/dx²) = -k²f̂_k

Truncated Fourier series

We now consider a finite number of harmonics: the truncated Fourier series is defined as:
S_l(f(x)) = ∑_k=-N/2^N/2 f̂_ke^i2πkx/λ_N where the number of harmonics is truncated to N+1. This truncation will introduce a truncation error as a function of N: ε = f(x) - S_N(f(x)).

Aliasing error and sampling

There is a second source of error: the aliasing, where the integral in the definition of Fourier coefficients:
f̂_n = ∫^Δ₀ f(x)e^-j2πkx/Δ dx is computed by using a finite number of points, thus introducing the discrete Fourier transform. When using N+1 discretization points, it is impossible to distinguish an harmonic with wavenumber k with another harmonic with a higher wavenumber (±).

In torsion flows, the largest is given by the Koluwo scale η: (k_n)_max = ^2π/_η

It's important to find how many sampling points M are needed to avoid aliasing error when simulating a function with harmonics limited to N/2. We need Fourier coefficients to be zero for all harmonics below -N/2 and above N/2. We discovered that we need at least M = N.

Finite difference method

We now consider N_c space N+1 points where the velocity u(x) in the discretized space becomes Ũ:(x_i) where x_i = (i-1)Δx, i=1,...,N+1, and Δx = ^Δ/_N. We want an algebraic version of differential operators. The continuous derivative is defined as: ∂u/∂x = lim_h->0 u(x+h) - u(x)/h

Additional information

Space discretization mercoledì 19 luglio 2023 14:16

Wavenumber: k_m = ^{2 π k}⁄_Δ is a measure of the spatial frequency of a wave

Wavelength: λ_m = ^{2 π}⁄_km

In Fourier space, a convolution integral becomes a simple multiplication:
(f * g) = ∫(f(x)g(x-d))dx → ₁⁄_Δ (ƒ_k ^{ƒ_k^})

Parseval's theorem: ∫(|f(x)|²)dx = ¹⁄_Δ Σ ^{∈ |ƒ_k|2} = Σ ^k=-∞⁄_&infin|ƒ_k|² = ƒ_k ƒ^*_k

Derivative property: The Fourier coefficients of g = df/dx is:
df⁄_dx = ^d⁄_dx Σ_k=-∞⁄^{k=∞ f_ei_kx = Σ_k=-∞∞j k_m ƒei_kx&is.;τcfunction = k of work τxƒ_k = -(df/dx) → j k_m ƒ_k}

The Fourier coefficients of g - d²f/dxdx is:
d²f/dx² = Σ_k=-∞^k=∞k²_m^f_ei^kx → ƒ^{^} g_k = -(df/_{dx²) = -k^2mƒk}

We now consider a finite number of harmonics; the truncated Fourier series is defined as:
S_N(f(x)) = Σ_k=-N/2