[Paper Review] A well posed acoustic analogy based on a moving acoustic medium

W. Möhring|arXiv (Cornell University)|Sep 20, 2010

Aerodynamics and Acoustics in Jet FlowsEngineering9 references22 citations

TL;DR

This paper proposes a well-posed acoustic analogy based on the stagnation enthalpy in a moving, irrotational, isentropic flow, using a self-adjoint convected wave operator that ensures reciprocity and energy conservation. It applies the formulation to model sound radiation from a convected vortex colliding with a rigid cylinder, demonstrating stable, numerically feasible solutions with non-dipolar radiation patterns at moderate Mach numbers.

ABSTRACT

For flows of a lossless gas, the stagnation enthalpy obeys a linear convected wave equation with coefficients which depend on the flow variables. This equation is self-adjoint and one has a reciprocity relation between source and observer. It fulfills for subsonic flow a quadratic conservation equation implying stability. It is taken as basis for an acoustic analogy and is applied to the sound generation by the collision of a convected vortex and a rigid cylinder.

Motivation & Objective

• To develop a mathematically well-posed acoustic analogy for sound generation in moving flows, avoiding issues with traditional analogies.
• To establish a self-adjoint operator for sound propagation in a moving medium, ensuring reciprocity and energy conservation.
• To model sound radiation from vortices convected in a potential flow, particularly focusing on non-dipolar radiation at non-small Mach numbers.
• To validate the approach numerically using a PDE solver for a vortex-cylinder interaction problem.

Proposed method

• The paper derives a linear, self-adjoint convected wave equation for stagnation enthalpy in a lossless, isentropic, irrotational flow, using material derivatives to ensure Galilean invariance.
• The governing equation is derived from a variational principle, leading to a quadratic conservation law (energy theorem) and a reciprocity relation between source and observer.
• The acoustic source terms are expressed as divergence terms involving entropy and vorticity inhomogeneities, which act as sources in the wave equation.
• A numerical solution is obtained using a general-purpose PDE solver (PDEase/2) on a PC, solving the equation for a two-dimensional vortex convected past a rigid cylinder.
• Boundary conditions include vanishing normal velocity at rigid walls and a non-reflecting condition based on the characteristic speed of the flow.
• The solution is validated by visualizing the stagnation enthalpy field over time, showing immediate sound radiation due to the inhomogeneous flow.

Experimental results

Research questions

• RQ1 Can a well-posed acoustic analogy be formulated for sound generation in moving flows using a self-adjoint operator?
• RQ2 Does the proposed analogy preserve energy conservation and reciprocity in subsonic, irrotational flows?
• RQ3 How does sound radiation from a convected vortex differ from dipole radiation, especially at moderate Mach numbers?
• RQ4 Can the formulation be numerically solved with standard PDE solvers on a personal computer?

Key findings

• The derived equation for stagnation enthalpy is self-adjoint and leads to a valid energy theorem and reciprocity principle, ensuring mathematical well-posedness.
• Passively convected vorticity and entropy inhomogeneities do not radiate sound, as shown by the solution of the homogeneous equation in the constant-flow case.
• Numerical simulations show immediate sound radiation upon vortex entry into the inhomogeneous flow field, even at early times, due to convection effects.
• The radiation pattern deviates significantly from a dipole character at moderate Mach numbers, indicating complex wave interactions.
• The numerical solution exhibits small-scale oscillations due to numerical errors, suggesting the need for higher-resolution or stabilized schemes.
• The approach is numerically feasible with a standard PDE solver on a PC, demonstrating practical applicability to complex aeroacoustic problems.