PERIODIC WATER · FOURIER PSEUDOSPECTRAL DNS

Spectral Flow
Laboratory

A direct numerical simulation of incompressible water in a repeating 3D cube. The velocity is represented with Fourier waves; no turbulence model is used.

CONNECTING

FLOW TRACERS

Each curve follows a moving parcel of water.

BLUE + CYAN — WATER PATHSGOLD — HIGHLIGHTED REFERENCE PATHS

CLICK FOR A NEW FLOW · DRAG TO ROTATE

PHYSICAL TIME0.0000 sdt —

REYNOLDS NUMBER—\(\mathrm{Re}=U_0L/\nu\)

DIMENSIONLESS DIVERGENCE—\((L/U_0)\max |\nabla\!\cdot\!\mathbf{u}|\)

THROUGHPUT—render / solver

THE MODEL BEHIND THE PICTURE

Incompressible Navier–Stokes

The solver advances velocity u while pressure p enforces zero divergence:

\[\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\!\cdot\!\nabla)\mathbf{u}=-\frac{1}{\rho}\nabla p+\nu\nabla^2\mathbf{u}+\mathbf{f}\]\[\nabla\!\cdot\!\mathbf{u}=0\]

PERIODIC 3D DOMAIN

\[\mathbf{u}(\mathbf{x}+L\mathbf{e}_i,t)=\mathbf{u}(\mathbf{x},t),\qquad L=0.01\ \mathrm{m}\]

Leaving one face of the cube means re-entering through the opposite face. The wireframe is not a solid wall.

FOURIER FORM USED IN CODE

\[\frac{\partial\widehat{\mathbf{u}}}{\partial t}=-\mathbf{P}(\mathbf{k})\,\widehat{(\mathbf{u}\!\cdot\!\nabla)\mathbf{u}}-\nu|\mathbf{k}|^2\widehat{\mathbf{u}}+\widehat{\mathbf{f}}\]\[\mathbf{P}(\mathbf{k})=\mathbf{I}-\frac{\mathbf{k}\mathbf{k}^{\mathsf T}}{|\mathbf{k}|^2}\]

The projection \(\mathbf{P}\) removes the pressure contribution and enforces \(\mathbf{k}\!\cdot\!\widehat{\mathbf{u}}=0\). The nonlinear term is evaluated pseudospectrally with 2/3 dealiasing and integrated with fourth-order Runge–Kutta.

FIXED WATER PROPERTIES

Water at 20 °C

\(\rho=998.2\ \mathrm{kg\,m^{-3}}\)Fixed to the density of liquid water at the stated reference condition.
\(\mu=1.0016\!\times\!10^{-3}\ \mathrm{Pa\,s}\)Fixed to the dynamic viscosity of water.
\(\nu=\mu/\rho=1.003406\!\times\!10^{-6}\ \mathrm{m^2\,s^{-1}}\)Fixed from the water density and dynamic viscosity above.
\(T=293.15\ \mathrm{K}\;(20\,^{\circ}\mathrm C)\)Fixed as the condition used for the water properties.
\(p_{\mathrm{ref}}=101{,}325\ \mathrm{Pa}\)Fixed as the reference condition for liquid water; the solver computes pressure fluctuations implicitly.

FIXED SIMULATION SETTINGS

Grid: \(16^3\) points · cube length: \(L=0.01\ \mathrm m\) · initial RMS speed: \(U_0=0.001\ \mathrm{m\,s^{-1}}\). These are model settings, not intrinsic properties of water.