Sound wave propagation and shock-sound interaction in a converging-diverging nozzle

This project involved solving a problem which was part of the Third Computational Aeroacoustics (CAA) Workshop on Benchmark Problems held at the Ohio Aerospace Institute in Cleveland, Ohio on November 8-10, 1999. The problem statement is given below.

Problem 1 : Propagation of sound waves through a transonic nozzle

In a transonic cascade, the local Mach number of the flow in the narrow passages may be close to sonic. The computation of sound propagating through such regions presents a challenging problem. To reduce the complexity of the problem, but retaining the basic physics and difficulties, we will model such propagation problems by a one-dimensional acoustic wave transmission problem through a nearly choked nozzle.

We will use the following as characteristic scales,

length scale = diameter of nozzle in the uniform region downstream of the throat (see figure),
velocity scale = speed of sound in the same region,
time scale =
density scale = mean density of gas in the same region,
pressure scale =

Consider a one-dimensional nozzle with an area distribution as follows,

The governing equations in dimensionless form are,

The Mach number in the uniform region downstream of the throat is .

Small amplitude acoustic waves, with angular frequency , is generated way downstream and propagate upstream through the narrow passage of the nozzle throat. Let the upstream propagating wave in the uniform region downstream of the nozzle throat be represented by,

where . Use a computation domain of size 20, 10 upstream and 10 downstream of the nozzle throat, to calculate the distribution of maximum acoustic pressure inside the nozzle.

This problem can, of course, be calculated accurately if a very large number of mesh points is used. But this is not always practical. It is recommended that no more than 400 mesh points be used. Report the locations of your mesh points and the pressure distribution. Also report the total number of mesh points used.

Problem 2 : Shock-sound interaction

In imperfectly expanded supersonic.jets, shock-cell structures are formed downstream of the nozzle exit. To simulate the shock-sound interactions, the problem is simplified as a sound wave passing through a shock in a quasi-1-D supersonic nozzle.

This problem uses the same geometry as Problem 1, but now there is a supersonic shock downstream of the throat.

In this problem, the quasi-1-D Euler equations are solved,

All quantities are non-dimensionalized using the upstream values,

length scale =
density scale =
velocity scale =
pressure scale =
time scale =

where D is the nozzle height and a is the speed of sound, .

As before, the domain is , and the area of the nozzle is given by,

At the inflow boundary, the conditions are,

where,

The pressure will be set at the outflow boundary to create a shock,

The data required for this problem is,

1. Grid used for the problem (i,x)
2. On the domain , give the steady mean distribution () and the perturbation at the start of a period ()
3. Over the period of the perturbation, give the pressure perturbation at the exit plane ()

The results that I obtained for Problem 1 are shown below. The exact solution obtained by Dr. Ray Hixon has been superimposed over the numerical solution.

A 251-point grid was used in both problems which was significantly less than the maximum permissible number of grid points (400) as stipulated in the problem statement. For spatial differencing, the optimized explicit DRP scheme by Tam and Webb was used and the fourth-order explicit Runge-Kutta scheme was used for time marching.

For Problem 2, exact solutions were available only for the pressure quantities for comparison. Hence only the solutions for mean pressure distribution over the domain, unsteady pressure perturbation at the start of a period and the pressure distribution at the exit plane through one period of the perturbation are shown. The graph below shows the mean pressure distribution over the domain.

The next plot shows the unsteady pressure perturbation at the start of a period.

The last plot shows the pressure distribution at the exit plane through one period of the perturbation.

The complete project report can be obtained here (Adobe Acrobat Reader is needed to view the file and can be downloaded for free here). To get a copy of the Matlab code, use the CONTACT ME link.