Table Of ContentProgress Report
on
NASA GRANT NAG 1-1776
JET AEROACOUSTICS: NOISE GENERATION
MECHANISM AND PREDICTION
Period covered by this report
January 1, 1998 to December 31, 1998
Principal Investigator
Dr. Christopher Tam
Department of Mathematics
Florida State University
Tallahassee, FL 32306-4510
NASA Technical Officer for this grant
John S. Preisser
Mail Stop 461
NASA Langley Research Center
Hampton, VA 23681-0001
JET AEROACOUSTICS: NOISE GENERATION MECHANISM AND PREDICTION
Progress Report
This report covers the third year research effort of the project. The research work
focussed on the fine scale mixing noise of both subsonic and supersonic jets and the
effects of nozzle geometry and tabs on subsonic jet noise. Technical results of the
research effort are reported in the following publications.
1. Tam, C.K.W. and Auriault, L. "Jet mixing noise from fine scale turbulence",
AIAA Paper 98-2354, to appear in the AIAA J. Feb. 1999.
2. Tam, C.K.W. and Zaman, K.B.M., "Subsonic jet noise from non-axisymmetric
and tabbed nozzles", AIAA Paper 99-0077, 1999.
(Copies of the above are attached to the end of this report.
In publication 1, a new semi-empirical theory of jet mixing noise from fine scale
turbulence is developed. By an analogy to gas kinetic theory, it is shown that the
source of noise is related to the time fluctuations of the turbulence kinetic theory.
On starting with the Reynolds Averaged Navier-Stokes equations, a formula for the
radiated noise is derived. An empirical model of the space-time correlation
function of the turbulence kinetic energy is adopted. The form of the model is in
good agreement with the space-time two-point velocity correlation function
measured by Davies and coworkers. The parameters of the correlation are related to
the parameters of the k-E turbulence model. Thus the theory is self-contained.
Extensive comparisons between the computed noise spectrum of the theory and
experimental measured have been carried out. The parameters include jet Mach
number from 0.3 to 2.0 and temperature ratio from 1.0 to 4.8. Excellent agreements
are found in the spectrum shape, noise intensity and directivity. It is envisaged that
the theory would supercede all semi-empirical and totally empirical jet noise
prediction methods in current use.
Publication 2 uses the two similarity spectra discovered by Tam, Golebiowski and
Seiner to demonstrate that the jet noise fields from circular, elliptic or rectangular
nozzles are nearly identical. Nozzle shape modification is not an effective way to
suppress jet noise. The effects of inserting tabs into a subsonic jet is also
investigated. A jetlets model is developed to correlate the noise from a jet with N
tabs to that of a single jet. The jetlets model predicts that the noise spectrum is
shifted to a higher frequency by a factor of N 1_2. This result is in excellent agreement
with experiments. This theory is the only theory on the noise of jets from tabbed
nozzles in the literature.
AIAA Paper 98-2354
Jet Mixing Noise from Fine Scale Turbulence
Christopher K. W. Tam and Laurent Auriault
Department of Mathematics
Florida State University
Tallahassee, FL 32306-4510
Paper to be presented at the 4th AIAA/CEAS Aeroacoustics
Conference, Toulous, France, June 2-4, 1998.
JET MIXING NOISE FROM FINE SCALE TURBULENCE t
Christopher K.W. Tam* and Laurent Auriault**
Florida State University
Tallahassee, FL 32306-4510
Abstract theories up to the late 1980's can be found in Ref.
[6]. Ref. [7] to [9] provide reviews on the same sub-
It isknown that turbulent mixing noisefrom
ject. However, they include some of the more recent
high-speedjetsconsistsof two components. They
works and the emphasis and perspectives are quite
arethe noisefrom largeturbulentstructuresinthe different.
form of Mach wave radiation and the less directional Recently, Tam, Golebiowski and Seiner l° sug-
fine scale turbulence noise. The Mach wave radia-
gested that, because there is no intrinsic charac-
tion dominates in the downstream direction. The
teristic length and time scales in the mixing layer
fine scale turbulence noise dominates in the side-
of a high Reynolds number jet (up to the end of
line and upstream directions. In this paper, a semi- the core region), not only the mean flow and the
empirical theory is developed for the prediction of turbulence statistics must exhibit self-similarity, the
the spectrum, intensity and directivity of the fine same must be true for the radiated noise. By ex-
scale turbulence noise. The prediction method is
amining the entire data bank (1900 spectra in all)
self-contained. The turbulence information is sup- of the Jet Noise Laboratory of the NASA Langley
plied by the k- E turbulence model. The theory Research Center, they found that turbulent mix-
contains three empirical constants beyond those of ing noise of high-speed jets consisted of two distinct
the k - e model. These constants are determined
components (see also Ref. [9]). Each component ex-
by best fit of the calculated noise spectra to exper- hibits self-similarity of its own. One component ra-
imental measurements. Extensive comparisons be- diates principally in the downstream direction. This
tween calculated and measured noise spectra over is consistent with Mach wave radiation from the
a wide range of directions of radiation, jet veloci- large turbulence structures/instability waves of the
ties and temperatures have been carried out. Excel- jet flow t1-13. The other component that has a rela-
lent agreements are found. It is believed that the tively uniform directivity is dominant in the sideline
present theory offers significant improvements over and upstream directions. These characteristics sug-
current empirical or semi-empirical jet noise predic- gest that it is the noise from the fine scale turbulence
tion methods in use. There is no first principle jet of the jet flow. Tam et al.1°succeeded in identifying
noise theory at the present time. two similarity spectra from the data. They demon-
strated that one of the spectra they found fitted the
1. Introduction
noise from the large turbulence structures and the
Sincethe pioneeringwork of LighthilIl'2,there other the noise from the fine scale turbulence regard-
have been numerous attempts todevelop ajetnoise less of the jet velocity, temperature and direction of
radiation. More recently, Tam 14 showed that even
predictiontheory.In the literatureo,ne findsmany
proposed theoriesand semi-empiricaltheories.The the noise spectra of non-axisymmetric jets including
main difficultiynpredictingjetnoiseisour lackof jets from rectangular, elliptic, plug and suppressor
nozzles fitted the same two empirically found simi-
understandingofthe turbulenceinjetflows.This is
trueeven today. However, many ofthe fundamen- larity spectra. This indicates that the noise sources
talphysicsofjetflowsthat affectthe propagation of these jets are similar to those of the circular jet.
and radiationofjetnoisewere recognizedveryearly. Their noise is also made up of two components.
They have been incorporatedintosome ofthe the- The primary objective of this work is to develop
ories.These effectsincludemean flowrefractio3n'4 a semi-empirical theory for the prediction of the fine
and sourcemotion 5.An excellentreviewonjetnoise scale turbulence noise from high-speed jets. The
present theory is self-contained. The turbulence in-
t Copyrigh©t1998byC.K.W.Tam. PublishebdytheCon- formation of the jet flow needed for noise prediction
federatioofnEuropeanAerospaceSocietiweisthpermission. are supplied by the k-e turbulence model. Since the
* DistinguishReedsearchProfessorD,epartmentofMathe- k - E model is a semi-empirical model, the present
maticsA.ssociatFeellowAIAA. theory is also semi-empirical. As will become clear
"*GraduatestudentD,epartmentofMathematics. later, the noise prediction formula developed here
containsthreeempiricaclonstantsT.heseconstants
on its surroundings. This pressure, following (1), is
aredeterminebdybestfit tothenoisedata.Once
equal to
theconstantsaredecidedt,heformulacanprovide
accuratenoisespectrumanddirectivitypredictions 2k
Pturb=q,=- P(v2) = 5P , (2)
overthejet velocityratiorangeof_ = 0.4to3.0
_loo
(uj and aoo are the fully expanded jet velocity and where k, = ½(v2) is the kinetic energy of the fine
ambient sound speed) and temperature ratio range scale turbulence per unit mass. The pressure given
by (2) is a macroscopic quantity valid for length scale
of _ -- 1.0 to 5.0 (Tr and Too are the jet reservoir
larger than the size of an individual blob of turbu-
and ambient temperature). This parametric range
brackets all known jet noise experiments and com- lent fluid. This pressure must be balanced by pres-
sure and momentum flux of the surrounding fluid.
mercial jet engine operating conditions.
The use of the k - _ turbulence model to predict If this pressure fluctuates in time, it will inevitably
jet noise is not new. Khavaran, Krejsa and Kim 15'16, give rise to compressions and rarefactions in the fluid
medium. This results in acoustic disturbances. Fol-
Bailly and coworkers 17-2° employed the k-e model
to provide turbulence statistics to their chosen ver- lowing this reasoning, one would expect the source
sion of the acoustic analogy theory for supersonic jet of fine scale turbulence noise to be equal to the time
noise prediction. Khavaran et al. tuned their empir- rate of change of Pturb or qs in the moving frame
of the fluid. In other words, oq, the convective
ical constants to fit the measured jet noise data at Dt '
Mach 1.4. However, even at this Mach number, the derivative of q,, is the noise source term in a turbu-
lent flow.
calculated spectra do not fit the measured data well
In section 2, the equations governing the gener-
at 90 degrees and in the forward direction for which
the fine scale turbulence noise dominates. Bailly and ation and propagation of acoustic disturbances in a
coworkers were more interested in using their model jet flow (generated by fine scale turbulence) are for-
to calculate Mach wave radiation from supersonic mally derived. Here we will begin with the Reynolds
jets. There is fair agreement between their com- Averaged Navier-Stokes equations (RANS). The re-
lationship between the radiated sound field and the
puted and measured directivity data but the calcu-
source of fine scale turbulence noise will be estab-
lated spectral distributions do not match well with
lished. In section 3, a semi-empirical model of the
experiments.
To develop a fine scale turbulence noise theory noise source space-time correlation function is de-
requires many steps. We will first turn to the ques- veloped. The parameters of the correlation function
tion of how fine scale turbulence in a jet generates are then related to the length and time scales of the
k-e turbulence model. By means of this source cor-
sound. For this purpose, we will use the gas kinetic
theory analogy. The use of gas kinetic theory to relation function, a formula for the spectrum of the
elucidate ideas in turbulence is well established _1. radiated noise is derived. Extensive comparisons be-
tween the calculated results of this formula and ex-
Consider the motion of gas molecules in a moving
frame fixed to the mean velocity _ as shown in fig- perimental measurements have been carried out. A
sample of these comparisons are reported in section
ure 1. Suppose m is the mass of a molecule, n is
4. Good agreements are found over both subsonic
the number density. Due to the random motion of
and supersonic/vlach number. Temperature depen-
the gas molecules, the gas exerts a pressure, p, on
dence over a wide range of jet to ambient tempera-
its surroundings. It is a simple matter to show, fol-
ture ratios is also correctly predicted.
lowing standard kinetic theory of gases 22'23, that p
is given by,
2. Formulation
1 1
Consider the flow of a turbulent compressible
(1)
p=-_mn(v.v)= p(v 2)
jet. A convenient way to account for the effect of
compressibility is to use Favre average variables 24.
where v is the random molecular velocity, p is the
For the present purpose, we may regard the average
density of the gas and ( ) is the ensemble average.
as a volume average 2s with a sharp Fourier cutoff
Now we will regard fine scale turbulence as small
filter; the volume being smaller than the large tur-
blobs of fluid moving randomly as shown in figure 2.
bulence structures of the jet flow but larger than
Again, let v be the random velocity of the fine scale
the fine scale turbulene. We will decompose a flow
turbulence measured in the mean flow moving frame.
variable into two parts,
By analogy to the gas molecules of figure 1, the fine
scale turbulence effectively exerts a pressure, Pturb, f = f+f" (3)
where f and f" are the Favre average and Favre is the unsteady pressure exerted by the fine scale
fluctuating components. On starting from the equa- turbulence. In (8), _i and -# are the mean veloc-
tions of motion, it is straightforward to derive the ity and density of the jet and ui, p are the acoustic
familiar Reynolds averaged Navier-Stokes equations field variables associated with kj. We will now sup-
(r_NS); e.g., see Ref. [26]. plement equation (8) with the linearized continuity
and energy equation (ignoring viscous and thermal
dissipations) to form a closed system of equations
-#\ ot + _s oxj ] (4) for the acoustic field. With respect to a cylindri-
cal coordinate system (r, ¢, x) centered at the nozzle
- --_x_ t- _(90' (_pU, Uj q__,j) exit (with the x-axis coinciding with the jet axis),
the full set of governing equations are,
where an overbar denotes a Reynolds averaged quan-
tity. The terms _ij and -pu" IiI ujII on the right side of -#[a-_=+_+ Ou vad, -]J +_=Op OO_q, (lOa)
(4) are the viscous and Reynolds stresses, respec-
tively. On following the generally accepted RANS
approach, we will adopt the Boussinesq eddy viscos- -#[O-6rT+_U_Ov]O+p a--;= Oaq_, (lOb)
ity model for the Reynolds stresses; i.e.,
lo"w_- + _ _O'zw] + lrap0¢ - lr a0q¢, (lOc)
-pu_'u_ = 2pt Sij - A$ij - "_-#k,_ij (5)
OOp'-7+ _uO-_p= + 7_ [! Oo(v-r--)7lo-w+ - + Ou]
where #t is the eddy viscosity and
=0. (lOd)
S_j = _ \ Ozj + Ox_) ' In these equations (u, v, w) are the velocity compo-
nents in the (x, r, ¢) directions. Also, in accordance
-pu_'u_' = 2-#k,. (7)
of the locally parallel flow approximation, p, u are
regarded as functions of r alone and _ = p=_ is con-
Of importance is that k, as defined by (7) is the
stant.
kinetic energy of the fine scale turbulence per unit
mass. 2.1. The Acoustic Field
As discussed in the previous section, k, in (5)
effectively represents a pressure field exerted by the Equations (10a) to (10d), except for the noise
fine scale turbulence on the surrounding fluid. The source terms on the right, are identical to the Lil-
time fluctuation of this pressure field causes com- ley's equation 4'6. As is well known, the Lilley's ap-
pressions and rarefactions in the fluid medium. In proach was designed to account for the mean flow
this way, sound is generated by the small scale tur- refraction effect. This effect is especially significant
bulence. Inside the jet flow, the acoustic field is a for the high frequency part of the noise spectrum.
very small part of the unsteady fluctuations. It is This is confirmed in the numerical results reported
sufficient to consider the acoustic field generated by in section 4.
the time dependent part of ks, denoted by k,, to A formal solution of (10) directly relating the
be given by the linearized form of equations (4) and radiated acoustic pressure field to the fine scale tur-
(5). The linearization is to be performed over the bulence intensity qs can be found by using the space-
mean flow of the jet. In addition, both molecular time Green's function of these equations. Since q,
and eddy viscosity terms will be ignored as they will appears on the right side of equations (10a), (10b)
have only a relatively small effect on the acoustic and (10c), three Green's functions are necessary. Let
disturbances. The linearized form of (4) with only (u,_,vn,w,_,p,_) with n = 1,2 and 3 be the Green's
k, terms retained on the right side, after applying functions corresponding to a source on the right
the locally parallel mean flow approximation, is side of equations (10a), (10b) or (10c), respectively.
These Green's functions satisfy the following nonho-
[Oui Oui O_j ] Op _ Oqs (8) mogeneous equations.
rOuo ou° Op°
where -#LOt + 5-_-= + v"_r . + O--T
2__ (9)
q,=-_p o ----6(x-- Xl)6(t --tl)$nl (11)
av.1 functions are related to the adjoint Green's function
+ OxJ + a---_- by2T,
(12) _(x, xl,,.) = _.(x_,x,_)
[Own OOw,,] loop. ,_2(xx,_,,o)= v,_(Xlx,,,,.,) (20)
_;(x,xl,,,,) = w,,Cx_x,,_).
= ,_(x- x_)_(t- q)_.3 (13)
ouo1
IOOWn It is worthwhile to point out that the arguments on
OO-+-5-i + "_-P Or + ;-_-+ oo=j
the two sides of equation (20) satisfy the reciprocity
=0 (14) relation. That is, the source point and the field point
of the adjoint are interchanged. It is clear from (20)
where n = 1,2, 3. Here 5( ) is the Dirac delta func- that there is another important advantage in using
tion and 5nm is the Kronecker delta. The Green's the adjoint Green's function. If one is interested in
functions depend on four variables. They are the the far acoustic field, namely, the pressure field, it is
field or observer coordinates, x, the source coordi- only necessary to solve the adjoint equations (16) to
nates, xt, the field or observation time, t, and the (19) once to obtain (ua, va, wa). On the other hand,
source time, tl. To avoid confusion, whenever it is to find P'I, P2 and P3 directly one has to solve the
necessary to display the arguments of the Green's original time harmonic Green's function equations
function, they will always be in exactly this order; three times.
e.g., p.(x, xl,t,tl). By means of the adjoint Green's function, the
The space-time Green's function is related to the pressure field generated by the source terms on the
time harmonic Green's function (un, v,, w, _',,) by right side of (10a) to (10d) is formally given by,
the Fourier inverse transform. For example, we have
c_ *lx"l:-ffffff[
p,(x, xl,t,tl) = f _,_(x, xl,w)e-i_(t-t')dw (15)
mC_
, .ooq.(x_,tl)
+ v,4xl, x,w) 0r---_
The equations governing the time harmonic Green's
l
functions are simply the Fourier transforms of (11) w,_(xl, x, w) OOq,(Xl,
+
to (14). Recently the present authors 27 have shown r 0¢ 1
that there is great computational advantage to use
•e-i'o(t-tOdwdtldXl.
the adjoint time harmonic Green's function instead
(21)
of the time harmonic Green's functions. On fol-
o o +o_
Let _71 - (_--_x_ex +o'_w ey ozt ez) be the gradient
lowing the work of Ref. [27], it is easy to find
with respect to the xl coordinates. (21) may be
that the adjoint time harmonic Green's function
rewritten in the form,
(ua,v_,w_,p_) (subscript a indicates the adjoint)
satisfies the following equations.
pcx,,,-:f//f/f (v.(x,,
[ oouo1 x,w)qs(xl,Q,)
--_ iwu,,+ ooxj--_,p-_z--z =0 (16)
_CX_
Ova _ Op_ -q, Cxl,tl) Vl .v, Cxz,x,w)] (22)
-_ i_.+_ oo= _=° -_p-b-T=o (17)
•e-i_o(t-tOdwdtldX_
oowlo _pOpo_ o. (is)
--_ iww. + _ Oz J r 0¢
where v. = (u,, va, wo). This formulacan be simpli-
fied in two ways. First, we can apply the divergence
- i_p_ - _-52=- o--7- + ; + o_ ]
theorem to the first term of the integrand. This con-
verts the volume integral dXl to a surface integral
= 18(x- xx) (19) outside the jet flow. But qs is zero outside the jet.
Thus, there is no contribution from the first term.
It is straightforward to show that the pressure Now for v, (Xl, x,to), the source is outside the jet flow
fields of the three original time harmonic Green's if x is in the far field. Thus by making use of (19),
4
(22)becomes,
By using f exp[i(w- w2)r]dr = 21r6(w- w2), it
--OO
vCx,t) is straightforward to derive from (24), the following
formula for S(x, w).
x outside the jet
oo
•pae-i°dt-t_)dw] dtldXl
=_ffffff -- CX_
-- 00 •<Dqs(xl,tl)DDtql,(xi,tl)D>tl (26)
•pae-i_(t-t*)dwJ dt 1dxl • •e-i(wl+u_2)t+iuJiti+i_2t2
Finally, by simple integration by parts, we find the •6(w - w2)dwlaw2dtl dt2d xl dx2.
desirable formula for p(x, t),
.<.,,SSYSS
3. Model Space-Time Correlation Function
To proceed further, we need a mathematical rep-
x outside the jet (23)
resentation of the noise source space-time correla-
I e_i_(t_t,)dw] JDqs (xl, tl) .. , tion function /\ ncl,(XDl,ttl 0 Dq,(DXr2_,t2) _." This correla-
_11 atlaXl tion function has never been measured before. How-
where n o 0 ever, two-point space-time correlation of the fluctu-
= _ + _ _ is the convective derivative
ating axial velocity component in jets has been mea-
following the mean flow.
sured and studied by Davies, Fisher and Barrett 2s
Equation (23) is the main result of this section.
and Chu 29. We believe that the mathematical form
Obviously, it is open to the interpretation that noise
from fine scale turbulence is generated by the time of ,,[Dq'Dq'\DDt_r21 should be similar to that of the mea-
sured two-point space-time correlation function of
rate of change of the turbulence kinetic energy or
the axial velocity component. Here we propose to
pressure in the moving frame following the convec-
adopt a model space-time correlation function char-
tion of the fine scale turbulence by the mean flow.
acterized by three parameters• We will show that
The term inside the square bracket is the adjoint
Green's function with the source located at x and with an appropriate choice of the parameters, the
model function fits the measured function of Davies
the field point at xl according to the reciprocity re-
lation. et al. (see figure 3) well.
Let _ = zl-z2, 0 = Yl -Y_., ¢ = zl - z_,
r = tl - t2. We will consider the following model
2.2. The Spectral Density of the Radiated
function
Sound
By means of (23), the autocorrelation function
< Dq_(xi,tl) Dq_(x2,t2) >
for a point in the acoustic far field may be formed; Dr1 Dr2 -- c2r_ (27)
i.e., •e- _- '%'t-(._-_')'+"_+¢'l
co
<.<.,,,.<.,,f+f .S,>f-.o(.,,x,.,,
In (27) _ is the mean flow velocity or the transport
velocity of the fine scale turbulence. The three pa-
--CO
rameters of the model are gs, rs and _s. ts is the
(24)
•Pa(X2 x,w.) -<Dqs(xi,t,) Dqs(xl,t2) > characteristic size of the fine scale turbulence in the
' D_ 1 Dt 2
moving frame of the mean flow. rs is the character-
. e-iWa(t-tt)-iw2(t-t2)-iw_rdwld_2dtldt2dXl dx2 •
istic decay time. Kr, is the decay distance. In the
In (24), ( ) is the ensemble average. limit _, r/,_, r -+ 0, (27) becomes
The spectral density of the radiated sound,
S(x,w) is the Fourier transform of the autocorre-
lation function or </Dq.____x,t))l) __ cg_''rqs_s (28)
oo
if
Thus _'_is a measure of the RMS value of the fluctu-
S(x,w) = _ (p(x,t)p(x,t+r))ei_'dr. (25)
ating kinetic energy of the fine scale turbulence and
crs represents a typical time scale of the fluctuation. model includes contributions from the large turbu-
The coefficient c is expected to be less than 1.0. This lence structures whereas l_ and rs are those of the
indicates that the fluctuating time is shorter than fine scale turbulence alone. On accounting for this
the turbulence decay time. difference, we propose to let,
Figure 3 shows a plot of the correlation func- k] k
& = cle = ct--, r, = cTr = cT- (30)
tion (27) with es = 0.1758 inches, r, = 447.4# sec.
and _ = 3.515 x 10-3 inch/# sec. As can be seen,
where ct and c_ (ct,cr < 1.0) are constants to be
the model function is in reasonably good agreement
determined empirically.
with the measurements of Davies et alfl s. This re-
In the above, we have noted that _'_ is a mea-
sult assures us that the model space-time correlation
sure of the intensity of the fluctuation of the kinetic
function does have the right functional character-
energy of the fine scale turbulence. _Ve, therefore,
istics. We will use this model correlation function
expect it to be proportional to q. (q = ]_k.) We
throughout this work.
will let
Model correlation function (27) has three pa-
rameters, namely, _,, r8 and _'_. They are the char- _- A2q2 (31)
acteristic parameters of the fine scale turbulence of
where A is the third empirical constant.
the jet flow. Here we propose to obtain the values
With (30) and (31), the two-point space-time
of these parameters through the k- e turbulence
correlation function (27) is known save for the three
model.
constants ct, cr and A. These constants are to be
It is generally known that the standard k - e determined by best fit of the calculated noise spectra
turbulence model does poorly in predicting the mean
to experimental measurements.
flow of jets 3°. Thies and Tam 31recognized the prob-
lem of applying the standard k-e model to jet flows• 4. A Formula for the Noise Spectrum
They proposed to modify the k-e model coefficients
and showed convincingly by comparison with a large On substitution of (27) into (26), the noise spec-
set of jet flow data over the jet Mach number range trum in the far field is given by
of 0.4 to 2.0 and temperature ratio of 1.0 (cold jet) co -,--2,
to 4.0 that their modified k- e model is reliable and
S(x,w) = • a(Xl,X, Wl)Pa(X2,X,_2) c2v ?
accurate. In this work, the modified k - _ model of
--co
Ref. [31] is used.
•e- _ - _'_[,(__'-_-_(t, =t_))_+(_,,-_)_+(_,-,_)_1
The k - e model provides only two pieces of in-
formation about the turbulence of the jet flow. They •e-iCwt+_')t+iw'tt+iw't" (32)
are k, the averaged turbulence kinetic energy, and e,
•_(_; - ta2)dtldt2dca2dw2d xxdx2
the dissipation rate. But with k and e known, it is
We will now show that most of the above integrals
possible to form a length e, characterizing the size
can be evaluated analytically. The entire expression
of the small scale turbulence and a decay time, r, of
can be reduced to a single volume integral over the
the turbulence as follows,
jet flow.
k-] k The first step is to integrate dt 1. The integration
e=--, _-=-. (29) can best be carried out by first making a change of
g
variable to s where
The parameters gand r of (29) are directly rele-
8 = (t 1--t2) -- (xl --x2)
vant to the parameters _ and r, of model two-point
space-time correlation function (27). To establish Next, the t2 integration can be performed resulting
their relationship, we performed a k - e model cal- in 2zr_(wl + w_). With _(wl + w_) and (f(w - w2) in
culation for the jet flow of the Davies et al. ex- the integrand, the wl and w_ integrals can be readily
periment. At the location of the measured correla- evaluated giving
tion function shown in figure 3, the value oft and r co .--2
obtained were very close to 0.176 inches and 447#
S(x,_) = 2_ i-_ "
sec., respectively. These are the values of i, and r,
--OO
used to match correlation function (27) to the mea-
_2l:2
surements. We, therefore, believe that t_ and r_ of •pa(xl, x, -w)Pa (x_, x,w)e- _- _ (33)
our model correlation function is directly related to
and 7- of the k-e model. However, the k-e •e- tt_2[(r/l-y_)2+(zt-z2./2l -._ dxl dx_.
