Philip C.E. Jorgenson

Glenn Research Center, Cleveland, Ohio

Ching Y. Loh

Taitech, Inc., Brook Park, Ohio

Computing Axisymmetric Jet Screech

Tones Using Unstructured Grids

NASA/TM—2002-211799

August 2002

AIAA–2002–3889
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Philip C.E. Jorgenson

Glenn Research Center, Cleveland, Ohio

Ching Y. Loh

Taitech, Inc., Brook Park, Ohio

Computing Axisymmetric Jet Screech

Tones Using Unstructured Grids

NASA/TM—2002-211799

August 2002

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

38th Joint Propulsion Conference and Exhibit

cosponsored by the AIAA, ASME, SAE, and ASEE

Indianapolis, Indiana, July 7–10, 2002

AIAA–2002–3889
Acknowledgments

This work was supported by the Supersonic Propulsion Technology Program

 at the NASA John H. Glenn Research Center at Lewis Field.

Available from

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National Technical Information Service

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The Aerospace Propulsion and Power Program at

NASA Glenn Research Center sponsored this work.

Available electronically at http://gltrs.grc.nasa.gov
COMPUTING AXISYMMETRIC JET SCREECH TONES USING UNSTRUCTURED

GRIDS

Abstract

The space-time conservation element and solution element (CE/SE) method is used to solve

the conservation law form of the compressible

axisymmetric Navier-Stokes equations. The

equations are time marched to predict the unsteady flow and the near-field screech tone

noise issuing from an underexpanded circular

jet. The CE/SE method uses an unstructured

grid based data structure. The unstructured

grids for these calculations are generated based

on the method of Delaunay triangulation.

The purpose of this paper is to show that an

acoustics solution with a feedback loop can be

obtained using truly unstructured grid technology. Numerical results are presented for two

different nozzle geometries. The first is considered to have a thin nozzle lip and the second

has a thick nozzle lip.

Comparisons with available experimental data

are shown for flows corresponding to several

different jet Mach numbers. Generally good

agreement is obtained in terms of flow physics,

screech tone frequency, and sound pressure

level.

1 Introduction

It is desirable to develop a general computer code that

can predict the complex flow structure around complicated geometries and the associated aeroacoustics. Many

computational fluid dynamics methods have been developed that take advantage of an inherent grid structure

which permits flow solutions to be obtained efficiently.

For computational aeroacoustics (CAA), another obvious advantage of the structured grid is its well-controlled

grid quality (size and shape) that helps to maintain the

high quality of the computed acoustic waves. This structure, which makes the solver so efficient, often makes

it difficult to generate grids about complex geometries

without an additional complication of blocked or overset

grid techniques.

An unstructured grid flow solver can alleviate many of

the problems associated with structured grids. Unlike a

structured grid, the unstructured grid neighborhood must

be provided to the solver explicitly. The simplest geometric figure that can cover a two-dimensional domain is

the triangle. The advantage of the triangle is that it can

easily be used to generate a grid over an arbitrary domain

without overlapping. It also has the advantage of clustering in high flow gradient regions without the concern of

the surrounding cell structure. The main disadvantage of

the unstructured grid is the additional memory requirements in the flow solver to store the grid. More of the

burden is placed on the computer code, thus the user input is reduced both during grid generation and flow solution such that more configurations can be tested in a

shorter period of time. For CAA, the grid quality of an

unstructured grid is less controllable and it may deteriorate the acoustic wave computations. The situation may

be even more critical if a complicated acoustic feedback

loop is present, which is the problem type considered in

this paper.

The problems under consideration in this paper are circular underexpanded supersonic jets [1-8]. These jets radiate a combination of mixing noise, broadband shockassociated noise, and under certain conditions screech

tones. Mixing noise from jets can be directly associated with large-scale structures, or instability waves in

the shear layer. The broadband shock-associated noise

and screech tones are associated with the interaction of

these instability waves with shock cell structure in the

jet core. The screech tones arise due to a feedback loop,

i.e., part of the acoustic waves generated by the instabilPhilip C.E. Jorgenson

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Ching Y. Loh

Taitech, Inc.

Brook Park, Ohio 44142

NASA/TM—2002-211799 1
ity wave/shock-cell interaction propagate upstream and

re-generate the instability waves at, or in the vicinity of,

the nozzle lip. The desire to control screech noise is

not only due to general noise reduction concerns. It can

also cause damage to the structure due to sonic fatigue.

The feedback loop is sensitive to small changes such as

lip thickness and other phenomena. Most of this understanding comes from experimental observations. Numerical methods that can predict screech tones will help in

the understanding and the eventual control of this noise

source.

A direct numerical simulation was computed by Shen

and Tam [9,10] for screech tones issuing from a circular

jet, based on the well-known DRP scheme, with a structured grid. A large computational domain of 35D long

in the streamwise direction and 17D in radial direction

was used in their computation, with D being the nozzle

diameter.

More recently the space-time conservation element

and solution element (CE/SE) method with a triangulated

structured grid [11-14] has been used to predict the nearfield screech tones from underexpanded supersonic jets

[15]. The near-field results compared well with the experimental data, [4-8]. This work also demonstrated the

capability of the CE/SE nonreflecting boundary condition. This allows the computational domain to be more

compact reducing the overall mesh memory size requirements. The present work is a companion work of [15].

Here a truly unstructured grid, generated by a point insertion Delaunay method rather than a triangulated structured grid, is employed. This is done with the idea that

more complex geometries will require a more flexible

grid generation technology. The purpose of this paper is

to show that an acoustics solution with a feedback loop

can be obtained using unstructured grid technology.

The governing equations are given in Section 2. Here

the nondimensionalized unsteady axisymmetric NavierStokes equations in conservation law form are presented.

Unstructured grid generation is discussed in Section 3.

The requirements of the grid for the flow solver will be

explained. Also the flow code was executed on a parallel

computer architecture, so the domain decomposition and

message passing libraries are briefly reviewed. The unstructured based axisymmetric CE/SE method used here

is briefly discussed in Section 4. Section 5 will provide

the problem description with the initial and boundary

condition requirements for the particular screech problems examined in this paper. In Section 6 numerical results are presented and and compared to previous numerical findings using structured grids and available experimental findings [4-8]. Conclusioning remarks will be

made in Section 7.

2 Governing Equations

The unsteady axisymmetric Navier-Stokes equations

are used to model the viscous problems in this study,

since experimentally for the thin lip nozzle [4-8] with

Mj 6 1.19 the flow is axisymmetric. In nondimensionalized differential form the conservation laws are written

as:

Ut + Fx + Gy = Q, (1)

where x, y, and t are the streamwise and radial coordinates and time, respectively. The conservative flow variable vector, U, and the flux vectors in the streamwise

direction, F, and radial direction, G, are given by:

U =





U1

U2

U3

U4



 , F =





F1

F2

F3

F4



 , G =





G1

G2

G3

G4



 ,

with

U1 = ρ, U2 = ρu, U3 = ρv,

U4 = p/(γ − 1) + ρ(u2 + v2)/2.

Here ρ, u, v, p, and γ are the density, streamwise velocity

component, radial velocity component, static pressure,

and constant specific heat ratio, respectively. The flux

vectors are further split into inviscid and viscous fluxes:

F = Fi − Fv, G = Gi − Gv,

where the inviscid fluxes are given as:

Fi1 = U2,

Fi2 = (γ − 1)U4 + 

(3 − γ)U2

2 − (γ − 1)U23



/2U1,

Fi3 = U2U3/U1,

Fi4 = γU2U4/U1 − (γ − 1)U2



U2

2 + U23



/2U2

1 ,

Gi1 = U3, Gi2 = U2U3/U1,

Gi3 = (γ − 1)U4 + 

(3 − γ)U2

3 − (γ − 1)U22



/2U1,

Gi4 = γU3U4/U1 − (γ − 1)U3



U2

2 + U23



/2U2

1 ,

and the viscous fluxes are:

Fv1 = 0, Fv2 = μ(2ux − 2

3∇·V),

Fv3 = μ(vx + uy),

Fv4 = μ[2uux + (uy + vx)v − 2

3 (∇·V)u+

γ

P r

∂

∂y (

U4

U1

− u2 + v2

2 )],

Gv1 = 0, Gv2 = μ(vx + uy),

Gv3 = μ(2vy − 2

3∇·V),

NASA/TM—2002-211799 2
Gv4 = μ[2vvy + (uy + vx)u − 2

3 (∇·V)v+

γ

P r

∂

∂y (

U4

U1

− u2 + v2

2 )],

where u, v, ux, uy, vx, vy are respectively the x and y

flow velocity components and their derivatives, which

can be written in terms of the conservative variables

U1, U2, U3 and U4. The Prandtl number and viscosity

are P r and μ, respectively. The velocity divergence is

∇·V = ux + vy + v/y.

The source term, Q, appearing on the right hand side

of the equation, Eq. (1), is a result of the axisymmetric

formulation and is given as:

Q =





Q1

Q2

Q3

Q4



 ,

where

Q1 = −U3/y, Q2 = −U2U3/U1y,

Q3 = −U2

3 /U1y, Q4 = −G4/y.

Considering (x, y, t) as coordinates of a threedimensional Euclidean space, E3, the Gauss divergence

theorem can be used to give the following equivalent integral form of the conservation laws, Eq. (1):

I

S(V )

Hm · dS =

Z

V

QmdV, m = 1, 2, 3, 4, (2)

where S(V ) denotes the surface of the volume V in E3

and Hm = (Fm, Gm, Um) represents the total flux leaving the boundary of space-time region. The integral form

of the conservation laws allows the capture of shocks

and other flow discontinuities naturally. This integral

form of the equations is solved numerically by the spacetime conservation element and solution element method

which is described in Section 4.

3 Grid Generation

An unstructured grid is used to show that aeroacoustics with a feedback loop can be predicted in complex

domain shapes. A triangular shaped cell will be used

as the geometric shape to cover a two-dimensional computational domain. The use of an unstructured grid removes the available implicit reference of the local computational neighborhood by incrementing an index as in

a structured grid. Instead, the neighborhood is provided

explicitly through a connectivity matrix. This connectivity matrix usually contains cell based information as well

as edge based information.

Figure 1: Triangulation of computational domain with

enlargement showing nozzle lip region.

The use of a triangular unstructured grid formulation

has some distinct advantages over a structured grid. One

advantage to using triangular cells is that with them it is

easy to generate grids about complex geometries. This

reduces the amount of time required to generate a suitable grid. Also grid adaptation can be done locally without adding unnecessary cells to other regions of the domain.

The method used in this work follows a path similar to

that of Holmes and Snyder [16] to triangulate a region.

An example of an unstructured grid for the thin lip nozzle

geometry generated in the above manner is shown in Fig.

1. The computation is axisymmetric, so only the upper

half of the domain needs to be considered. An enlargement of the nozzle lip region gives an indication of the

grid resolution used in these aeroacoustic feedback loop

problems.

The flow code requirements dictate the type of output

that the grid generation scheme must provide. A connectivity array must be generated for an unstructured grid

so that a cell neighborhood is completely defined for the

flow code. The code in the current work is based on a

cell centered scheme. Here the triangle itself is the control volume used in the finite volume formulation.

Connectivity is determined by cell nodes, cell edges,

and edge cells. Cell nodes are the nodes at the vertices

of a triangle. The cell edges are the edges of a triangle

NASA/TM—2002-211799 3
that connect those nodes. The edge cells are the cells

that lie on either side of a given edge. Typically the x,

y coordinates of the cell nodes are the only grid floating

point numbers required as input by a flow code. The grid

connectivity is defined through integer arrays.

Boundary condition information must be defined explicitly. Ghost cells are used across boundaries such that

specific boundary conditions can be applied. A boundary

condition such as solid wall, inlet, exit, or any other user

defined condition can be prescribed along any cell edge

through the ghost cell.

For aeroacoustic problems, large numbers of cells are

needed to resolve the unsteady waves. The grids are

made more manageable through the use of parallel computer architecture. The publically available code METIS

[17] is used to decompose the unstructured mesh into a

predetermined number of subdomains so that it can take

advantage of computers with multiple processors. The

mesh partitioning code then load balances, minimizes the

communication between processors, and minimizes the

maximum number of subdomains adjacent to any other

subdomain. Each processor then operates on a subset

of the larger domain. It is necessary to communicate the

subdomain boundary information to the neighboring processor so that the numerical algorithm can perform computations that require data from its immediate neighbor.

The message passing libraries provided by the standardized package MPI [18] is used for point-to-point communication between processors. For unstructured grid

based solvers a list must be maintained that provides the

boundary cells so that the appropriate data may be sent

and received by a processor.

A windowing technique is used to pack points in regions of the computational domain where more resolution is desired. To a limited extent the smoothness of the

grid is sacrificed for an increase in resolution using this

technique. A grid generated by this method will be referred to as windowed Delaunay point insertion (WDPI).

The unstructured grid generator Gridgen [19] is also

used to generate a grid in the same domain for comparison purposes. The stretching techniques in this grid generator results in a much smoother grid than that provided

by the windowing technique.

4 Discretization Technique

The space-time conservation element and solution element method is a finite volume method where the elements include space and time coordinate dimensions.

The original two-dimensional CE/SE method was derived for triangular space-time grids. To date the grids

used in computing the aeroacoustics from jet flows using this method have been triangulated structured grids.

Here the method is applied to more general unstructured

grids generated by the Delaunay method described in the

A

B

C

A’

B’

C’

x

y

t

D

E

F

D’

E’

F’

O’

O

Figure 2: Conservation elements used for updating solution.

previous section. In this section a general review of the

CE/SE method is given as it is applied to jet noise calculations.

There are three conservation elements (CE’s)that are

associated with a given triangular space-time element,

Fig. 2. The three conservation elements are shared geometrically by the three adjacent triangular space-time

elements. The geometric center of the triangular faces

adjacent to face ∆ABC are shown as points D, E, and

F. Each conservation element has six faces that form

a quadrilateral cylinder in space-time. The bottom and

top faces are formed by connecting the centroids of the

triangular bottom and top faces (of the space-time triangular element) to the nodes that make up that face, respectively. For a given space-time triangular element,

this construction produces three quadrilateral faces on

each of two spatial planes (i.e., A, D, B, O). These are

nonoverlapping elements. For a given space-time triangular cylinder, the union of the three conservation elements forms a hexagonal cylinder of volume V (the

lower face of this hexagonal cylinder is composed of

nodal points A, D, B, E, C, F). The conservation law

Eq. (2) is applied to this hexagonal cylinder. The resulting solution is associated with the center of the hexagon

at the new time level (not necessarily coincident with

O0

).

The solution elements (SE) used here are the interfaces of the conservation elements that form the hexagonal cylinder, Fig. 2.

The integral equation, Eq. (2), will provide a solution

at the geometric center of the top hexagonal face. This

does not necessarily coincide with the triangle cell center

(the solution element) location, E. Discretization of the

integral conservation laws, Eq. (2) leads to

X

S(V )

H∗

m · ∆S = V (Qm)

n+1

j , (3)

NASA/TM—2002-211799 4
for m = 1, 2, 3, 4, where the space-time flux at the surface centers of S(V ), H∗

m = (F∗

m, G∗m, U∗m), can be obtained. The source term on the right-hand side of Eq. (3)

is computed at the center of the hexagon A0

D0B0E0C0F0

at the new time level.

To evaluate the resulting system of algebraic equations, Eq. (3), the values must be computed at the solution element faces (actually the face centers). These face

centered values are approximated by the linear Taylor series expansions. The unknowns in the integral equation,

Eq. (3), are located at the cell center of the hexagon associated with O0

. Expanding the integral equation for a

given space-time element results in 12 scalar equations

for the 12 unknowns at O0

. Each of the three conservation elements contributes four scalar equations. More

details of the solution method can be found in [14].

The simple non-reflecting boundary conditions described previously [20,21, 15] still work well with a Delaunay based unstructured grid. The weighted a − 

CE/SE scheme is used in this work. To suppress wiggles around shocks the flux derivatives use a weighted

average similar to that of van Albada [22]. Details are

provided in [14].

The source term evaluation was presented in a previous paper [15]. Briefly, the source term Q is a function

of the unknown U; therefore, a local Newton iterative

procedure is needed. The procedure takes about 2-3 iterations to converge.

A simplified implicit LES procedure similar to those

used in [23] is adopted here to account for the strong

momentum exchange in the shear layer. In the LES, a

simple Smagorinsky subgrid scale model is used for the

eddy viscosity:

μt = (Cs∆)2(2SijSij )

1/2,

where

Sij = 1

2 (

∂ui

∂xj

+

∂uj

∂xi

),

with ∆ = (∆x∆y)1/2 and Cs = 0.1. Then, μ + μt

replaces μ in the actual computation. In practice, the

LES model was only applied for radii less than 1.5D and

4D downstream of the nozzle exit plane.

5 Problem Description

The physical domain of interest for the screech problem is shown in Fig. 3. A slice of the axisymmetric domain shows the thin and thick lip nozzle geometries and

the near-field boundaries where boundary conditions are

specified in the flow code. The circular jet flow enters

through the nozzle from the left and expands into the stationary ambient atmosphere. The flow condition specified at the nozzle exit at the start of the computations is

choked.

recessed nozzle 

exit plane

0.2D thick lip

0.625D thick lip

axis of symmetry

nozzle wall

Inlet

BC

NRBC

Outflow

NRBC

0.5D

2D 6D

6D

Figure 3: Geometry of the computational domain.

Here we are interested in predicting the source of the

noise rather than its propagation into the far-field. The diameter, D, of the jet nozzle is chosen as the length scale.

The density, ρ0, speed of sound, a0, and temperature, T0,

in the ambient flow are taken as scales for the dependent

variables.

The Delaunay grid generator is used to cluster the triangular cells near the nozzle lip to provide good resolution for the shock cells and the acoustic feedback loop

path. Cell stretching is used for these computations. We

will compare flow solutions computed on a Delaunay

based unstructured grid, described in Section 3 of this

paper, with those previously computed on a triangulated

structured grid [15]. The triangulated structured mesh is

constructed by dividing each rectangular cell of the structured mesh into four triangles. An unstructured connectivity matrix is then produced. More specifics about the

numbers of cells and flow code parameters will be provided in the the results section of this paper.

To clearly display the upstream propagating screech

waves, the computational domain was extended 2D upstream of the nozzle exit. The full computational domain is a circular cylinder of 8D axial length and 6D

radius. At the nozzle exit, the inflow plane is recessed

by two cells so as not to numerically restrict or influence the feedback loop. The first nozzle geometry definition follows that given by Ponton et al. [8] where a

straight nozzle lip of 0.2D in thickness is adopted. The

second nozzle geometry has the same computational domain, however the nozzle lip thickness is increased to

0.625D. The computational domain is modified by extending the upstream, downstream, and radial boundaries

by 2D to determine if the boundary locations has an effect on the solutions. However, no discernible effect is

realized. Only one case was used to test the effect of the

extended computational domain.

NASA/TM—2002-211799 5
5.1 Initial Conditions

Initially, the flow in the computation domain is set to ambient flow conditions, i.e., (using nondimensional variables)

ρ0 = 1, p0 = 1

γ

, u0 = 0, v0 = 0.

5.2 Boundary Conditions

At the inlet boundary, the conservative flow variables and

their spatial derivatives are specified to be those of the

ambient flow, except at the nozzle exit, where an elevated

pressure is imposed, i.e., the jet is under-expanded, as in

the physical experiments. By using the ideal gas isentropic relations, it follows that the nondimensional flow

variables at the nozzle exit, with Me = 1, are given by

ρe = γ(γ + 1)pe

2Tr

,

pe = 1

γ

"

2 + (γ − 1)M2

j

γ + 1 # γ

γ−1

,

ue =

 2Tr

γ + 11/2

, ve = 0,

where Tr is the reservoir (plenum) temperature. We will

also follow the experimental cold-flow condition where

the reservoir temperature is set to the ambient temperature, i.e., Tr = 1.

At the symmetry axis, i.e., y = 0, a simple reflective

boundary condition is applied. At the top and outflow

boundaries, the Type I and Type II CE/SE non-reflecting

boundary conditions as described in the next subsection

are imposed, respectively. The no-slip boundary condition is applied on the nozzle walls.

5.3 Non-Reflecting Boundary Conditions

In the CE/SE scheme, non-reflecting boundary conditions (NRBC) are constructed to allow fluxes from the

interior domain of a boundary CE to smoothly exit the

domain. There are various implementations of the nonreflecting boundary condition and in general they have

proven to be well suited for aeroacoustic problems [15].

The following NRBCs are employed in this paper.

For a grid node (j, n) lying on the outer radius of the

domain, the non-reflective boundary condition (Type I)

requires that

(Ux)

n

j = (Uy)

n

j = 0,

while Un

j is kept fixed at its initial steady boundary

value. At the downstream boundary, where there are

substantial gradients in the radial direction, the nonreflective boundary condition (Type II) requires that

(Ux)

n

j = 0,

while Un

j and (Uy)n

j are now defined by simple extrapolation from the nearest interior node j0

, i.e.,

Un

j = Un−1/2

j0 (Uy)

n

j = (Uy)

n−1/2

j0 .

These are identical boundary condition specifications

used in [15] for the results computed on a triangulated

structured grid.

6 Results

The CE/SE method is applied to the underexpanded

axisymmetric circular jet, with two nozzle lip thicknesses. The thin lip nozzle has a wall thickness of 20%

of the nozzle jet exit diameter. This case is used to compare the results obtained on a Delaunay based unstructured grid with experimental data [8] as well as with results previously obtained using a triangulated structured

grid [15]. The flow solver for both grids is identical.

The flow data is compared at a jet Mach number, Mj ,

of 1.19. Screech frequency and sound pressure levels

over a range of jet Mach numbers are also compared for

this thin lip nozzle configuration. The thick lip nozzle

has a wall thickness of 62.5% of the nozzle jet exit diameter. This case is compared with expermental results

[8] over a range of jet Mach numbers. A large number of

times steps are required to achieve appropriate accuracy

in the Fourier analysis of the time series data. Typically

80K time steps are needed to remove the initial transients from the computational domain. Then, another

330K time steps are required to record the time series

data.

It is important to emphasize that no harmonic forcing

is imposed in the numerical simulation. The initial impact of the boundary condition at the nozzle exit stimulates the jet shear layer and triggers the feedback loop

that generates the screech waves.

In the following, the computed jet flow data and acoustic screech data are presented separately, and comparisons are made with available experimental results. The

thin lip nozzle is examined first and comparisons are

made with the experimental and the previously obtained

numerical results. The results of the thick lip nozzle are

then presented and compared with the experimental results.

6.1 Flow and Shock-Cell Structure for the Thin

Lip Nozzle

For the flow data, the CE/SE method is applied to the

thin lip nozzle case at a Mj of 1.19. The resulting shock

cell structure for the axisymmetric flow calculation after 200K time steps is shown as a Schlieren (densitygradient modulus) contour plot, Fig. 4. The experimental [4] shock cell structure is shown in Fig. 5, and the

numerical results show good agreement with the shockcell structure in the experiment. For example, the shock

NASA/TM—2002-211799 6
Figure 4: Instantaneous snapshot of shock cell structure

for thin lip nozzle, (numerical Schlieren) Mj = 1.19.

Figure 5: Experimental Schlieren plot showing shock

cell structure, Mj = 1.19.

cell width is about 0.8D in the streamwise direction. In

both plots, it is observed that the first two shock cells

appear to be sharp and clear since the shear-layer instability wave is too weak at these locations to significantly

affect the shock cell. However, it interacts strongly with

the shock cells once it has gained a sufficient amplitude

through its streamwise growth, which is evident by the

deformation of the third and fourth shock cells. The deformation and movement of these shock cells are also

seen in the experiments [4,5]. This is, as pointed out by

Seiner [2], the mechanism by which the acoustic waves

that produce the broadband shock-associated noise and

the screech tones are generated. Furthermore, the fifth

shock cell and those further downstream are so severely

deformed that they almost disappear.

As demonstrated in the previous numerical results

[15], the screech wave length of 1.6D is predicted in the

computations. This is in good agreement with the experimental results (e.g.[4]) and the theoretical predictions of

Seiner [2] and Tam [1].

The thin lip nozzle flow solution on the WDPI and

Gridgen based grids compare well with the solution obtained previously on the triangulated structured grid [15].

The triangulated structured grid used a grid consisting of

132K cells. The WDPI grid has 167K cells. The Gridgen has a 136K cell grid. Since the flow code is the

same for all of the grids presented here, there is no clear

advantage to using a triangulated structured grid. In fact,

a truly unstructured grid should be much more effective

when the CE/SE method is applied to complex geometries.

1.08 1.1 1.12 1.14 1.16 1.18 1.2

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Panda (A1 mode)

Panda (A2 mode)

Ponton et al. (A1 mode)

Ponton et al. (A2 mode)

CE/SE triangulated structured (A1 mode)

CE/SE triangulated structured (A2 mode)

CE/SE unstructured WDPI (A1 mode)

CE/SE unstructured WDPI (A2 mode)

CE/SE unstructured Gridgen (A1 mode)

CE/SE unstructured Gridgen (A2 mode)

Fr

e

q

u

e

n

cy

(Hz

)

Mj

Thin Lip Nozzle

Figure 6: Frequency of thin lip nozzle over a range of

Mj .

6.2 Screech Frequency and Sound Pressure

Level

Because the data is unsteady, many of the same techniques used to analyze experimental data are also used

to analyze the numerical results. At a point on the nozzle

exit lip (x = 2.0, y = 0.642) the time history of pressure

is stored over a large number of time steps. This data

is then post processed by Fast Fourier Transform (FFT)

techniques to obtain spectral information.

The resulting screech frequencies are plotted over a

range of jet Mach number in Fig. 6. Two experimental

data sets are provided by Panda et al. [6] and Ponton

et al. [8]. The nozzles in the experiments have the same

diameters and operating Mach number, however they differ slightly in shape and thickness. The geometry used in

the numerical predictions is that of Ponton et al. [8]. The

CE/SE method is used to compute a solution on the same

jet geometry but on three different grids. The first series

of solutions are computed on the triangulated structured

grid [15]. The second set of numerical data are computed

on a grid generated by WDPI. This screech frequency

compares quite well with that in [15]. They slightly underpredict the experimental data at the lower jet Mach

numbers but they do predict the same trend. At the higher

levels of Mj the screech frequencies compare well with

experiment.

The third set of solutions are computed on a grid generated by Gridgen [19]. The frequencies produced by the

solution on the Gridgen grid are lower than those produced on the the other two grids. At a low jet Mach number, Mj , of 1.09 there are no detectable acoustic waves.

NASA/TM—2002-211799 7
1.08 1.1 1.12 1.14 1.16 1.18 1.2

0

20

40

60

80

100

120

140

160

180

Thin Lip Nozzle

Mj

SP

L

(dB)

CE/SE triangulated structured (A1 mode)

Panda (A1 mode)

Panda (A2 mode)

Ponton et al. (A1 mode)

Ponton et al. (A2 mode)

CE/SE triangulated structured (A2 mode)

CE/SE unstructured WDPI (A1 mode)

CE/SE unstructured WDPI (A2 mode)

CE/SE unstructured Gridgen (A1 mode)

CE/SE unstructured Gridgen (A2 mode)

Figure 7: Sound pressure level of thin lip nozzle over a

range of Mj .

The computations at the other jet Mach numbers underpredict the A1 mode screech frequencies observed in the

experiment, but they do predict the same trend. The A2

mode screech frequencies give a close match to the experimental data.

The sound pressure level (SPL) is shown in Fig. 7. The

computed results are in general lower than the experimental data. They are closer to the experimental results

of Ponton et al. [8]. The solutions on the WDPI grid

compare well with the triangulated structured grid. The

solutions on the grid generated by Gridgen [19] underpredicts the SPL of the A2 mode significantly. This difference could be due to the lack of resolution in that grid

downstream of the nozzle.

In general it is felt that it is necessary to cluster cells

in critical flow areas. The grid generated by WDPI is

probably better in this case since points are forced into

the region of the flow where the interaction between the

shock cell and shear layer is strongest. A careful choice

of α and time step is necessary to obtain the appropriate

amount of dissipation.

6.3 Flow and Shock-Cell Structure for the

Thick Lip Nozzle

For the thick lip nozzle case with a Mj = 1.19, the

computed shock cell structure is shown in Fig. 8. This

is an instantaneous snapshot of isobars combined with

a numerical Schlieren. Unlike the thin lip nozzle case,

the shock cell location which interacts with the instability wave begins between the second and third shock cell.

This is much closer to the nozzle lip than is observed in

nozzle

shock cells

entrainment

vortex

Figure 8: Instantaneous snapshot of screech wave radiation for thick lip nozzle, (isobars and Schlieren) Mj =

1.19. Solution shows a strong subharmonic.

the thin lip nozzle case. This is probably due to a stronger

feedback mechanism due to the thick lip. Figure 8 is the

larger computational domain, alluded to earlier in this

paper, to test the effect of boundary location. The nozzle

exit location is not changed, so the pressure data is still

taken at x = 2.0, y = 0.642.

The thick lip case shows a strong fluid flow entrainment which produces a counter rotating vortex just upstream of the nozzle lip, Fig. 8. This entrainment vortex

is observed for over 500K time steps. The location of

the vortex practically does not move. The vortex passes

cleanly through and breaks the acoustic waves as they

move upstream.

A strong subharmonic wave is observed in this calculation. It propagates from a region in the flow where the

pairing of vortices occurs and out through the upstream

boundary. The subharmonic wave is also split by the entrainment vortex. Due to the two-dimensional axisymmetric aspect of the computation, the subharmonic wave

may be much stronger than the physical one.

The total number of grid cells for the WDPI grid is

184K cells. The grid is clustered in the region of the

nozzle lip using the windowing technique described earlier. The Gridgen grid contains 140K cells. This grid is

also clustered but with smoother results than that given

by windowing. Figure 8 has 250K cells with grid clustering in the vicinity of the nozzle lip.

NASA/TM—2002-211799 8
For this geometry an α of 0.5 is used for the WDPI

solution with a time step of 0.0005. Any larger time step

requires a correspondly larger α to the point where it affects the acoustic solution. The Gridgen based solution

still could be run with an α of 1.0 and a time step size of

0.002.

6.4 Screech Frequency and Sound Pressure

Level

The FFT is used to analyze the unsteady feedback loop

that is generated between the shock cell structure and

the nozzle lip. The pressure history is recorded over a

large number of time steps, Fig. 9. The pressure level

is recorded every twentieth time step after a time period

such that the start-up transients have washed out of the

computational domain. Figure 9 represents the time trace

of pressure (nondimensional) at the point on the nozzle

exit lip, x = 2.0, y = 0.642 for a Mj = 1.15 using the

WDPI grid. This signal is then analyzed by FFT. The

resulting SPL spectrum is shown in Fig. 10, where only

the frequency range of 0 to 20000 Hz is displayed. There

are two SPL spikes in the spectrum that correspond to estimated screech frequencies of 7195 Hz (SPL = 142dB,

A1 mode) and 9169 Hz (SPL = 136dB, A2 mode) with

a binwidth (or bandwidth) of 326 Hz. This indicates that

there is either two screech tone frequencies occurring at

the same time or that screech mode switching is present.

A strong subharmonic is also shown in this case.

The screech frequencies at the lower jet Mach numbers matched the results of Ponton et al. [8] very well,

Fig. 11. At higher jet Mach numbers, Mj = 1.15, 1.17,

and 1.19, the screech frequencies are overpredicted.

The sound pressure levels for the thick lip nozzle in the

experiment has a higher level for the A2 mode than for

the A1 mode, Fig. 12. This is not seen in the numerical

predictions.

A major advantage of the grid produced by Gridgen

is that it requires less input from the user than does the

WDPI grid. The grid resolution could be augmented considerably in the feedback region of the flow domain with

proportionately better results. The smoothness characteristics of the Gridgen grid may also play an important

role in getting a better solution.

The axisymmetric computation may be predicting a

much larger subharmonic frequency than would be seen

in an experiment or in a three-dimensional computation.

This may be due in part to the fact that the flow is being

artificially confined as axisymmetric.

7 Concluding Remarks

In this paper, the unstructured CE/SE Navier-Stokes

method is applied to underexpanded supersonic axisymmetric jets over a range of jet Mach numbers for thin lip

0.002 0.003 0.004 0.005 0.006 0.007

0.7

0.705

0.71

0.715

0.72

0.725

0.73

Time(sec)

P

Thick Lip Nozzle

Figure 9: Time history of unsteady nondimensional pressure at nozzle exit (x = 2.0, y = 0.642), Mj = 1.15.

0 5000 10000 15000 20000

90

100

110

120

130

140

150

Frequency(Hz)

SP

L

(dB)

Thick Lip Nozzle

Figure 10: Sound pressure level at nozzle exit (x =

2.0, y = 0.642), Mj = 1.15, 326 Hz binwidth. Note

the strong subharmonics to the left.

NASA/TM—2002-211799 9
1.08 1.1 1.12 1.14 1.16 1.18 1.2

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Fr

e

q

u

e

n

cy

(Hz

)

Mj

Ponton et al. (A2 mode)

Thick Lip Nozzle

Ponton et al. (A1 mode)

CE/SE unstructured WDPI (A1 mode)

CE/SE unstructured WDPI (A2 mode)

CE/SE unstructured Gridgen (A1 mode)

CE/SE unstructured Gridgen (A2 mode) α=1.0, ∆Τ=0.002

α=0.5, ∆Τ=0.0005

α=0.5, ∆Τ=0.0005

α=1.0, ∆Τ=0.002

Figure 11: Frequency of thick lip nozzle over a range of

Mj .

1.08 1.1 1.12 1.14 1.16 1.18 1.2

0

20

40

60

80

100

120

140

160

180

Thick Lip Nozzle

Mj

SP

L

(dB)

Ponton et al. (A1 mode)

Ponton et al. (A2 mode)

CE/SE unstructured WDPI (A2 mode)

CE/SE unstructured Gridgen (A2 mode)

CE/SE unstructured Gridgen (A1 mode)

CE/SE unstructured WDPI (A1 mode)

α=1.0, ∆Τ=0.002

α=0.5, ∆Τ=0.0005

α=0.5, ∆Τ=0.0005

α=1.0, ∆Τ=0.002

Figure 12: Sound pressure level of thick lip nozzle over

a range of Mj .

and thick lip nozzle geometries. The resulting screech

tones are analyzed.

Truly unstructured grids instead of triangulated structured grids are used to show the flexibility of the CE/SE

method and to test its crucial capability of using unstructured grids to capture the physics in aeroacoustic problems with a feedback loop. The shock cell structure and

many acoustic aspects of the flows compare well with the

available experimental data, [4-8] and previous numerical results, [15] including the variation of screech frequency with jet Mach number for the A1 and A2 modes.

The thick lip case shows a strong fluid flow entrainment which produced a counter rotating vortex just upstream of the nozzle lip. This vortex is not seen in the

thin lip case.

The difference in the total number of computational

cells used for a given domain by the solver with a triangulated structured grid and the truly unstructured grid is

not significant, so it is possible to consider aeroacoustic

problems with more complex geometries using truly unstructured grids. Great care must be exercised in generating an unstructured grid. There were several cases where

lack of appropriate grid resolution in critical regions of

the computational domain produced a flow solution but

no acoustics.

It is necessary to control the grid clustering to capture the aeroacoustics in these jet flows with a feedback

loop. Grid qualities like smoothness, size, and shape,

are important for the dissipative aspects of the numerical

method.

References

[1] Tam, C. K .W., “Supersonic Jet Noise,” Ann. Rev.

Fluid Mech. vol. 27, pp. 17-43 (1995).

[2] Seiner, J. M., “Advances in High Speed Jet Aeroacoustics,” AIAA Paper 84-2275 (1984).

[3] Tam, C. K. W., “Jet Noise Generated by Large Scale

Coherent Motion,” NASA RP-1258, pp. 311-390

(1991).

[4] Panda, J., “Shock Oscillation in Underexpanded

Screeching Jets,” J. Fluid Mech., vol. 363, pp. 173-

198 (1998).

[5] Panda, J., “An Experimental Investigation of

Screech Noise Generation,” J. Fluid Mech., vol.

378, pp. 71-96 (1999).

NASA/TM—2002-211799 10
[6] Panda, J., Raman, G. and Zaman, K. B. M. Q., “ Underexpanded Screeching Jets from Circular, Rectangular and Elliptic Nozzles,” AIAA paper 97-1623

(1997).

[7] Panda, J. and Seasholtz, R.G, “Measurement of

Shock Structure and Shock-Vortex Interaction in

Underexpanded Jets Using Rayleigh Scattering,”

Phys. Fluids, vol. 11, pp. 3761-3777 (1999).

[8] Ponton, M. K., Seiner, J. M. and Brown, M. C., “

Near Field Pressure Fluctuations in the Exit Plane

of a Choked Axisymmetric Nozzle,” NASA TM

113137 (1997).

[9] Shen, H. and Tam, C. K. W., “Numerical Simulation of the Generation of Axisymmetric Mode Jet

Screech Tones,” AIAA Paper 98-0283 (1998).

[10] Shen, H. and Tam, C. K. W., “ Three-Dimensional

Numerical Simulation of the Jet Screech Phenomenon,” AIAA Paper 2001-0820 (2001).

[11] Chang, S. C., “The Method of Space-Time Conservation Element and Solution Element—A New

Approach for Solving the Navier-Stokes and Euler

Equations,” Journal of Computational Physics. vol.

119, pp. 295-324 (1995).

[12] Chang, S. C., Yu, S. T., Himansu, A., Wang, X.

Y., Chow, C. Y. and Loh, C. Y., “The Method

of Space-Time Conservation Element and Solution

Element—A New Paradigm for Numerical Solution

of Conservation Laws,” Computational Fluid Dynamics Review, eds. M. M. Hafez and K. Oshima,

Wiley (1997).

[13] Chang, S.-C., Wang, X.-Y. and Chow, C.-Y., “The

Space-Time Conservation Element and Solution

Element Method—A New High Resolution and

Genuinely Multidimensional Paradigm for Solving

Conservation Laws,” J. Comp. Phys. vol. 159, pp.

89-136 (1999).

[14] Wang, X.-Y. and Chang S.-C., “ A 2-D Nonsplitting Unstructured Triangular Mesh Euler

Solver Based on the Space-Time Conservation Element and Solution Element Method,” C.F.D. J. vol.

8, pp309-325 (1999).

[15] Loh, C. Y., Hultgren, L. S., and Jorgenson, P. C. E.,

“Near Field Screech Noise Computation for an Underexpanded Supersonic Jet by the CE/SE Method,”

AIAA Paper 2001-2252 (2001).

[16] Holmes, D. G., and Snyder, D. D., “The Generation of Unstructured Triangular Meshes Using Delaunay Triangulation,” Numerical Grid Generation

in Computational Fluid Mechanics ’88, Pineridge

Press, Miami, 1988. pp 643-652.

[17] Karypis, G., and Kumar, V., “Multilevel k-way

Partitioning Scheme for Irregular Graphs,” University of Minnesota, Department of Computer Science/Army HPC Research Center., Technical Report 95-064.

[18] Message Passing Interface Forum, “MPI: A Message Passing Interface Standard,” International

Journal of Supercomputer Applications, Vol. 8, no.

3/4, pp. 165-414, 1994.

[19] Gridgen Version 13, User Manual, Pointwise, Inc.,

1998, Bedford, Texas.

[20] Loh, C. Y., Hultgren, L. S. and Chang S.-C.,

“Computing Waves in Compressible Flow Using

the Space-Time Conservation Element Solution Element Method,” AIAA J., Vol. 39, pp. 794-801

(2001).

[21] Loh, C. Y., Hultgren, L. S., Chang, S.-C. and Jorgenson, P. C. E., “Vortex Dynamics Simulation in

Aeroacoustics by the Space-Time Conservation Element Solution Element Method,” AIAA Paper 99-

0359 (1999).

[22] van Albada, G. D., van Leer, B., Roberts, W. W.,

“A Comparative Study of Computational Methods

in Cosmic Gas Dynamics,” Astronomy and Astrophysics., Vol. 108, pp. 76-84, 1982.

[23] Bogey, C., Bailly, C. and Juve, D., “Computation of  ́

the Sound Radiated by a 3-D Jet Using Large Eddy

Simulation,” AIAA Paper 2000-2009 (2000).

NASA/TM—2002-211799 11
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NASA TM—2002-211799

AIAA–2002–3889

E–13494

WU–708–90–43–00

17

Computing Axisymmetric Jet Screech Tones Using Unstructured Grids

Philip C.E. Jorgenson and Ching Y. Loh

Computational fluid dynamics; Nozzle geometry; Screech tones

Unclassified -Unlimited

Subject Category: 02 Distribution: Nonstandard

Prepared for the 38th Joint Propulsion Conference and Exhibit cosponsored by the AIAA, ASME, SAE, and ASEE,

Indianapolis, Indiana, July 7–10, 2002. Philip C.E. Jorgenson, NASA Glenn Research Center, and Ching Y. Loh,

Taitech, Inc., Brook Park, Ohio. Responsible person, Philip C.E. Jorgenson, organization code 5880, 216–433–5386.

The space-time conservation element and solution element (CE/SE) method is used to solve the conservation law form of

the compressible axisymmetric Navier-Stokes equations. The equations are time marched to predict the unsteady flow and

the near-field screech tone noise issuing from an underexpanded circular jet. The CE/SE method uses an unstructured grid

based data structure. The unstructured grids for these calculations are generated based on the method of Delaunay

triangulation. The purpose of this paper is to show that an acoustics solution with a feedback loop can be obtained using

truly unstructured grid technology. Numerical results are presented for two different nozzle geometries. The first is

considered to have a thin nozzle lip and the second has a thick nozzle lip. Comparisons with available experimental data

are shown for flows corresponding to several different jet Mach numbers. Generally good agreement is obtained in terms

of flow physics, screech tone frequency, and sound pressure level.