Numerical Study of Air-jet Instrument Sounds using Hydrodynamic-Acoustic Splitting

Jin Woo Lee

Music Technology, MIT

Benjamin Sabatini

Dept. of Materials Science and Eng., MIT

Jared Katz

Museum of Fine Arts, Boston

Nate Steele

Museum of Fine Arts, Boston

Mark Rau

Music Technology, MIT

Air-jet Instruments

  • Recorders
    Alto recorder
    [Accession Number 17.1809]
  • Flutes
    Alto flute in G (Boehm system)      End-blown flute (shakuhachi) [Accession Number 1986.800]       [Accession Number 17.2154]
  • Vessel Flutes
    Soprano ocarina
    [Accession Number 1984.309]
  • Whistles
    Boatswain’s whistle
    [Accession Number 17.1830]

All images courtesy of the Museum of Fine Arts, Boston. www.mfa.org

Current Study

  • Direct Numerical Simulation of Air-jet Instruments
    • Instrument-agnostic formulation
    • Computationally intensive
      • Fluid-driven → Nonlinear PDE → Heavy Computation
      • Sound Speed → Small time steps → Heavy Computation
  • Our Approach: Hydrodynamic-Acoustic Splitting
    • Efficient
      • Hydrodynamics → large time steps with incompressible fluid
      • Acoustics → small time steps + linear wave (GPU parallel)
    • Applications on: recorder & vessel flute

2D Experiment Sandbox

Hydrodynamic-Acoustic Splitting

Flow Behind a Cylinder \[\underbrace{p(\mathbf{x},t)}_{\mathrm{Total~Pressure}} = \underbrace{P(\mathbf{x},t)}_{\mathrm{Fluid}} + \underbrace{p'(\mathbf{x},t)}_{\mathrm{Acoustic}}\]

  • \(P\): incompressible Navier–Stokes

  • \(p'\): inhomogeneous wave equation

P(\mathbf{x},t) p'(\mathbf{x},t)

Incompressible Navier–Stokes (INS)

\[\begin{align} \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} &= -\frac{1}{\rho_0}\nabla P + \nu\,\Delta \mathbf{u} + \mathbf{f} & \mathrm{Momentum}\\ \nabla \cdot \mathbf{u} &= 0 & \mathrm{Continuity} \end{align}\]

  • \(\mathbf{u}\): velocity field
  • \(P\): hydrodynamic pressure
  • \(\rho_0\): constant density
  • \(\nu\): kinematic viscosity
  • \(\mathbf{f}\): external source (jet)

INS Update Rule

\[\begin{align} \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} &= -\frac{1}{\rho_0}\nabla P + \nu\,\Delta \mathbf{u} + \mathbf{f} & \mathrm{Momentum}\\ \nabla \cdot \mathbf{u} &= 0 & \mathrm{Continuity} \end{align}\]

  • Advection–Diffusion prediction \[\small \mathbf{u}^* = \mathbf{u}^n + T_1 \left( -\mathbf{u}^n \cdot \nabla \mathbf{u}^n + \nu \nabla^2 \mathbf{u}^n \right)\]
  • Pressure Poisson solve \[\small \nabla^2 P^{n+1} = \tfrac{\rho_0}{T_1} \nabla \cdot \mathbf{u}^*\]
  • Velocity correction \[\small \mathbf{u}^{n+1} = \mathbf{u}^* - \tfrac{T_1}{\rho_0} \nabla P^{n+1}\]
  • Hydrodynamic time step (3D) \[T_1 = \min\!\left(\frac{\lambda_1}{\sqrt{3}}\frac{X_i}{U_0},\;\frac{X_i^2}{6\nu}\right)\]
    • \(\lambda_1\): Courant number
    • \(X_i\): grid spacing
    • \(U_0\): jet speed (\(\ll c_0\))
    • \(\nu\): kinematic viscosity

Moving on to the acoustic part…

Flow Behind a Cylinder \[\underbrace{p(\mathbf{x},t)}_{\mathrm{Total~Pressure}} = \underbrace{P(\mathbf{x},t)}_{\mathrm{Fluid}} + \underbrace{p'(\mathbf{x},t)}_{\mathrm{Acoustic}}\]

  • \(P\): incompressible Navier–Stokes

  • \(p'\): inhomogeneous wave equation

P(\mathbf{x},t) p'(\mathbf{x},t)

Lighthill’s Acoustic Analogy

The acoustic field described in an inhomogeneous wave equation: \[\begin{equation} \frac{1}{c_0^2}\,\frac{\partial^2 p'}{\partial t^2} - \Delta p' = \frac{\partial^2 L_{ij}}{\partial x_i\,\partial x_j} \end{equation}\]

  • \(c_0\): ambient speed of sound

  • \(L_{ij}\): Lighthill stress tensor

\[\begin{equation} \frac{\partial^2 L_{ij}}{\partial x_i\,\partial x_j} \approx \rho_0\sum_{i,j}G_{ij}G_{ji} \end{equation}\]

  • \(G_{ij} = \partial u_i/\partial x_j\): velocity-gradient

Acoustic Update Rule

The acoustic field described in an inhomogeneous wave equation: \[\begin{equation} \frac{1}{c_0^2}\,\frac{\partial^2 p'}{\partial t^2} - \Delta p' = \frac{\partial^2 L_{ij}}{\partial x_i\,\partial x_j} \end{equation}\]

  • Störmer–Verlet \[\small\begin{align} {p'}^{n+1} &= 2{p'}^n - {p'}^{n-1} \\ &+ (c_0 T_2)^2\nabla^2{p'}^n + S^n \end{align}\]
  • Acoustic source \(\small S^n = -\rho_0\,c_0^2\,\frac{T_2^2}{K\,T_1}\,\bigl(D_{k+1} - D_k\bigr)\) \(\small D = \nabla \cdot \mathbf{u}^*\)

Acoustic Update Rule

The acoustic field described in an inhomogeneous wave equation: \[\begin{equation} \frac{1}{c_0^2}\,\frac{\partial^2 p'}{\partial t^2} - \Delta p' = \frac{\partial^2 L_{ij}}{\partial x_i\,\partial x_j} \end{equation}\]

  • Acoustic time step \[\begin{align} T_2 &= \tfrac{\lambda_2}{\sqrt{3}}\tfrac{X_i}{\color{red}{c_0}} \\ &\approx \color{red}{3.4\times10^{-7}} \end{align}\] \[(c_0=343\ \mathrm{m/s},\ \lambda_2=1.0)\]

  • Recall: Hydrodynamic time step \[\begin{align} T_1 &= \min\!\left(\tfrac{\lambda_1}{\sqrt{3}}\tfrac{X_i}{\color{red}{U_0}},\;\tfrac{X_i^2}{6\nu}\right) \\ &\approx \color{red}{2.8\times10^{-6}} \end{align}\] \[(U_0\approx17\ \mathrm{m/s},\ \lambda_1=0.4)\]

Result: Tenor Recorder

Result: Historic Vessel Flute

Soprano ocarina photo courtesy of the Museum of Fine Arts, Boston. www.mfa.org

Result: Historic Vessel Flute

Simulated Results at \(t=0.17\ \mathrm{ms}\)

  • Colors: Streamwise Velocity
  • Dashed lines: Cut Planes
  • Green Region: Inflow
  • Colors: Acoustic Pressure
  • Arrows: Streamline

Future Directions

  • Validation
    • with parametric recorder models
    • with experimental data
  • Extension to more complex geometries
    • with more diverse shapes
    • with more diverse materials

Acknowledgments


← Presentation slides
https://jin-woo-lee.github.io/assets/slides/26-isma/slides.html

Web demo →
https://jin-woo-lee.github.io/demos