National Aeronautics and Space AdministrationLangley Research Center • Hampton, Virginia 23681-0001

NASA Technical Paper 3596

Numerical Simulation of Jet AerodynamicsUsing the Three-Dimensional Navier-StokesCode PAB3DS. Paul PaoLangley Research Center • Hampton, Virginia

Khaled S. Abdol-HamidAnalytical Services & Materials, Inc. • Hampton, Virginia

September 1996

Printed copies available from the following:

NASA Center for AeroSpace Information National Technical Information Service (NTIS)800 Elkridge Landing Road 5285 Port Royal RoadLinthicum Heights, MD 21090-2934 Springfield, VA 22161-2171(301) 621-0390 (703) 487-4650

Available electronically at the following URL address: http://techreports.larc.nasa.gov/ltrs/ltrs.html

iii

Contents

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Jones-Launder Two-Equation Turbulence Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Grid Adaption Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Solver Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Thek-ε Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Multiblock Structure and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Adaptive Grid Algorithm in the PAB3D Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

On-Design Circular Jet Plumes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Off-Design Jets Containing Weak Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Off-Design Supersonic Jets Containing a Mach Disk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Adaptive Grid Computations of Nonaxisymmetric Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Introduction

Knowledge of jet mixing aerodynamics is vital toseveral areas of commercial and military aircraft design,such as jet propulsion efficiency, propulsion integration,aeroacoustics, and jet interference with aircraft structure.Initial jet flow conditions are determined by nozzle exitpressure, temperature, Mach number, and nozzle geome-try. Once the flow leaves the jet nozzle, the jet flowbecomes a free shear layer. The action of turbulencedominates flow developments farther downstream. Assuch, jet flow properties are difficult to measure or pre-dict analytically.

When Prandtl introduced his mixing length hypo-thesis for turbulent flows, a brief analysis of a fullymixed jet was given as an example. Early analyses of jetmixing behavior were based mainly on this mixinglength hypothesis and one-dimensional momentum the-ory. (See refs. 1–4.) Mean flow properties derived fromthese analytical models compare well with experimentalmeasurements of jets at low subsonic speeds. However,data from jet flow measurements in the high subsonicand supersonic speed ranges (ref. 5) indicate signifi-cantdeparture from the results obtained by using one-dimensional momentum theory.

Jet flow contains a rich combination of flow inter-actions and flow physics. These combinations includeturbulent mixing and compressibility effects such asisentropic expansion and shock. Other factors mayinclude chemical reactions or shear layer instability.

Subsonic jet flow features are relatively simple. Themain variable in the flow is shear layer developmentalong the streamwise direction. The static pressure valueis almost constant with the ambient pressure. In theabsence of a pressure gradient, no significant inviscidflow feature will appear in a subsonic jet. According toreported experimental measurements, all turbulent axi-symmetric subsonic jets below Mach 0.6 are similar if

the flow variables are normalized by jet density and noz-zle exit velocity.

On the other hand, supersonic jet flow features canbe very complex. Because of the supersonic nozzle exitMach number, jet exit pressure can differ from ambientpressure. This pressure difference between the jet and theambient fluid must be resolved locally either across anoblique shock, by a prominent streamline curvature at thejet boundary, or by a Mach disk inside the jet. In addi-tion, shocks formed near the nozzle exit may reflectrepeatedly at the sonic line in the shear layer. Althoughthe convected turbulence interacts with shocks in the jet,the position of the reflected shock depends on the loca-tion of the sonic line in the turbulent shear layer. Suchinterdependence of flow interactions can become verycomplex.

Earlier jet flow analysis codes, with or withoutchemical interactions included, were formulated withsimplified assumptions of the Navier-Stokes equationsand the turbulence model to provide the best jet flowsimulation within modest limits of computing resourcesavailable during this time. Analytical methods and simu-lation codes developed by this approach have been suc-cessfully applied to problems in air-breathing enginedevelopment, acoustics, and rocket propulsion. (Seerefs.6–12.) However, there are some drawbacks to thisapproach. First, simplified assumptions are often difficultto justify. Second, application of the simplified formula-tions is limited to jet flow simulation. The formulationsare difficult to integrate with computational codes for air-frame aerodynamics when performing propulsion air-frame integration analysis. It is preferable in such casesto perform the analysis with the three-dimensionalNavier-Stokes equations without empirical assumptionsfor jet flow alone.

For general use of jet flow simulation, some basicrequirements must be met. The Navier-Stokes codeshould be upwind biased to capture internal shocks and

Abstract

This report presents a unified method for subsonic and supersonic jet analysisusing the three-dimensional Navier-Stokes code PAB3D. The Navier-Stokes code wasused to obtain solutions for axisymmetric jets with on-design operating conditions atMach numbers ranging from 0.6 to 3.0, supersonic jets containing weak shocks andMach disks, and supersonic jets with nonaxisymmetric nozzle exit geometries. Thisreport discusses computational methods, code implementation, computed results, andcomparisons with available experimental data. Very good agreement is shownbetween the numerical solutions and available experimental data over a wide rangeof operating conditions. The Navier-Stokes method using the standard Jones-Laundertwo-equation k-ε turbulence model can accurately predict jet flow, and such predic-tions are made without any modification to the published constants for the turbulencemodel.

2

other jet flow discontinuities. The code should also befully three-dimensional in space because the relationsbetween turbulent kinetic energy and Reynolds stressesare basically three-dimensional. The turbulence modelshould be capable of providing a time scale and aconsistent description of the production and transportproperties of turbulent kinetic energy. Therefore, a two-equation turbulence closure model is required.

Many upwind-biased three-dimensional Navier-Stokes codes are available that meet the jet flow simula-tion requirement. However, the availability of codes witha robust two-equation turbulence mode in this class islimited. In this report, the PAB3D code is used for all jetflow computations. The purpose of this report is to showthe feasiblity of establishing a unified method for sub-sonic and supersonic jet analysis with a general purposethree-dimensional Navier-Stokes code.

The PAB3D code is developed to obtain numericalsolutions to the Reynolds averaged Navier-Stokes equa-tions in three-dimensional spatial domain. The mainsolver algorithm is the upwind Roe scheme, for whichthe numerical dissipation is small. The Jones-Launder(ref. 13) two-equationk-ε turbulence closure model isused to compute the turbulent stresses in the flow. Thisapproach is chosen for jet flow analysis because it is con-sistent in tracking production and transport properties ofturbulence kinetic energy and dissipation scale length inthe shear flow. In the Jones-Launderk-ε turbulencemodel, several empirical constants are required. Only thepublished values for these constants are implemented inthe PAB3D code. These values are fixed for all computa-tional applications of the PAB3D code.

This report describes the mathematical formulationof governing equations, the turbulence model, and theadaptive grid generation algorithm, along with thenumerical implementation of each. The adaptive gridgeneration algorithm is designed especially for nonaxi-symmetric jet computations.

Several categories of jet flow computations aredescribed separately in the section “Results and Discus-sion.” The first category describes axisymmetric jetsoperating at on-design exit conditions so that the jet exitpressure matches the ambient static pressure. Results areobtained for jet exit Mach numbers ranging from 0.6to 3.0. Computed velocity and turbulence intensity distri-butions in the jet are compared with experimental data.The second category presents results for supersonic jetswith internal weak shocks. The discussion includes com-puted results for jet exit pressures above and below theambient static pressure to show characteristics of theshock-containing supersonic jets. Computed results arecompared with available experimental data. The thirdcategory of computed cases is axisymmetric supersonic

jets with embedded Mach disks. Flow conditions forthese jets are the result of a supersonic jet nozzle operat-ing at pressures far from nozzle design value. One partic-ular case details a Mach 1.5 nozzle operating at a nozzlepressure ratio 3.15 times greater than the design value forthis nozzle. The last category includes supersonic jetswith nonaxisymmetric initial cross sections. Shear layerdevelopment of these jets is very different from a typicalaxisymmetric jet because of added geometrical degreesof freedom. The development of elliptic, rectangular, andsquare jets operating at the same exit pressure and Machnumber is compared.

Symbols

a local speed of sound

e internal energy per unit mass

C1, C2, Cµ constants in two-equation turbulencemodel

F, G, H inviscid flux components in Navier-Stokesequations

Fv, Gv, Hv viscous flux components in Navier-Stokesequations

total flux vectors (inviscid plus viscous) inNavier-Stokes equations

f1, f2, fn monitoring function for grid adaptation

i, j, k grid index inξ-, η-, ζ-directions

J Jacobian of coordinate transformation

k turbulent kinetic energy

Lc jet potential core length

l1 shock cell length measured from nozzleexit to first shock intersection at jetcenterline

M Mach number

NPR nozzle pressure ratio,

n distance in a direction normal to a solidwall

production term for turbulent kineticenergy

p pressure

pe jet exit static pressure

po ambient static pressure

pt jet total pressure

conservative variable vector in Navier-Stokes equations

R jet exit radius or area equivalent radius

F G H, ,

pt

po------

P

Q

3

S source function in Navier-Stokesequations

T temperature

t time

Ue jet exit velocity

Uc jet centerline velocity

u, v, w velocity components inx-, y-, andz-directions

components of turbulence velocityfluctuation

urms root-mean-square value of turbulence

velocity fluctuation,

ut turbulence velocity fluctuation

W shock cell length measured from nozzle lipto position of first shock reflection in shearlayer

x, y, z spatial coordinates

Γ compressibility correction factor

δij Kronecker delta

ε turbulent kinetic energy dissipation rate

µ dynamic coefficient of viscosity

ν kinematic coefficient of viscosity

ξ, η, ζ generalized coordinate as function ofx, y,andz

ρ density

σε, σk constants in two-equation turbulencemodel

τ shear and normal stress components

Subscripts and superscripts:

e jet exit condition

k turbulent kinetic energy

L laminar quantities

o free-stream condition

T turbulence related quantities

v viscous component of flux vectors

ε turbulent energy dissipation

Governing Equations

The governing equations of the Reynolds averagedNavier-Stokes formulation include the conservationequations for mass, momentum, and energy and theequation of state for gas. In this study, the perfect gas lawis chosen to represent the properties of air. For a turbu-lent flow, the Reynolds stresses can be represented by

turbulence closure models for practical applications.Because one of the dominant factors governing jetdynamics is turbulent shear layer mixing, the turbulenceclosure model is essential for realistic jet flow simulationwhen using Navier-Stokes methods. The Jones-Launder(ref. 13) two-equationk-ε turbulence model is used inthis study. The Navier-Stokes equations and the mathe-matical representation of the two-equation turbulencemodel are described briefly in separate subsections ofthis report.

For computation of nonaxisymmetric jet flows, aspecial requirement in grid generation arises. High griddensity is required for regions occupied by the shearlayer and the embedded shock so that high gradients ofmean flow and turbulence quantities can be accuratelyrepresented in the numerical solution. However, the posi-tion of the shear layer and the shock positions of a non-axisymmetric jet are not known in advance. This specialrequirement can be met by using an adaptive grid. Theanalytical basis for an adaptive grid is described in thereport section “Grid Adaptation Strategy” followingdiscussions of Navier-Stokes equations and the Jones-Launderk-ε turbulence model.

Navier-Stokes Equations

The mass, momentum, and energy conservationequations of the Reynolds averaged Navier-Stokes equa-tions can be written in terms of generalized coordinatesand in a conservative form as follows:

(1)

wheret, ξ, η, andζ are the independent variable for timeand the general curvilinear coordinates in the griddomain, is the conservative flow variable vector (ρ,ρu, ρv, ρw, ρe) in generalized coordinates, arethe total generalized flux vectors including inviscid andviscous components, and the source termS is zero for theNavier-Stokes equations in this form. This equation isintroduced here mainly to indicate the relationshipbetween the basic Navier-Stokes equations and the two-equation turbulence model equations. Reference 14 pre-sents details of the Navier-Stokes equations as applied inthe PAB3D code. A simplified form of the Navier-Stokesequations which omits all the streamwise derivatives ofthe Reynolds stresses is used in the PAB3D code. Omis-sion of these terms is done for computational economyand does not introduce significant computation error.The remaining cross stream derivatives are numericallyimplemented at several levels in PAB3D. The thin layerNavier-Stokes approximation is one option for the user.This study uses the option of uncoupled Reynolds stressderivatives in two directions.

ui′

ui′ ui′

∂Q∂t------- ∂F

∂ξ------- ∂G

∂ζ------- ∂H

∂η-------+ + + S=

QF G H, ,

4

Jones-Launder Two-Equation Turbulence Model

The Jones-Launder formulation for the two-equationturbulence model uses the turbulent kinetic energyk andthe dissipation rateε as the principal variables. Thisstudy uses the expanded three-dimensional form (ref. 15)of the original Jones-Launder model. This modified for-mulation is fully three-dimensional, and the governingequations are written in a conservative form as general-ized coordinates. The governing equations can be cast inthe same form as the Navier-Stokes equations, where

Here, is the full three-dimensional production termdefined as

or is expanded to

where

In the definitions ofSε andSk, the termsLε andLk arenear-wall effects which are not important to free jet cal-culations, and denotes derivatives in a directionnormal to the solid wall boundary. However, these termsare included in the PAB3D code. The functionΓ is thecompressibility correction function. Several correctionshave been developed by different authors. Among thewidely used compressibility correction functionsΓ arethose proposed by Sarkar et al. (ref. 16) and by Wilcox(ref. 17).

Sarkar model (ref. 16):

(2)

Wilcox model (ref. 17):

(3)

whereH is the Heaviside function, isthe local turbulence Mach number,a is the local speed ofsound, and is a cutoff turbulence Mach number.The commonly accepted value is used inthe PAB3D code. The compressibility correction factoris required when the local flow Mach number is greaterthan 1.0. In the Sarkar model, the compressibility correc-tion is activated everywhere in the flow field whenapplied for a given computation. The Wilcox model is amodification of the Sarkar model so thatΓ is nonzeroonly for values of greater than This condi-tion implies that compressibility correction is activatedfor local flow Mach numbers near or greater than 1, withno correction otherwise.

Grid Adaption Strategy

For an accurate representation of the flow field, suf-ficient grid density must be provided in the mixingregion. Unlike an axisymmetric jet, the nonaxisymmetricjet is not self-similar and can evolve in dramatically dif-ferent fashion in different sectors of the jet cross section.Because the position of the shear layer is not known inadvance, a large number of predetermined grid points

Q ρερk

= F ρuερuk

= G ρvερvk

= H ρwερwk

=

Fv

µε∂ε∂x------

µk∂k∂x------

= Gv

µε∂ε∂y-----

µk∂k∂y------

= Hv

µε∂ε∂z-----

µk∂k∂z------

=

P τxxT ∂u

∂x------ τyy

T ∂v∂y----- τzz

T ∂w∂z------- τxy

T ∂u∂y------ ∂v

∂x------+

+ + +=

τyzT ∂v

∂z----- ∂w

∂y-------+

τzxT ∂w

∂x------- ∂u

∂z------+

+ +

SSε

Sk

=

Sε C1Pεk-- C2ρε

k-- ε 2ν ∂ k

∂n----------

2

– Lε+–=

Sk P ρ 1 Γ+( )ε Lk+–=

P

P τxxT ∂u

∂x------ τyy

T ∂v∂y----- τzz

T ∂w∂z------- τxy

T ∂u∂y------ ∂v

∂x------+

+ + +=

τyzT ∂v

∂z----- ∂w

∂y-------+

τzxT ∂w

∂x------- ∂u

∂z------+

+ +

P µT ∂u∂y------ ∂v

∂x------+

2 ∂v

∂z----- ∂w

∂y-------+

2 ∂w

∂x------- ∂u

∂z------+

2

+ +

=

2∂u∂x------

2 ∂v

∂y-----

2 ∂w

∂z-------

2

+ +23--- ∂u

∂x------ ∂v

∂y----- ∂w

∂z-------+ +

2–

+

23---ρk

∂u∂x------ ∂v

∂y----- ∂w

∂z-------+ +

–

µT Cµρk2

ε-----= µε µL µT

σε------+= µk µL µT

σk------+=

Cµ 0.09=

C1 1.44= C2 1.92= σε 1.3= σk 1.0=

τi jT µT

∂ui

∂xj--------

∂uj

∂xi--------+

23---

∂uk

∂xk--------δi j–

23---ρkδi j–=

∂/∂n

Γ MT2=

Γ MT2 MT o,

2–( )H MT MT o,–( )=

MT k ρ⁄ a⁄=

MT o,MT o, 0.25=

MT MT o, .

5

are required to provide high density coverage of thethree-dimensional space if a fixed grid is used for thecomputations. An alternative is to provide high grid den-sity in the appropriate locations by using an adaptive gridstrategy.

For jet plume analysis, high grid density is requiredin high velocity gradient regions in the shear layer andin high pressure gradient regions near shock fronts. Thenumber of grid points in each direction of a structuredgrid is fixed. Local grid density can be varied by redis-tributing the available grid points in the computationaldomain to match selected flow characteristics such aspressure and velocity gradients. Various methods can beused to redistribute grid density according to given multi-ple functional requirements. In this study, the equi-distribution principle and the alternate direction gridadaption method published by Eiseman et al. (refs.15,18, and 19) are chosen as the basis for adaptive gridimplementation in the PAB3D code.

In the equidistribution approach, a monitoring func-tion which governs grid density over the computationdomain is defined. The monitoring function can be geo-metrically represented as a hypersurface in a space withdimensions that are one higher than the spatial dimen-sions of the computational domain. The process of gridadaption begins by constructing a uniform mesh overthemonitoring surface. For a one-dimensional case, themonitoring surface is a curve over the linear spatialdomain. Equidistribution is simply a uniform distributionof points at equal arc distances on the entire length of themonitoring curve. When this distribution of points is pro-jected back to the one-dimensional baseline in the physi-cal domain as adapted grid points, the grid density isproportional to the gradient of the monitoring function.For a two-dimensional grid domain, the monitoring sur-face is a curved surface in three dimensions over the two-dimensional physical space. The equidistribution processinvolves constructing a mesh system over the curvedmonitoring surface so that all the grid cells encloseapproximately equal surface areas. Once the equidistri-bution construction is complete, the mesh pattern on themonitoring surface is projected onto the original physicaldomain. Similar to the one-dimensional case, high griddensities are again obtained in regions where the moni-toring function has high gradients.

If the monitoring surface is assumed to representmountains and valleys in a landscape, the previouslymentioned process is similar to making a contour map ofthis landscape. Steep slopes in the landscape are natu-rally represented by tightly packed contour lines on themap, which is a horizontal projection of the originalthree-dimensional surface. Visualization of the monitor-ing surface can be difficult for a three-dimensional spa-

tial domain. However, the algebra and the geometry forthe adaption process remain unchanged. In addition toequidistribution of arc lengths or areas on the hyper-surface, normal curvature of the monitoring surface canalso be used as a weighting function to provide additionalmesh density control.

The alternative direction adaption proposed by Eise-man simplifies the equidistribution procedure by per-forming arc length equidistribution on the monitoringsurface along each family of coordinate lines. If cellskewness remains within reasonable limits, equidistribu-tion of all sides of the grid cells will also distribute thecell area or volume into approximately equal sizes. How-ever, orthogonality is not enforced in this procedure. Thedegree of grid concentration for given values of the gra-dients of a monitoring function is controlled by a propor-tional constant. Since orthogonality is not enforced in thealternate direction equidistribution procedure, excessivecell skewness and cell collapse can occur if the propor-tional constant is given too high a value.

For grid adaption to more than one flow quantity,multiple monitoring functions can be used. A simpleapproach is to combine all monitoring functions as asingle weighted sum. The approach of Eisman andBrockelie (ref. 20) treats each monitoring function as anadditional geometrical dimension (which is orthogonal toall previous dimensions). In this approach, grid featuresrepresented in each monitoring function remain distinct.The differential arc length element can be given as

(4)

whereds0 is the arc length in the physical or grid domain,ds1 is the arc length on the monitoring surface, graddenotes a component of the gradient in the tangentialdirection of the coordinate curves, andw(s0) is anoptional weighting function which is proportional to thecurvature of the monitoring surface.

A modified approach called the sequential adaptionmethod is used in this paper. Assuming there areN monitoring functions, the monitoring functions areapplied sequentially. After each step of adaption, themesh on the previous monitoring surface is treated as a“stretched” uniform mesh to support the next monitoringfunction. The arc length increments on each of the moni-toring surfaces can be written as

(5)

Once the adaption process is completed over the lastmonitoring function, the mesh coordinates are projectedsequentially back to all previous base surfaces. The

ds1 1 w s0( )+ 1 grad f 1( ) 2 grad f 2( ) 2 …+ + + ds0=

dsn 1 wn sn 1–( )+ 1 grad f n( ) 2+ dsn 1–= n 1 2 … N, , ,=( )

6

last one is the physical space where an adapted grid isestablished.

Where the curvature weighting functionsw1, …, wnare zero, the sequential adaption method and the vectormonitoring function method are mathematically thesame. Only their geometrical interpretation and computa-tional implementation are different.

Computational Methods

The simplified Reynolds averaged Navier-Stokesequations and the Jones-Launderk-ε turbulence modelare implemented in the PAB3D code for general fluiddynamics analysis in three-dimensional space. Distinc-tive features of this code include provisions to accept amultiblock grid with patched interface, compact memoryrequirement, and solver options. In particular, a space-marching solver with adaptive grid capability is availablefor jet flow computation when flow conditions meet thespace-marching scheme criterion. For such cases, thespace-marching solution accuracy is indistinguishablefrom accuracy obtained by using the time-marchingsolver algorithm. The space-marching procedure cancomplete a converged solution in approximately one-twentieth the computer time required by the time-marching solver to obtain a solution for the same flowconditions.

Solver Algorithm

Three numerical schemes have been implemented inthe PAB3D code as solvers for the Navier-Stokes equa-tions: the Van Leer flux-vector splitting scheme, the Roeflux-difference splitting scheme, and a space-marchingscheme that is a modified version of the basic Roescheme. These schemes are implicit, upwind, and con-structed by using the finite volume approach. Only the

inviscid portion of the flux vectors , , and is sub-jected to the splitting and upwind procedures. The diffu-sion terms of the Navier-Stokes equations are centrallydifferenced. Reference 14 details mathematical descrip-tion of these schemes.

The flux-vector scheme and the flux-difference split-ting scheme are used in all three computational direc-tions. An updated solution at each iteration is obtained byusing an implicit procedure in the meshη,ζ-planes atconstant values ofξ and a relaxation procedure in theξ-direction consisting of a forward and a backwardsweep. This particular implementation strategy has anadvantage for computational efficiency. Since the met-rics for the implicit procedure are required for only up tothree planes, the metric constants are recomputed oneplane at a time at the advancing front of the prevalentsweep direction instead of being stored for the entire grid

domain. Moderate or large mesh sizes require an averageof only 22 words of memory for each gird point. Thishighly efficient use of computer memory is obtained at amodest cost of approximately a 3-percent increase incomputer time per iteration. The overall computer timerequirement per iteration per grid point is similar to othercodes of this type.

For time-marching solutions using the Van Leer orthe Roe scheme, each iteration count consists of either aforward or a backward sweep in theξ-direction with onestep of implicit update of the solution in each of the crossplanes. The inviscid terms in the Navier-Stokes equa-tions in the Roe scheme are cast in the form of anapproximate Riemann problem. The interface flux in thestreamwise direction is determined by separate terms,depending on the quantities on the left (upstream) and theright (downstream) sides of the interface. For a fullysupersonic flow, the information can travel only in theflow direction. Such information is carried by the termsrepresenting upstream dependence. The terms whichcarry the downstream dependence can be ignored with-out introducing significant flow solution error. This stateof information transfer in the Roe scheme solver is truefor a broad category of subsonic and supersonic jet flowswhere the streamwise pressure gradient is small. Byignoring the downstream dependence terms in the Roescheme, the solver becomes the space-marching scheme.Under this modified scheme, a solution is obtained planeby plane from upstream to downstream by carrying out asufficient number of implicit iterations in each planeuntil the convergence criterion is met. A solution for theentire computational domain is established in a singleforward sweep.

The k-ε Turbulence Model

The governing equations of the Jones-Launder for-mulation of thek-ε turbulence model are written as a pairof coupled transport equations in conservative form. Inprinciple, this pair can be implemented together with theNavier-Stokes equations as either a set of seven coupledequations or a separate pair uncoupled from the Navier-Stokes equations. The fully coupled approach wouldcause serious problems such as a significant increase incomputational effort and working space in the computermemory and numerical stiffness of the coupled set ofseven equations. In the PAB3D code, solutions of thekand ε equations are decoupled from the Navier-Stokesequations and from each other. Time step differencesremain in this uncoupled system of flow and turbulenceequations. However, the problem is circumvented bysolving thesek and ε equations with a CFL (Courant,Fredricks, and Levy) number that is smaller by at leasta factor of 2. The potential difference in timewise devel-opment of the flow variables and turbulence variables

F G H

7

has not led to any noticeable effect in either the overallconvergence rate or the quality of the solutions.

Multiblock Structure and Boundary Conditions

The PAB3D code is designed to handle complexconfigurations by using several types of multizone,multiblock grid topologies. A restricted option, which issuitable for jet plume calculation with the space-marching schemes, calls for streamwise division of thecomputational domain into zones in theξ-direction. Thegrid space in each zone can further be divided into blocksin theη- andζ-directions. Otherwise, the code supports ageneral multiblock scheme where the computationaldomain can be divided into any collection of blocks.Number of blocks, block size, and parametric orientationare not restricted. The concept of zones is not relevant inthis general scheme. General patched block interfacecommunication is allowed. The only restriction for thisgeneral multiblock connectivity scheme is that the con-nected block interfaces are contiguous. A grid partitionfeature is available in the PAB3D code for the conve-nience of turbulence modeling. If different viscous stressmodels are employed within a block, theξ-direction ofthe block can be partitioned by choosing a starting indexfor each viscous stress model.

The boundary conditions often used for jet computa-tions include inflow, outflow, free stream, solid walls,and geometrical symmetry. Three types of inflow bound-ary conditions are provided: Riemannian characteristics,fixed total temperature and total pressure, and a com-pletely fixed set of five flow parameters. Two outflowboundary conditions are needed: constant pressure forsubsonic flows and extrapolation for supersonic flows.The Riemannian characteristics boundary condition isused at free stream boundaries. On a solid boundary,either a no-slip or an inviscid-slip boundary conditioncan be specified. Finally, the symmetry boundary condi-tions include mirror imaging across a plane in any orien-tation and polar symmetry around an axis in thestreamwise direction.

In addition, a universal high-order symmetry bound-ary condition for Navier-Stokes code applications isdeveloped in the course of this jet plume study. This uni-versal symmetry boundary condition provides a simplemethod for the user to specify a symmetry boundary con-dition at a grid plane not aligned to a surface with a con-stant physical coordinate value. Reference 14 details thisboundary condition.

Adaptive Grid Algorithm in the PAB3D Code

For nonaxisymmetric jet calculations in this report,one quarter of the jet cross section is represented in thegrid domain. Hence, flow symmetry across both the hori-

zontal and the vertical axis is assumed. In each plane, thegrid is divided into two parts: a high density grid in thenear field of the jet flow and a low density grid for the farfield. Only the high density grid near the jet flow isadapted to the flow solution. A Cartesian topology ischosen for the initial unadapted high density grid. Gridadaption proceeds from plane to plane in the streamwisedirection. Two monitoring functions are used for adapt-ing the grid to the velocity and pressure gradients of theflow solution. The monitoring functions are normalizedso that one constant is used for each function to controlthe intensity of adaption. (See ref.21.) A single grid wasused in reference 21 to cover both the near field and thefar field. A third monitoring function was employed toredistribute a uniform Cartesian grid to form a dense gridzone in the near field and a sparse grid distribution in thefar field.

The adaptive grid procedure is coupled to the space-marching solver in PAB3D. Grid indicesi, j, andk areassigned to theξ, η, andζ coordinates of the generalizedcoordinates. In the space-marching algorithm, a numeri-cally converged solution is computed at eachj, k-planethrough multiple iterations. This solution is then coupledto the next plane downstream, and the computationalprocess is repeated.

For jet flow computations considered in this report,initial conditions for the first plane representing the flowcondition at the nozzle exit are prescribed according toknown nozzle operating conditions. The initial grid at thefirst plane is generated externally according to the initialflow conditions by using the same grid adaption proce-dure as the one implemented in the PAB3D code. Oncethe solution process is started, grid adaptions for subse-quent grid planes are computed within the code. A grid iscreated for the (i + 1)-plane by adapting the grid to thenumerically converged solution in thei-plane. The adap-tive grid procedure is implemented as efficientlyas possible to match the high efficiency of the space-marching solver. The computational efficiency of thismultiple function grid adaption procedure is analyzedduring this study. The time taken for grid adaption isapproximately 4 percent of the total time required for theflow solver. Only 1 cycle of adaption is used for eachplane. The flow solution at each plane normally takesapproximately 20 to 30 iterations before nominal conver-gence criterion is met.

Results and Discussion

Results of jet flow computations are presentedin several groups: jets operating at on-design conditions,supersonic jets containing weak shocks, supersonic jetscontaining strong shocks, and nonaxisymmetric super-sonic jets. For these groups of computation, the initial

8

flow condition for the jet is specified at the inflowboundary of the computational grid. The internal flowupstream of the jet nozzle is not modeled. Figure 1 showsa sketch of the jet flow configuration and a typical com-putational grid for on-design axisymmetrical jets.

On-design operation of a jet is defined as the condi-tion for which the jet exit static pressure is identical tothe ambient static pressure. For a subsonic jet, the exitstatic pressure is naturally adjusted to the ambient staticpressure. A supersonic jet flow is established by using aconvergent-divergent nozzle designed for a fixed Machnumber. The on-design nozzle pressure ratio (NPR),defined aspt/po, is a fixed value for each given Machnumber. For jets operating at on-design NPR, pressuregradients are very small in the entire flow domain.Shocks in the flow domain are typically weak or absentin on-design supersonic jets. The principal driving mech-anism for on-design jet plume development is turbulentmixing in the jet shear layer. This report computes on-design jet flows by using the space-marching scheme inPAB3D.

At off-design operating conditions, the initial jetflow condition is either overexpanded or underexpanded.Shock waves will appear in the jet flow. For a given noz-zle geometry, the exit jet Mach number is fixed regard-less of NPR (assuming that the NPR is high enough tofully establish supersonic flow at the nozzle exit). AtNPR values sufficiently close to the design point, onlyweak shocks are present in the jet flow and jet flowdevelopment can be computed using the space-marchingsolver in PAB3D.

Once strong shocks in the form of Mach disks appearin the jet plume, the flow downstream of the shockbecomes subsonic. Furthermore, high static pressureimmediately downstream of the shock leads to rapidacceleration and expansion of the subsonic flow. Hence,a strong pressure gradient exists. The time-marchingsolvers in the PAB3D code must then be used becauseconditions permitting the use of the space-marchingscheme are violated in this region. However, the space-marching method alone cannot detect the occurrence of aMach disk. The decision to use either the space-marchingor time-marching options in the PAB3D code for a par-ticular case must be guided by tabulated experimentaldata or by theoretical estimates. Reference 5 gives anexcellent reference for Mach disk formation in axisym-metric jet plumes.

Jet exhaust nozzles of practical interest in propulsionsystems may have a nonaxisymmetric exit cross section.The dynamic characteristics of nonaxisymmetric jets aresignificantly more complex than those for axisymmetricjets because of the added degrees of freedom in jet flowgeometry. The adaptive grid option in the PAB3D code

is used to provide appropriate grid density distributionfor the shear layer and shock regions in the jet flow. Thefollowing subsections give detailed discussions of resultsin each group of jet flow computations.

On-Design Circular Jet Plumes

Flow solutions for on-design circular jets within jetexit Mach numbers ranging from 0.6 to 3.0 are computedusing the space-marching method in the PAB3D code. Inthis series of jet flow simulations, the initial flow condi-tion for the jet plume is specified at the inflow boundaryof the computational domain. A small velocity compo-nent in the ambient air parallel to the jet flow direction isrequired in the PAB3D code for maintaining numericalstability of the space-marching scheme. A freestreamMach number of 0.001 is sufficient in fulfilling thisnumerical requirement.

The on-design jet grid is constructed as a single layerwedge which covers a sector of 2.5° in the circumferen-tial direction. Figure 1 shows general layout of this grid.There are 400 uniformly sized grid cells in the stream-wise direction covering a distance ofx/R= 40 and 48grid cells in the transverse direction covering a radial dis-tance ofy/R= 8. At the inflow station of the jet, the jetplume is defined by 18 grid cells, and the remaining 30grid cells cover the distance fromy/R= 1.0 to 8.0. Theinitial shear layer region near the nozzle exit plane iscovered by 24 grid cells centered above and below thenozzle lip. High grid density is provided in the shearlayer to capture the turbulent mixing process. As the jetflow spreads downstream, approximately 30 grid cellsare located within the jet flow. For computational conve-nience, the grid domain is divided into four blocks in thestreamwise direction. The grid domain can easily beextended in the streamwise direction by adding moreblocks.

General features of jet flow computation using thePAB3D code with the two-equationk-ε turbulence clo-sure model are illustrated by the solutions of a typicalsubsonic jet atM = 0.6. Compressibility correction forthe k-ε turbulence model is not needed in this com-putation. Figure 2 shows the computed centerline veloc-ity profile for the M = 0.6 jet. The centerline velocitymaintains its exit value for a distance up to approxi-mately x/R= 12 and decreases farther downstream as aresult of turbulence mixing. The classical relation ofvelocity decay is given by

(6)

whereLc is the intercept of thex−1 decay curve and hori-zontal line uc(x)/Ue = 1.0 (referred to as the potential

uc x( )Ue

-------------Lc

x-----=

9

core length in jet flow literature). Figure 2 shows boththe computed centerline velocity profile and the classicalvelocity decay as indicated by equation (6). Good agree-ment is shown between the PAB3D solution and theresult obtained with equation (6).

Figure 2 also shows the computed centerline velocityprofiles for a Mach 2.0 jet operating on-design using theJones-Launderk-ε turbulence model with three differentmethods of compressibility corrections. The compress-ibility correction factor in thek-ε turbulence model has astrong influence on jet flow development. Turbulencemixing is strongest in the jet flow when no compressibil-ity correction is applied. For this case, the potential corelength isLc/R= 17.2. The action of turbulence mixing inthe jet is weaker when compressibility corrections areapplied. The value ofLc/R is 22.6 and 25.2 for the Sarkarand the Wilcox methods, respectively. The velocitydecay downstream of the end of the potential core is alsocomputed according to equation (6) and the value ofLc/Rfor each case. The results are shown in figure 2.

Good agreement is observed between the PAB3Dsolutions using the Sarkar and the Wilcox correctionsand their corresponding results using equation (6) foruc/Ue greater than 0.7. Foruc/Ue less than 0.7, thePAB3D solution begins to deviate from the classical 1/xdecay rate. For the solution without compressibility cor-rection, the decay rate starts to deviate from the 1/x decayat approximatelyuc/Ue = 0.8. Without compressibilitycorrection in thek-ε turbulence model, the predicted tur-bulence level is too high for the Mach 2.0 supersonic jetsolution. This steep velocity decay is an indication ofexcess mixing in the jet shear layer.

Figure 3 shows the downstream evolution of theM = 0.6 jet velocity cross section. Atx/R= 0, the initialvelocity profile across the entire width of the jet nozzleexit has a prescribed constant value ofUe. The cross sec-tion atx/R= 5 (fig. 3) shows the initial development of athin shear layer, and the width of the potential core isnarrower than its width at the jet exit. The cross sectionat x/R= 15 is located just downstream of the end of thepotential core. The velocity profile atx/R= 15 has notyet attained a Gaussian distribution. However, the Gauss-ian velocity distribution has been established atx/R= 25.Figure 4 shows the turbulence intensity distributions atthe correspondingx/R stations. The peaks of the turbu-lence intensity distributions atx/R = 5 and 15 are locatedin the middle of the shear layer where the velocity gradi-ent is the highest. Although the centerline turbulencelevel atx/R= 25 is significantly higher, the peak turbu-lence intensity remains off center, and the turbulenceintensity distribution across the jet is not Gaussian.According to measured data by Wygnanski and Fiedler

(ref. 22), self similarity of the turbulence intensity is usu-ally established atx/R values between 50 and 70.

Figure 5 shows computed centerline velocity profilesfor aM = 2.22 jet and the experimental data measured byEggers. (See ref. 23.) Like the centerline velocity profilesfor a Mach 2.0 jet shown in figure 2, the solutionsobtained by using different compressibility correctionsare different. With no compressibility correction, thepotential core length is underpredicted. The location ofthe end of the potential core appears to agree with thecenterline velocity profile predicted using the Wilcoxmodel. However, centerline velocity computed by usingequation (6) and the potential core length of the Sarkarsolution Lc/R= 27.15 agrees very well with the dataobtained farther downstream. (See ref. 23.) The agree-ment between computational and measured data is muchbetter when compressibility corrections are applied,although a small difference exists between the Sarkarmodel and the Wilcox solutions. Figure 6 shows the cor-responding results of velocity distributions in thejetcross section at x/R= 25. The importance ofcompressibility correction for supersonic jets is furtherillustrated here, as the compressibility-corrected compu-tations come very close to the measured data, whereasthe uncorrected computation underpredicts the centerlinevelocity by nearly 40 percent.

A group of on-design jet plumes with exit Machnumbers ranging from 0.6 to 3.0 is computed to illustratethe trend of turbulent mixing as a function of Mach num-ber. Figure 7 shows typical turbulence intensity distribu-tions ut/Ue in the longitudinal plane of symmetry of thejet at three different Mach numbers: 0.8, 1.2, and 1.6.The contours in figure 7 show that turbulence is absent inthe potential core region. Intense levels of turbulencestart to develop at the lip of the jet nozzle exit. The posi-tion of the maximum turbulence intensity in the initialzone of the shear layer occurs near the lip line of the jet.As the shear layer evolves farther downstream, the posi-tion of maximum turbulence intensity migrates towardsthe jet centerline. This general pattern remains the samefor all on-design circular jets computed within the Machnumber range from 0.6 to 3.0. The length of the potentialcore and the value of maximum turbulence intensity varyas a function of Mach number. Figure 8 summarizes thecomputed turbulence intensity distributions along the jetcenterline as a function of Mach number. The Wilcoxcompressibility correction is used for these computationsbecause the definition of the Wilcox correction providesa consistent blending of compressibility correction forsubsonic and supersonic flow regions.

The potential core length is usually defined as thedistance from the jet exit to the beginning of centerlinevelocity decay. An important equation for potential core

10

length as a function of jet Mach number is given by Lau,Morris, and Fisher in reference 24 as:

(7)

Core length is obtained by an empirical curve fit to alarge collection of measured values for potential corelength in subsonic and supersonic jets.

Figure 9 shows the potential core lengths computedwith the standard Jones-Launder two-equationk-ε turbu-lence model with the Sarkar and the Wilcox compress-ibility corrections. The core length is defined as the pointwhere the value of the centerline jet velocity has droppedto 0.99 times the jet exit velocity. The potential corelength derived with the Wilcox compressibility correc-tion is higher than the value computed in equation (7) forthe entire Mach number range. Values obtained by usingthe Sarkar compressibility correction are higher than theWilcox results. However, the trends of core length varia-tion as a function of Mach number are similar in all threesets of results. An alternate value of the potential corelengths can be obtained from the computed jet flow solu-tion when the end of potential core in the jet flow isdefined as the point where turbulence intensity levelexceeds a threshold ofurms= 0.01 along the jet center-line. The core lengths obtained by the turbulence inten-sity definition (also shown in fig. 9) agree very well withthe values obtained with the velocity decay criterion.

The difference between the computed and the empir-ical curve fit formula based on measured values origi-nates from several sources. In the work by Lau, Morris,and Fisher, the experimental database contains measuredpotential core length values for jets operating at differenttemperatures. Equation (7) is a curve fit for isothermaljets where the jet static temperature is the same as theambient temperature, whereas the jet total temperature ishigher than the ambient total temperature. Many datapoints for cold jets, where the jet total temperature is thesame as the ambient air temperature and therefore the jetstatic temperature is colder than the ambient temperature,are above the curve fit of equation (7). In this report, thejet flows are computed as cold jets.

A second source of discrepancy may come frommodeling boundary conditions in the computations. Forjet flows in the laboratory, the boundary layer within thejet nozzle has a finite thickness at the nozzle exit. Theinitial turbulence level and the thickness of the nozzleboundary layer give the jet mixing layer an earlier start inits development. Therefore, the computed core lengthwill be shorter if the initial boundary layer at the jet noz-zle exit is included in the computations. In addition tothese circumstances, grid density and accuracy of turbu-lence modeling are important factors to be considered for

further refinements of the computational method for jetflow predictions.

It is significant that the mean flow and turbulencelevels of on-design circular jet plumes are predicted oversuch a wide range of Mach numbers by using the stan-dard Jones-Launderk-ε turbulence model and the Wilcoxcompressibility correction without changing the pub-lished constants for the turbulence model. In a broadercontext, the modeling of jet plumes is often required inpropulsion and airframe integration. A consistent compu-tational analysis for such jet plume modeling using theNavier-Stokes method should not permit ad hoc changesto the turbulence model. The results of this parametricinvestigation indicate that ad hoc modifications to thestandard Jones-Launder turbulence model are not neededfor jet flow analysis.

Off-Design Jets Containing Weak Shocks

This section shows the flow properties of a Mach 2jet containing weak shocks using solutions obtainedwithin a limited range of nozzle pressure ratios. Thespace-marching solver procedure in the PAB3D code isused to compute these jet flows. At Mach 2, the jet flowis free from Mach disk formation for values of NPRbetween 4.6 and 13.8, which correspond to a ratio of jetexit to ambient pressurepe/po between 0.6 and 1.8. Fig-ure 10 shows a density contour for a typical under-expanded jet. At the jet exit, a curved shock near the lipline of the jet nozzle is formed to resolve the pressuredifference between the ambient flow and the flow insidethe jet. An internal weak shock system which reflectsrepeatedly between the shear layer and the jet centerlinealso exists in the jet.

Figure 11 shows the computed pressure distributionalong the jet axis forpe/po = 0.8, 1.2, 1.4, 1.6, and 1.8.Only one overexpanded case is included in this collection(pe/po = 0.8). The pressure distribution of the over-expanded jet is characterized by a sharp shock front andpressure peak close to the jet exit. This feature is notfound in underexpanded jets. Beginning with the secondpeak, the features of overexpanded jet pressure oscilla-tions on the jet centerline follow the same trend as thepatterns shown for underexpanded jets. For the under-expanded jets, the cycle of pressure oscillation beginswith a smooth expansion. The flow is then recompressedtowards the first pressure peak in two stages. The pres-sure rises sharply about halfway then recompresses grad-ually the rest of the way. Although not shown inthe figure, expansion and recompression processes aresmooth for subsequent periods of oscillation. Refer-ence25 gives further discussion of the centerline pres-sure distributions of off-design jet flows at Mach 2.

Lc

R----- 8.4 2.2M2+=

11

Extensive flow visualization measurements of super-sonic jets at off-design conditions were obtained by Loveet al. (See ref. 5.) Shock formation in the jet flow is char-acterized by two lengths:l1, the distance from the jet exitto the first shock intersection with the jet centerline; andW, the distance from the jet exit to the first shock inter-section with the sonic line in the shear layer. Figure 10shows definitions of these lengths. Figure 12 shows thecomputed values and measured values ofl1 and W(ref. 5) for several overexpanded and underexpanded val-ues of NPR. Excellent agreement is demonstrated by theresults in figure 12.

Figure 13 details computed centerline pressure dis-tributions and experimental data for aMach 2.0 jetat pe/po = 1.445 (NPR= 11.3). Three solutions areobtained by using the basick-ε turbulence model with nocompressibility correction, the Sarkar correction, and theWilcox correction. The solution without compressibilitycorrection shows that the amplitude of pressure oscilla-tions diminishes rapidly downstream and predictedwavelength is much shorter than the experimental data inthe downstream region of the jet. The solutions obtainedwith a compressibility-correctedk-ε turbulence modelshow excellent agreement with measured data. Differ-ences between the Sarkar and Wilcox corrections aresmall. The amplitudes of the computed solutions closelyfollow the test data, but their phase relations with re-spect to the measured data are somewhat different.At x/R= 40, the Wilcox solution leads the measured databy approximately one sixth of one period, whereas theSarkar solution lags behind the measurements by approx-imately half that amount. All three solutions are verysimilar near the jet exit. However, the amplitude of thefirst pressure peak near the jet exit is underpredicted byapproximately 15 percent.

Figure 14 shows the computed values for the axialturbulence velocity component and the measured dataobtained by Seiner, Dash, and Wolf (ref.26). The inter-action between the repeated shock-cell structure and theturbulence produces a significant periodic modulation ofthe axial turbulence velocity component. The magnitudeof the fluctuation is in phase with the pressure fluctuationin the jet. (See fig. 13.) Good agreement in both phaseand amplitude is seen between the computed solutionsand the measured data. The compressibility-correctedsolutions provide better agreement with the measure-ments than the uncorrected solution. It is encouraging tofind from this comparison that the standard Jones-Laun-der k-ε model is capable of accurate predictions of theturbulence velocity in a shock-containing supersonic jet.For practical applications such as jet noise prediction, anestimate of turbulence intensity in the jet flow is needed.A computational capability for predicting turbulenceintensity distributions in the jet flow is highly desirable

because measurement of turbulence in high speed flow isexceedingly difficult.

Better predictions of the turbulent velocity fluctua-tions in a supersonic jet can be obtained with furtherimprovement of the turbulence model. In standardk-εturbulence models, the local turbulence kinetic energy isattributed equally to all three turbulence velocity compo-nents. However, it is known that the magnitude of theaxial component is higher than those of the transversecomponents in the jet shear layer. Therefore, a betterredistribution relationship of the turbulent kinetic energyand the Reynolds stress tensor components would raisethe value of the computed . Furthermore, the mod-ulation of the axial turbulence velocity component by theinternal shock waves would be stronger, since the ampli-tude of shock turbulence interaction is roughly propor-tional to the shock strength and the magnitude of theaxial component of the velocity fluctuations.

Use of the space-marching algorithm to obtain a jetflow solution requires less than 100 seconds of CPU timeon the Cray Y-MP computer at the Langley ResearchCenter. Use of the time-marching solver to obtain a con-verged solution for the same cases typically requires2000 seconds of CPU time. The ratio of computer timerequired when using the time-marching solver increasesby a factor of 20. Figure 15 presents jetcenterline pres-sure distributions obtained by using the space-marchingand time-marching solvers. The flow solutions obtainedby these two different procedures are practically thesame. Detailed discussion of this comparison can befound in reference 27.

Off-Design Supersonic Jets Containing a MachDisk

Mach disks may appear in a supersonic jet if theoperating NPR is significantly different from NPRdesign value. The conditions for Mach disk formation fora given nozzle depend on nozzle design Mach numberand details of the nozzle geometry, such as the nozzlewall exit angle. Mach disk formation can occur in bothoverexpanded and underexpanded conditions. For aMach 2.0 nozzle with on-design NPR of 7.82, Mach diskappears if the operating NPR is less than 4.6 or greaterthan 13.8. For a nozzle withM = 1.5 with an on-designNPR value of 3.67, a Mach disk will form in the jet forNPR less than 2.7 or greater than 6.1.

For jet flow computations where Mach disk forma-tion is expected, the time-marching solver in the PAB3Dcode is used. A different computational grid is alsorequired. When the case of a Mach disk containing jetflow is originally computed with the on-design jet grid,the Mach disk is never formed in the converged solution.The shock front initiated at the nozzle exit propagates as

ut/Ue

12

a weak shock all the way to the jet centerline and thencontinues as a regular reflected weak shock. In theon-design grid, the cell streamwise versus radial aspectratio is 4. Although the PAB3D code solver is designedas an upwind algorithm, certain numerical errors in thetransonic regime prevent the proper formation of a Machdisk in the flow solution. Because general patched gridcapability is available in the PAB3D code, a new grid iscreated so that the grid forx/R from 0 to 2 has a grid dis-tribution similar to the on-design jet grid but with doublethe density in each direction. Forx/R from 2 to 10, auniformly sized grid distribution is retained in thestreamwise direction. In the radial direction, a uniformgrid distribution is provided fory/R from 0 to 2 so thatcell aspect ratio in the entire region is 1.0. An exponen-tially expanding grid is used fory/R from 2 to 8 to coverthe free-stream domain outside the jet flow. Figure 16shows a sketch of this revised grid. The overall griddomain is divided into four blocks for computationalconvenience. The correct Mach disk containing solutionis obtained by using this revised grid.

A solution for a Mach 1.5 jet operating atNPR =11.6 (pe/po = 3.15) is obtained. Figure 17 showsthe density and Mach number distributions in the jet. Awell-formed Mach disk is located atx/R= 4.4. Theradius of the Mach disk is approximately 0.68R in thecomputed solution. The location of this Mach disk agreeswith the measurements given in reference 5. However,the computed radius of the Mach disk is smaller than thecorresponding measured value. The reflected weak shockand a slip line initiated at the outer edge of the Mach diskis clearly shown by the computed density contours. Thecontour value indicates that the Mach number upstreamof the Mach disk has accelerated to values greaterthan 4.0, whereas the Mach number downstream of thefirst Mach disk is reduced to values below 0.2. Down-stream of the first Mach disk, the flow near the centerlineagain accelerates to supersonic speeds nearx/R = 8.0. Asecond Mach disk is subsequently formed atx/R= 8.6.Though much weaker, the second Mach disk can be seenin a schlieren photograph for a jet operating at nearly thesame jet initial conditions. (See ref. 5.)

Since the time-marching computations for the Machdisk case are executed by using grid sequencing, con-verged solutions at three grid levels are obtained duringthe process. Figure 18 shows the Mach number contoursusing the one-fourth and one-half linear grid density inthe j- andk-directions. Even at the quarter density gridlevel, the first Mach disk is captured in the solution. Boththe location and the radius of this Mach disk are estab-lished in this coarse grid. In the half density grid, the sec-ond Mach disk emerges in the solution. Only minorchanges in the flow physics are detectable between the

half density grid solution and the full density grid solu-tion, as shown earlier in figure 17.

Adaptive Grid Computations ofNonaxisymmetric Jets

For jet flow computations using an adaptive grid, aquarter plane symmetry for the jet is assumed. The grid isdivided into two domains: a high grid density innerdomain near the jet flow, and an outer domain withreduced grid density to cover the free-stream domainaway from the jet flow. A Cartesian grid topology is usedin the inner domain to accommodate a wide range of jetexit geometries. The outer domain has a polar topologywith significantly less grid density than the inner domain.Figure 19 shows a sketch of the grid cross section. Refer-ence 28 shows that the computational simulation of a cir-cular jet remains perfectly axisymmetric even though thegrid is Cartesian. Furthermore, the adaptive grid proce-dure provides adequate grid densities to support accuratecomputations in the jet shear layer and in regions near ashock front.

This section discusses computed solutions for anelliptic, a rectangular, and a square jet using the adaptivegrid method. The elliptic jet is known for its unusualmixing characteristics. The rectangular jet family, whichincludes the square jet as a special case, is widely usedfor propulsion integration in advanced aircraft systems.Both elliptic and rectangular jets are capable of switchingtheir major and minor axis directions in different crosssections along the jet. An initial Mach number of 2.0 andan operating NPR of 11.31 are chosen for all three con-figurations. The operating NPR corresponds to an exitstatic pressure ratio ofpe/po = 1.445. In addition, theMach 2 elliptic jet is also computed at its on-design flowcondition at NPR = 7.82.

Figure 20 shows the computed Mach number con-tours for the underexpanded elliptic jet in the major andthe minor planes of symmetry. The elliptic cross sectionat the nozzle exit plane has an aspect ratio of 2.0. Theshock fronts are clearly defined by the Mach numbercontours in the major plane of symmetry. The shockreflection pattern in the core region of the jet is quite reg-ular. In the plane containing the minor axis, the shock isinitially reflected in the shear layer with a scale propor-tional to the minor axis length. However, the short wavepattern quickly disappears at approximatelyx/R= 6. Far-ther downstream in the jet, only the long wave patterndominated by the length scale of the major axis remains.Another unique feature in figure 20 is the expansion rateof the outer jet boundary. In the plane containing themajor axis, the jet boundary expands slowly in the radialdirection. In contrast, the jet boundary in the planecontaining the minor axis expands rapidly in the radial

13

direction. By approximatelyx/R= 15, the width of theelliptic jet in the original minor axis direction hasexceeded the jet width in the original major axis direc-tion. Hence, this computation indicates an axis switchingphenomenon for a supersonic elliptic jet.

When using similarity analysis, the major and theminor axes of the initial cross section are considered twoindependent reference scale lengths for the jet flow. Asimple consequence of assuming two independent refer-ence scales would be that the internal shock reflectionpattern would repeat in two directions along two differ-ent scales. On the other hand, the difference betweenthese two scales must be resolved within the jet flow to acommon scale since shock fronts cannot cross each otherwithout some type of interaction. The computed resultdemonstrates the complexity of such aerodynamic inter-actions. In the plane of symmetry containing the minoraxis, the internal shock wave length is initially governedby the minor axis scale. However, the shear layer posi-tion expands rapidly outward in the minor plane of sym-metry; thus, the reflection length scale of the downstreamshock wave pattern is changed. In the major plane ofsymmetry, the width of the jet in this plane remainsapproximately constant; thus, the reference scale of thejet in the minor axis direction is allowed to catch up. Thenonlinear interaction within the jet flow eventually leadsto a unified scale length for the shock cell system.

Figure 21 shows the Mach number contours in anon-design elliptic jet at Mach 2.0. In the absence of ashock structure in the jet, the Mach number distributionin both the major and minor planes of symmetry issmooth and indistinguishable from the previously com-puted Mach number distributions in circular jets. Similarto the underexpanded elliptic jet case, the shear layergrowth in the minor plane of symmetry is faster than thegrowth in the major plane of symmetry. Atx/R= 40, thewidths of the jet as seen in both planes of symmetry arealmost equal. However, axis switching does not occur inthe on-design case.

In order to examine the possibility of axis switching,cross section Mach number contours are shown for thesefour jets in figures 22–25. The exit cross section aspectratio for the elliptic and the rectangular jet exit shapesis 2.0. The Mach number contours in each of thecross sections show only a narrow band fromM = 0.8to M = 1.2 with a contour interval of 0.1 to highlight theshape of the cross section. Figure 22 shows the evolutionof the elliptic underexpanded jet cross sections. At theexit, the major axis of the elliptic cross section is orientedin the horizontal direction. The jet boundary growsrapidly in the vertical direction. Atx/R= 30, the majoraxis of the elliptic cross section has clearly switched

to the vertical direction. The aspect ratio of the ellipse atx/R= 30 is approximately 1.50.

Figure 23 shows the equivalent sequence for a rect-angular jet. Jet boundary growth in the vertical directionis even faster than that of the elliptic jet. Atx/R= 30, theaspect ratio of the shape is approximately 1.70. The cor-ners of the original rectangular shape have been roundedoff in the process of turbulent mixing.

Figure 24 shows the evolution sequence for a squarejet. In this case, the original square shape for the Machnumber contours evolves rapidly in the jet flow. Atx/R= 12, the corners of the square are actually trans-posed by 45°, with the corner sharpness well preserved.As the shear layer grows thicker farther downstream, theshape of the jet cross section quickly losses its distinctionas a square and eventually becomes circular.

Figure 25 shows the shape evolution sequence for anelliptic jet with on-design exit pressure ratio. Althoughthe jet grows mainly in the vertical direction, axis switch-ing does not occur in this on-design jet flow. The jet sim-ply becomes a near-circular jet atx/R= 30. At the lastcomputed streamwise position atx/R= 40, the cross sec-tions for all four jets simply retain their geometrical char-acters similar to those established atx/R= 30.

Figure 26 shows the Mach number contours in theplane of symmetry of the square jet and its centerlineMach number distribution. Qualitatively, these distribu-tions are very similar to the corresponding distributionsshown in figure 27 for a circular jet operating at the sameexit Mach number and NPR. The visually strikingdynamic behavior of the square jet axis (fig. 24) showsthat switching apparently has little influence on flowdevelopment near the jet centerline. It should be pointedout that the square jet and the circular jet, with a commonreference dimension of 1.0, have different jet exit areas.In order to compare the streamwise jet flow developmenton a normalized basis, the length scale for the square jetin figure 24 should be reduced by a factor of

. With this scale adjustment, the phaseand amplitude of the centerline Mach number oscilla-tions of the square jet and the circular jet agree almostexactly starting from the second peak.

In order to provide some validation for the adaptivegrid computation procedure for nonaxisymmetric jets, aMach 2.0 on-design circular jet solution computed byusing the axisymmetric grid is compared with the samejet computed by using a three-dimensional adaptive grid.Two levels of adaptive grid densities are also used to ver-ify grid convergence: 40× 40 cells and 56× 56 cells forthe inner high density grid cross sections. Figure 28shows the axisymmetric grid solution forM = 2.0. Theadaptive grid results are shown in figures 29 and 30.

π 4⁄ 0.8862=

14

Nearly identical solutions for the turbulence intensitydistribution in the meridian plane and along the center-line of the jet are obtained for the two adaptive griddensities. For example, the maximum turbulence level inthe jet plume is 0.134 for both the 40× 40 cell and the56 × 56 cell solutions. The solution using an axisymmet-ric grid differs slightly from the adaptive grid results intwo aspects. First, the maximum turbulence level in themiddle of the shear layer is slightly higher, with a valueof 0.142. Second, the turbulence intensity profile alongthe centerline is shifted upstream by approximatelyx/R= 2.0.

The reason for the different maximum intensity inthe shear layer is not clear. However, the spatial shift ofthe turbulence intensity profile along the centerline has ageometrical explanation. It is difficult for the grid adap-tion algorithm to handle very large velocity gradientssuch as those occurring near the jet exit. Consequently, itis not possible to specify an initial shear layer thicknessof less than 0.05 jet radius at the jet exit plane. Since theinitial shear layer is thicker, the inner boundary of theturbulent shear layer intersects with the centerline at asmaller value ofx/R. In fact, with a downstream shift ofthe adaptive grid centerline turbulence intensity profile,the turbulence intensity profile can be matched perfectlywith results using the axisymmetric grid.

Concluding Remarks

The main purpose of this report is to establish a uni-fied method for jet flow prediction using the Navier-Stokes method with a two-equationk-ε turbulence clo-sure model. Although the jet flow may contain a varietyof complex flow physics features, the Navier-Stokesmethod simply requires that the initial condition andboundary conditions of the jet operating conditions bespecified for the problem. Detailed flow physics devel-opments in the jet are predicted by the Navier-Stokesmethod. The validity of this approach is demonstrated bythe high quality jet flow solutions obtained with thePAB3D code.

This study examines several categories of jet flowconditions. For on-design subsonic and supersonic axi-symmetric jets, the flow field is dominated entirely byturbulent mixing. Numerical solutions within a Machnumber range of 0.6 to 3.0 are accurate when comparedwith available experiment data for parameters such asmean velocity and turbulence intensity distributions inthe jet, centerline velocity decay, and the potential corelength variation as a function of Mach number.

For off-design supersonic jet flows containing weakshocks, flow predictions are compared with experimentaldata. Good agreement is obtained between the computedresults and experimental data for key parameters, includ-ing first shock-cell lengths and centerline pressure distri-bution. The predicted distributions of the streamwisecomponent of turbulence velocity fluctuation in anunderexpanded Mach 2.0 jet show good agreement withmeasured data.

Turbulence intensity in the jet flow is an importantquantity for jet noise prediction. Since direct measure-ment of turbulence in a supersonic jet is very difficult tomake, a predictive capability provided by the PAB3Dcode is very useful for practical applications. Goodagreement between predictions and experimental mea-surements has also been obtained for a Mach 1.5 jet oper-ating at 3.15 times its design nozzle pressure ratio whereMach disks are present in the jet flow.

Many of the modern propulsion jet nozzles employnonaxisymmetrical exit geometries. The adaptive gridmethod examined in this study has produced good resultsfor elliptic, rectangular, and square jets. However, thecomputed results are not verified for lack of experimentaldata. The accuracy of the adaptive grid procedure is illus-trated by a comparison between an adaptive grid solutionof an axisymmetric jet and a solution for the same jetusing a single cell wedge grid. Although the adaptivegrid has a Cartesian-topology and the single-cell wedgegrid has cylindrical symmetry boundary conditions, thesolutions are essentially identical.

For most jet flows where strong shocks are absent inthe computational domain, the space-marching solver inthe PAB3D code can be used. When the space-marchingoption is used for jet flow computation as conditions per-mit, the computer time is one twentieth of the timerequired for obtaining a time-marching solution with thesame flow conditions. The accuracy of the solutionsobtained by these different solvers is practically indistin-guishable. Substantial savings in computer time can berealized by using the space-marching method in thePAB3D code if the analyses of many cases of jet flowconditions are required for design applications.

NASA Langley Research CenterHampton, VA 23681-0001May 23, 1996

15

References

1. Reichardt, Hans:Gesetzmässigkeiten der freien Turbulenz.VDI-Forschungsh. 414, 1942.

2. Görtler, H.: Berechnug von Aufgaben der freien Turbulenz aufGrund eines neuen Näherungsansatzes.Z. Angew. Math.Mech., Bd. 22, Nr. 5, Oct. 1942, pp. 244–254.

3. Warren, Walter R., Jr.: An Analytical and ExperimentalStudy of Compressible Free Jets. Publ. No.: 23,885, Univ.Microfilms, Inc., 1957.

4. Abramovich, G. N.: The Theory of Turbulent Jets. M.I.T.Press, 1963.

5. Love, Eugene S.; Grigsby, Carl E.; Lee, Louise P.; andWoodling, Mildred J.:Experimental and Theoretical Studiesof Axisymmetric Free Jets. NASA TR R-6, 1959. (SupersedesNACA RM L54L31 by Love and Grigsby; RM L55J14 byLove; RM L56G18 by Love, Woodling, and Lee; and TN 4195by Love and Lee.)

6. Dash, Sanford M.; and Wolf, David E.:Fully-Coupled Analy-sis of Jet Mixing Problems, Part I: Shock-Capturing Model,SCIPVIS. NASA CR-3716, 1984.

7. Dash, Sanford M.; Pergament, Harold S.; and Thorpe,Roger D.: Computational Models for the Viscous/InviscidAnalysis of Jet Aircraft Exhaust Plumes. NASA CR-3289,1980.

8. Seiner, John M.: Advances in High Speed Jet Aeroacoustics.AIAA-84-2275, Oct. 1984.

9. Seiner, J. M.; and Norum, T. D.: Aerodynamic Aspects ofShock Containing Jet Plumes. AIAA-80-0965, June 1980.

10. Abdol-Hamid, Khaled S.; and Wilmoth, Richard G.: Multi-scale Turbulence Effects in Underexpanded Supersonic Jets.AIAA J., vol. 27, Mar. 1989, pp. 315–322.

11. Abdol-Hamid, Khaled S.: The Application of 3D MarchingScheme for the Prediction of Supersonic Free Jets.AIAA/ASME/SAE/ and ASEE 25th Joint Propulsion Conference, July1989. (Available as AIAA-89-2897.)

12. Abdol-Hamid, Khaled S.; Uenishi, Kenji; and Turner,William: Three-Dimensional Upwinding Navier-Stokes CodeWith k-ε Model for Supersonic Flows. AIAA-91-1669, June1991.

13. Jones, W. P.; and Launder, B. E.: The Prediction of Laminar-ization With a Two-Equation Model of Turbulence.Int. J. Heat& Mass Transf., vol. 15, no. 2, Feb. 1972, pp. 301–314.

14. Abdol-Hamid, Khaled S.; Lakshmanan, B.; and Carlson, JohnR.: Application of Navier-Stokes Code PAB3D With k-εTurbulence Model to Attached and Separated Flows. NASATP-3480, 1995.

15. Eiseman, Peter R.: Alternating Direction Adaptive Grid Gen-eration.AIAA J., vol. 23, no. 4, Apr. 1985, pp. 551–560.

16. Sarkar, S.; Erlebacher, G.; Hussaini, M. Y.; and Kreiss, H. O.:The Analysis and Modelling of Dilatational Terms in Com-pressible Turbulence.J. Fluid Mech., vol. 227, June 1991,pp. 473–493.

17. Wilcox, David C.: Progress in Hypersonic Turbulence Model-ing. AIAA-91-1785, June 1991.

18. Eiseman, Peter R.; and Erlebacher, Gordon:Grid Generationfor the Solution of Partial Differential Equations. NASACR-178365, ICASE Rep. No. 87-57, 1987.

19. Eiseman, Peter R.: Adaptive Grid Generation.Comput. Meth-ods Appl. Mech. & Eng., vol. 64, nos. 1–3, Oct. 1987,pp. 321–376.

20. Eiseman, P.; and Bockelie, M.: The Control and Application ofAdaptive Grid Movement.Finite Element Analysis in Fluids,Proceedings of the Seventh International Conference on FiniteElement Methods in Flow Problems, Univ. of Alabama inHuntsville Press, 1989, pp. 1025–1032.

21. Pao, S. Paul; and Abdol-Hamid, Khaled S.: Application of aNew Adaptive Grid for Aerodynamic Analysis of Shock Con-taining Single Jets. AIAA-90-2025, July 1990.

22. Wygnanski, I.; and Fiedler, H.: Some Measurements in theSelf-Preserving Jet.J. Fluid Mech., vol. 38, pt. 3, Sept. 18,1969, pp. 577–612.

23. Eggers, James M.:Velocity Profiles and Eddy Viscosity Distri-butions Downstream of a Mach 2.22 Nozzle Exhausting toQuiescent Air. NASA TN D-3601, 1966.

24. Lau, Jark C.; Morris, Philip J.; and Fisher, Michael J.: Mea-surements in Subsonic and Supersonic Free Jets Using a LaserVelocimeter.J. Fluid Mech., vol. 93, pt. 1, July 12, 1979,pp. 1–27.

25. Abdol-Hamid, Khaled S.: Three-Dimensional Calculations forUnderexpanded and Overexpanded Supersonic Jet Flows.AIAA-89-2196, July–Aug. 1989.

26. Seiner, J. M.; Dash, S. M.; and Wolf, D. E.: Analysis of Turbu-lent Underexpanded Jets, Part II: Shock Noise Features UsingSCIPVIS.AIAA J., vol. 23, no. 5, May 1985, pp. 669–677.

27. Abdol-Hamid, Khaled S.: The Application of 3D MarchingScheme for the Prediction of Supersonic Free Jets. AIAA-89-2897, July 1989.

28. Pao, S. Paul; and Abdol-Hamid, Khaled S.: Application of aNew Adaptive Grid for Aerodynamic Analysis of Shock Con-taining Single Jets. AIAA-90-2025, July 1990.

16

(a) Typical on-design jet flow configuration and terminology.

(b) Single cell wedge grid for on-design jet flow computations.

Figure 1. Sketch of typical axisymmetric on-design jet flow and computational grid.

Outflow boundaryInflow boundary

Computational domain

Turbulent shear mixing layer

Jet centerline

Potential core

Nozzle

17

Figure 2. Centerline velocity decay for subsonic and supersonic jet flows computed with standard Jones-Launderk-εturbulence model with different compressibility corrections.

10 20 30 400

.2

.4

.6

.8

1.0

1.2

x/R

uc/Ue

M = 0.6

M = 2.0, no correction

M = 2.0, Wilcox correction (ref. 17)

M = 2.0, Sarkar correction (ref. 16)

M = 0.6, Lc/x velocity decay (eq. (6))

M = 2.0, Lc/x velocity decay (eq. (6))

M = 2.0, Lc/x velocity decay (eq. (6))

M = 2.0, Lc/x velocity decay (eq. (6))

18

Figure 3. Axial velocity component distribution in cross sections in Mach 0.6 circular jet flow.

Figure 4. Turbulence intensity distribution in cross sections in Mach 0.6 circular jet flow.

2 4 6 80

.2

.4

.6

.8

1.0

1.2

y/R

x/R = 5.0

x/R = 15.0

x/R = 25.0

u/Ue

.05

.10

.15

.20

2 4 6 80y/R

x/R = 5.0

x/R = 15.0

x/R = 25.0

Turbulence intensity,

ut /Ue

19

Figure 5. Centerline velocity distribution for supersonic jet usingk-ε turbulence model with different compressibilitycorrections.

Figure 6. Axial velocity component distribution along radial direction atx/R= 25 usingk-ε turbulence model withdifferent compressibility corrections.

10 20 30 40 50 600

.2

.4

.6

.8

1.0

1.2

x/R

M = 2.22, k-ε—no correctionM = 2.22, k-ε—Sarkar model (ref. 16)M = 2.22, k-ε—Wilcox model (ref. 17)M = 2.22, test data from Eggers (ref. 23)Lc/x velocity decay—Lc/R = 27.15

uc/Ue

2 4 6 80

.2

.4

.6

.8

1.0

1.2

y/R

M = 2.22M = 2.22M = 2.22M = 2.22

no correctionSarkar model (ref. 16)Wilcox model (ref. 17)Test data from Eggers (ref. 23)

Axial velocity

component, u/Ue

20

Figure 7. Typical turbulence intensity distribution in axisymmetric jets computed by using Jones-Launderk-ε turbu-lence model with Wilcox model of compressibility correction forut/Ue contours.

5 10 15 20 25 30 35 400

2

4

6

8

0.010 0.020 0.0300.040

0.0600.080

0.1000.110

0.1200.1300.1400.150

Jet Mach number = 1.2

y/R

5 10 15 20 25 30 35 400

2

4

6

8

0.010 0.020 0.0300.040

0.0500.060

0.0800.100

0.110

0.1200.1300.1400.1500.160

Jet Mach number = 0.8

x/R

x/R

y/R

5 10 15 20 25 30 35 400

2

4

6

8

0.010 0.020 0.030 0.040

0.1200.1300.140

0.1000.110

0.0800.060

x/R

Jet Mach number = 1.6

y/R

21

Figure 8. Centerline turbulence intensity profiles of axisymmetric jets at various subsonic and supersonic Mach num-bers computed by using Jones-Launder two-equationk-ε turbulence model with Wilcox compressibility correction.

10 20 30 400

.05

.10

.15

.20

Jet Mach number = 0.6

0.8

1.0

1.2

1.4

1.6

2.0

1.8

2.2

2.5

3.0

x/R

Turbulence intensity,

ut /Ue

22

Figure 9. Potential core length as function of Mach number for subsonic and supersonic jets computed with Jones-Launderk-ε turbulence model and different compressibility corrections.

Figure 10. Typical density contours and first shock-cell length definitions for circular underexpanded supersonic jet. Jetexit Mach number = 1.50;pe/po = 1.445.

0 1 2 3

12

8

16

20

24

28

32

Mach number, M

Potential core length,

Lc/R

Sarkar, uc/Ue = 0.99

Wilcox, uc/Ue = 0.99

Wilcox, ut/Ue = 0.01 (eq. (7))

Lau, Morris, Fisher formula (ref. 24)

4 8 12 16 200

1

2

3

4

z/R

W

l1

x/R

23

Figure 11. Computed centerline pressure distribution with different exit pressure ratios for Mach 2.0 jet computed withJones-Launderk-ε turbulence model.

Figure 12. Computed and measured first shock-cell lengths for Mach 2.0 jet computed with Jones-Launderk-ε turbu-lence model.

2 4 6 8 100

.4

.8

1.2

1.6

2.0

2.4

x/R

Centerline pressure

coefficient, p/pe

0.800

pe/po

Exit pressure ratio,

1.200

1.445

1.600

1.800

.5 1.0Exit pressure ratio, pe/po

First shock- cell lengths,

l1/R and W/R

1.5 2.00

2

4

6

8

l1/R measured data (ref. 5)

W1/R measured data (ref. 5)

l1/R

W1/R

24

Figure 13. Centerline pressure distribution computed with Jones-Launderk-ε model with different compressibility cor-rections forpe/po = 1.445.

Figure 14. Centerline turbulence intensity computed with Jones-Launderk-ε model with different compressibility cor-rections forpe/po = 1.445.

10 20 30 400

.5

1.0

1.5

2.0

2.5M=2.0, J-L k-ε—no correction

x/R

Centerline pressure

coefficient, p/po

M=2.0, J-L k-ε—Sarkar (ref. 16)M=2.0, J-L k-ε—Wilcox (ref. 17)M=2.0, test data from Seiner (ref. 26)

5 10 15 200

.05

.10

.15

.20

x/R

Turbulence intensity,

ut /Ue

M=2.0, J-L k-ε—no correctionM=2.0, J-L k-ε—Sarkar (ref. 16)M=2.0, J-L k-ε—Wilcox (ref. 17)M=2.0, test data from Seiner (ref. 26)

25

Figure 15. Space-marching and time-marching solutions for underexpanded Mach 2.0 supersonic jet computed withJones-Launderk-ε turbulence model.

10 20 30 400

.5

1.0

1.5

2.0

x/R

Centerline pressure

coefficient, p/po

Space-marching solution

Time-marching solution

26

(a) Mach number contours of typical underexpanded supersonic jet containing multiple Mach disks.

(b) Multiblock single cell wedge grid for jet flows containing multiple Mach disks.

Figure 16. Flow configuration and computational grid for underexpanded jet containing one or more Mach disks.

2 4 6 8 100

1

2

3

4

x/R

y/R

W

l1

2 4 6 8 100

1

2

3

4

x/R

y/R

27

(a) Density contours; Interval = 0.40 kg/m3.

(b) Mach number contour; Interval = 0.25.

Figure 17. Density and Mach number contours for underexpanded circular jet containing multiple Mach disks.Exit Mach number = 1.50;pe/po = 3.15; fine grid solution.

2 4 6 8 100

1

2

3

4

1.20

0.80 1.60 1.601.60 0.80 1.60

2.40

1.20

2.001.60

0.405.60

x/R

y/R

2 4 6 8 100

1

2

3

4

2.50

1.502.00

2.50

2.50 2.25

3.00

3.50

0.501.25

2.00

0.50

0.50

4.00

x/R

y/R

28

(a) One-fourth density grid Mach number contours; Interval = 0.25.

(b) One-half density grid Mach number contours; Interval = 0.25.

Figure 18. Gridstudy for underexpanded circular jet containing multiple Mach disks. Jet exit Mach number = 1.50;pe/po = 3.15.

2 4 6 8 100

1

2

3

4

2.25

1.251.50

2.00

2.50

3.00

3.50

2.50 2.251.75

0.50

2.50

0.50

x/R

y/R

x/R

y/R

2 4 6 8 100

1

2

3

4

2.50

1.502.00

2.50

2.25 2.25

3.00

3.50

0.751.00

1.75

0.50

0.75

29

(a) Typical density contours of underexpanded elliptic jet in major plane of symmetry.

(b) Adapted grid cross section at inflow plane.

(c) Adapted grid longitudinal profile in plane of symmetry containing major axis of initial jet cross section.

Figure 19. Adapted grid geometry for elliptic supersonic jet. Shape aspect ratio = 2.0.

4 8 12 16 200

1

2

3

x/R

y/R

30

(a) Plane of symmetry containing major axis; Interval = 0.20.

(b) Plane of symmetry containing minor axis; Interval = 0.20.

Figure 20. Mach number contours for underexpanded supersonic jet with elliptic exit cross section. Shape aspectratio = 2.0; Exit Mach number = 2.00; NPR = 11.12;pe/po = 1.445.

5 10 15 20 25 30 35 400

2

4

6

2.60 2.202.200.80

0.200.40 0.60 0.60

1.00

y/R

x/R

5 10 15 20 25 30 35 400

2

4

6

2.40

1.00

1.20 1.002.20

0.200.40

0.800.60

x/R

z/R

31

(a) Plane of symmetry containing major axis; Interval = 0.20.

(b) Plane of symmetry containing minor axis; Interval = 0.20.

Figure 21. Mach number contours in major and minor planes of symmetry of elliptic jet shape ratio of 2.0; On-designNPR = 7.82.

5 10 15 20 25 30 35 400

2

4

6

0.20

1.002.00

0.400.60

0.80

y/R

x/R

z/R

5 10 15 20 25 30 35 400

2

4

6

1.00

0.200.40

0.600.80

x/R

32

Figure 22. Cross-section shape evolution in streamwise direction of elliptic underexpanded supersonic jet. Shape aspectratio = 2.0; Jet exit Mach number = 2.0; NPR = 11.12;pe/po = 1.445.

.4 .8 1.2 1.6 2.0

y/R

M = 1.2 M = 0.8

x/R = 20

.4 .8 1.2 1.6 2.00

.4

.8

1.2

1.6

2.0

y/R

M = 1.2 M = 0.8

z/R

0

.4

.8

1.2

1.6

2.0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

.4 .8 1.2 1.6 2.0

y/R

M = 1.2 M = 0.8

x/R = 30

.4 .8 1.2 1.6 2.0

y/R

x/R = 10x/R = 0

M = 0.8M = 1.2

33

Figure 23. Cross-section shape evolution in streamwise direction of rectangular underexpanded supersonic jet. Shapeaspect ratio = 2.0; Jet exit Mach number = 2.0; NPR = 11.12;pe/po = 1.445.

.4 .8 1.2 1.6 2.0

y/R

M = 1.2 M = 0.8

x/R = 20

.4 .8 1.2 1.6 2.00

.4

.8

1.2

1.6

2.0

y/R

M = 1.2 M = 0.8

x/R = 0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

.4 .8 1.2 1.6 2.0

y/R

M = 1.2 M = 0.8

x/R = 30

.4 .8 1.2 1.6 2.0

y/R

x/R = 10

M = 0.8M = 1.2

34

Figure 24. Cross-section shape evolution in streamwise direction of square underexpanded supersonic jet. Jet exit Machnumber = 2.0; NPR = 11.12;pe/po = 1.445.

.4 .8 1.2 1.6 2.0

y/R

M = 1.2 M = 0.8

x/R = 24

.4 .8 1.2 1.6 2.00

.4

.8

1.2

1.6

2.0

y/R

M = 1.2 M = 0.8

x/R = 0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

.4 .8 1.2 1.6 2.0

y/R

M = 1.2 M = 0.8

x/R = 36

.4 .8 1.2 1.6 2.0

y/R

x/R = 12

M = 0.8M = 1.2

35

Figure 25. Cross-section shape evolution in streamwise direction of on-design elliptic supersonic jet. Shape aspectratio = 2.0; Jet exit Mach number = 2.0; NPR = 7.82.

.4 .8 1.2 1.6 2.0

y/R

M = 1.2 M = 0.8

x/R = 20

.4 .8 1.2 1.6 2.00

.4

.8

1.2

1.6

2.0

y/R

M = 1.2 M = 0.8

x/R = 0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

0

.4

.8

1.2

1.6

2.0

z/R

.4 .8 1.2 1.6 2.0

y/R

M = 1.2 M = 0.8

x/R = 30

.4 .8 1.2 1.6 2.0

y/R

x/R = 10

M = 0.8M = 1.2

36

(a) Mach number contours in plane of symmetry; Interval = 0.50.

(b) Centerline Mach number profile.

Figure 26. Mach number distribution in underexpanded square jet. Exit Mach number = 2.0; NPR = 11.12.

5 10 15 20 25 30 35 400

2

4

6

8

1.502.50 2.00 2.00

0.501.00

y/R

x/R

Mach number

0 5 10 15 20 25 30 35 401.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

x/R

37

(a) Mach number contours in plane of symmetry; Interval = 0.50.

(b) Centerline Mach number distribution.

Figure 27. Mach number distribution in underexpanded circular jet. Exit Mach number = 2.0, NPR = 11.12.

0.501.00

1.502.50 2.00 2.50

5 10 15 20 25 30 35 400

2

4

6

8

y/R

x/R

Mach number

0 5 10 15 20 25 30 35 401.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

x/R

38

(a) Turbulence intensity distribution in plane of symmetry; contours.

(b) Centerline turbulence intensity distribution.

Figure 28. Turbulence intensity distribution in circular jet computed by using single cell wedge grid. Exit Machnumber = 2.0; on-design NPR = 7.82.

5 10 15 20 25 30 35 40

2

0

4

6

8

0.02 0.040.06

0.080.10

0.12

y/R

x/R

ut′ /Ue

5 10 15 20 25 30 35 400

.02

.04

.06

.08

.10

.12

x/R

Turbulence intensity,

ut /Ue

39

(a) Turbulence intensity distribution in plane of symmetry, contours.

(b) Centerline turbulence intensity distribution.

Figure 29. Turbulence intensity distribution in circular jet computed by using three-dimensional adaptive grids. ExitMach number = 2.0; on-design NPR = 7.82 ; medium grid density:j, k = 40.

0.020.04

0.060.08

0.10

0.12

5 10 15 20 25 30 35 40

2

0

4

6

8

y/R

x/R

ut′ /Ue

5 10 15 20 25 30 35 400

.02

.04

.06

.08

.10

.12

x/R

Turbulence intensity,

ut /Ue

40

(a) Turbulence intensity distribution in plane of symmetry, contours.

(b) Centerline turbulence intensity distribution.

Figure 30. Turbulence intensity distribution in circular jet computed by using three-dimensional adaptive grids. ExitMach number = 2.0; NPR = 7.82; high density adaptive grid:j, k = 56.

0.020.04

0.060.08

0.100.12

5 10 15 20 25 30 35 40

2

0

4

6

8

y/R

x/R

ut′ /Ue

5 10 15 20 25 30 35 400

.02

.04

.06

.08

.10

.12

x/R

Turbulence intensity,

ut /Ue

41

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REPORT DOCUMENTATION PAGE

September 1996 Technical Paper

Numerical Simulation of Jet Aerodynamics Using the Three-DimensionalNavier-Stokes Code PAB3D WU 505-59-70-04

S. Paul Pao and Khaled S. Abdol-Hamid

L-17516

NASA TP-3596

Pao: Langley Research Center, Hampton, VA; Abdol-Hamid: Analytical Services & Materials, Inc., Hampton, VA.

This report presents a unified method for subsonic and supersonic jet analysis using the three-dimensional Navier-Stokes code PAB3D. The Navier-Stokes code was used to obtain solutions for axisymmetric jets with on-designoperating conditions at Mach numbers ranging from 0.6 to 3.0, supersonic jets containing weak shocks and Machdisks, and supersonic jets with nonaxisymmetric nozzle exit geometries. This report discusses computationalmethods, code implementation, computed results, and comparisons with available experimental data. Very goodagreement is shown between the numerical solutions and available experimental data over a wide range of operat-ing conditions. The Navier-Stokes method using the standard Jones-Launder two-equationk-ε turbulence modelcan accurately predict jet flow, and such predictions are made without any modification to the published constantsfor the turbulence model.

Jet; Computation; Fluid dynamics 43

A03

NASA Langley Research CenterHampton, VA 23681-0001

National Aeronautics and Space AdministrationWashington, DC 20546-0001

Unclassified–UnlimitedSubject Category 02Availability: NASA CASI (301) 621-0390

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