Point Source in 2D Jet:Radiation and refraction of sound waves through a 2D shear layer

Model Gallery #16685

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Background and Motivation

• This is a benchmark model for the Linearized Euler Physics Interface. The model is from the NASA "Fourth Computational Aeroacoustics (CAA) Workshop on Benchmark Problems (2004)". The model results are compared to an analytical solution by Agarwal et al. (AIAA Vol. 42, No. 1, January 2004).

• A Gauss shaped point-like source is located in a narrow 2D jet of Mach 0.76. The model investigates the propagation of acoustic and non-acoustic waves (instability waves) in the flow when solving the linearized Euler equations in both the frequency and the time domain.

Linearized Euler Model 1

• The Linearized Euler (LE) equations are used to model the propagation of linear small amplitude waves in a non-uniform background (ideal gas like) flow.

• Because there is no damping in the LE formulation the equations also support the propagation and growth of non-acoustic instability waves in the transient model. These are known as Kelvin-Helmholtz instabilities. In real flows these are damped by nonlinearities in the governing equations (Navier-Stokes) and loss mechanisms (viscosity and thermal conduction).

• Typically we are only interested in the acoustic signal which can propagate over long distances.

Linearized Euler Model 2

• When the LE equations are solved in the frequency domain the instabilities are not excited as we restrict the model to only have a real valued frequency component. This means that only the acoustic signal (at the given frequency) is modeled.

• The current model compares the frequency and transient versions of the LE interface. It shows how an acoustic source interacts with a narrow jet with high gradients.

Geometry and Operating Conditions

Gauss point-like source S

xSymmetry line y = 0

PML (frequency domain only)

u0(y)

y

Jet

𝑆=𝐴𝑒(−𝐵𝑥 ∙𝑥2−𝐵𝑦 ∙𝑦

2 )cos (𝜔𝑡)𝒖0 (𝑦 )=𝑢 𝑗𝑒

(− log (2) ∙( 𝑦/𝑏)2)

Modeling Interfaces

• The model uses the Linearized Euler, Frequency Domain and the Linearized Euler, Transient interfaces.

Load model parameters andanalytical solution from files

Load model variables (jet and source)Set up PMLs for the frequency domain model

The Linearized Euler, Frequency Domain interface set up the Domain Source. The PMLs will act as open boundaries.

The Linearized Euler, Transient interface set up the Domain Source. The model is here terminated by a Rigid Wall condition.

Modeling Interfaces

• The equations solved are:Frequency Domain:Continuity EquationMomentum Equation (Euler’s)Energy Equation

Time Domain:Continuity EquationMomentum Equation (Euler’s)Energy Equation

• Dependent variables: density , velocity field u, pressure p• Background flow properties have subscript 0.• Sources are on the RHS of the equations. In this model the

source is defined by Se.

Model Setup

• The Linearized Euler Model

Here you set the backlground mean flow fields. In this model they are all user defined and stem from an analytical expression. In other models these values can automatically be picked up from a CFD model (requires the CFD Module).

Set the material parameters for the fluid. In the Linearized Euler interfaces the fluid is assumed to be an Ideal Gas.

Mesh

Use a structured Mapped mesh for the domain.

The mesh has to be well resoved nest the symmetry axis y = 0 to resolve the gradients on the very narrow background jet.

Close up of the mesh near y = 0.

Study

• Use the default settings for the frequency domain model in Study 1.

• Define the time step used in the transient model in Study 2:

The timestep Dt = 1/10*1/f0, defined under parameters, sets the maximal timestep taken by the solver to one tenth of the source periode.

Results: Frequency Domain

• Pressure Distribution: real(p)

Jet

Results: Frequency Domain

• Pressure Contours: real(p)

Jet

Results: Frequency Domain

• Instantaneous Velocity Components

y-velcotiy

x-velcotiy

Results: Frequency Domain

• Sound Pressure Level

Damping of outgoing waves in the PML layer.

Results: Frequency Domain

• Pressure at Cut Lines y = 15 m and y = 0: real(p)

Results: Frequency Domain

• Pressure at Cut Lines compared to the analytical solution: real(p) and imag(p)

Results: Transient

• Pressure distribution after 5 periods

Kelvin-Helmholtz type instabilities growing in the transient model.

Reflections at the wall.

Results: Transient

• Pressure at Cut Lines y = 15 m and y = 0

Kelvin-Helmholtz type instabilities growing in the transient model (at y = 0). These are created by the interaction with the background flow so they do not show at y = 15 which is outside of the jet.

Results: Comparison

• Pressure in Cut Line y = 15 m: comparison of frequency domain, analytical and transient models

Results: Comparison

• Pressure at Cut Line x = 0: comparison of frequency domain, analytical and transient models

t = 0.04 s

t = 0.41 s

The Kelvin-Helmholtz type instabilities are created when the wave is interacting with the jet (red line). The instabilities (green curve from transient model) propagate on top of the acoustic wave (blue curve from the frequency domain model) in the region outside of the jet.