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1
Chapter 1
Introduction
Performing experimental studies of jets is often expensive and difficult. The problems move
up a notch especially if supersonic jets are involved in those experiments, as it is usually
difficult to generate perfectly expanded supersonic jets. Even a minor under/over estimation
of the boundary layer displacement thickness while designing the nozzle could result in non-
ideal expansion. Apart from that, there are several other parameters like the plenum chamber
pressure, etc. that need to be controlled carefully while performing experiments with
supersonic jets in order to achieve near perfect expansion. Even analytical approaches have
their short comings due the steady-state and low Reynolds number assumptions that often
accompany them. Computational Fluid Dynamics (CFD) and Computational Aero Acoustics
(CAA) have proved to be excellent alternatives for jet flow and noise simulations.
Prior to the last decade, most of the turbulent flow simulations were carried out using
Reynolds Averaged Navier-Stokes (RANS) equations. The most commonly used turbulence
models for RANS simulations of jet flows are the two equation k-ε and k-ω models. RANS
equations are obtained by time averaging the Navier-Stokes equations and the Reynolds
stress tensor is modeled using the Boussinesq approximation where the tensor is given by the
product of an eddy viscosity and the strain rate tensor. Also, isotropic turbulence is assumed
in RANS approach. Due to the inherent approximations in the method, RANS approaches are
limited in their application for simulating turbulent jets, especially in aero-acoustics analysis,
where both the mean flow and the time history of the flow parameters are necessary for
adequate assessment of jet noise. There have been considerable improvements in the
turbulence modeling for RANS approach like, the work of Sarkar [1] to account for the
effects of compressibility, the k-ε model proposed by Thies and Tam [2], Pope's [3] vortex
stretching correction for improved prediction of round jets, the Chien k-ε model and the
famous two layered Menter SST (Shear Stress Transport) model [4] that switches from a k-ω
model to a k-ε model in regions outside the shear layer. But, in general, the results obtained
from RANS simulations are only the time average of the flow field. Hence, RANS approach
is inadequate for the prediction of turbulent jet flow features and jet noise.
The most straight-forward method to simulate a turbulent flow field is to perform Direct
Numerical Simulation (DNS). In DNS, the Navier-Stokes equations are solved in a time
accurate manner without any approximations, but in order to resolve the turbulence
accurately, in both space and time right down to the Kolmogorov scale, the grid spacing must
be no larger than the Kolmogorov scale. Since, the Kolmogorov scales are a function of thre
turbulent Reynolds number (Ret), one would require a total of Ret9/4
grid points to perform the
simulation and the cost of simulation is of the order of Ret3
. The Reynolds number limitation
can be overcome to a certain extent by using higher order numerical schemes which can
capture the small scale features using fewer grid points, but this also increases the
computational cost. With DNS the size of the computational grid that can be used is restricted
2
by the available computer memory and since the time step is proportional to the grid spacing,
the number of time steps possible in the simulation is restricted by the processing speed of the
computer. Hence, most of the DNS data available in current literature is for low Reynolds
number jets and these are very small compared to nozzles of practical use.
While RANS simulations are the cheapest in terms of computational cost, DNS is the
costliest. Large-Eddy Simulation (LES) is kind of a compromise between RANS and DNS.
Due to the growth in computing power over the last decade, LES has become very popular
among the CFD community for turbulent flow simulations. LES has enhanced accuracy
compared to RANS simulations as it resolves the large scale turbulent fluctuations directly
which carry most of the energy and momentum, while the unresolved small scales which are
isotropic and dissipative are modelled using eddy viscosity or sub-grid scale models. In LES
the Navier-Stokes equations are spatially filtered which results in equations with resolved
(large scale) and unresolved (small scale) terms. These unresolved or sub-grid scale terms in
the filtered equations are then modelled. Smagorinsky's eddy viscosity model [5] is the
simplest sub-grid scale model available. Other models are based on the works by Moin et. al.
[6], Vreman et. al. [7] and the very successful dynamic sub-grid scale eddy viscosity model
developed by Germano et.al. [8] where the model coefficient is computed dynamically based
on the local flow conditions as the simulation progresses. Debonis [9] also suggested that, as
in the case of DNS, numerical schemes with order of accuracy greater than two are necessary
for LES in order to resolve the large scale structures properly and efficiently.
Using axisymmetric Navier-Stokes equations complemented with a simplified LES model,
Loh et al. [10] obtained the instantaneous and time averaged flow fields for an under-
expanded screeching jet. They implemented the Conservation Element and Solution Element
(CE/SE) numerical scheme for investigating the near field screech noise of an under-
expanded axisymmetric jet and their results are in good agreement with the experimental
data. DeBonis [9] also used LES along with Smagorinsky's subgrid scale model (for the eddy
viscosity) [5] to simulate a turbulent compressible circular jet and later he and Scott [11]
further analyzed turbulent compressible round jets using sub-grid scale LES model. The
advantage with LES is that, it provides the time history of turbulent fluctuations which is
required for unsteady analysis. So, it has proved to be a promising tool for turbulent flow
simulations and jet noise predictions with its ability to capture the nonlinear sound generation
process of the jet near field.
In the past there have been investigations regarding jet noise sources and generation
mechanisms for both subsonic and supersonic jets by Tam [12], Hileman and Samimy [13],
Kastner et. al. [14], Arndt et. al. [15] and many more, but their works are mostly experimental
and/or theoretical. Freund [16], Kastner et. al. [14] and Bogey et. al. [17] performed noise
investigations computationally, but those studies were for subsonic jets. Tam and Auriault
[18] developed a semi-empirical theory for predicting fine scale turbulence noise from
supersonic jets using RANS simulations with a k-ε turbulence model and Bailly et. al. [19]
computed the radiated acoustic fields of subsonic and supersonic jets using a combination of
k-ε turbulence closure with three acoustic analogies. Recently, Bogey and Bailly [20] used
3
LES to compute the radiated noise of a Mach 0.9 jet. The fact that LES being an appropriate
tool for investigating turbulent flows and noise sources and the lack of appreciable amount of
information in current literature regarding attempts at locating dominant noise sources in
supersonic jets numerically, has been the motivation for the present work.
The current work utilizes an axisymmetric Navier-Stokes solver based on a fifth order
accurate low dissipation WENO finite difference scheme prescribed by Shu [21] and
developed by Dharani [22]. This solver has an optimized second order TVD Runge-Kutta
time stepping [23] and to implement LES to this solver, the unsteady compressible
axisymmetric Navier-Stokes equations are Favre-filtered [9]. A sub-grid scale model based
on the works of Moin et. al. [6] and Vreman et. al [7] has been used for closure.
In the present work, the stream-wise location of the dominant noise source of three ideally
expanded supersonic jets with Mach numbers, M = 1.3, 1.4 and 2 has been determined using
the near-field pressure model developed by Arndt et. al. [15] and derived from the point-
source solution of the spherical wave equation. Spectral analysis of the near-field pressure
signal has been performed at various stream-wise locations for locating the noise source. The
near-field pressure model has previously been used by Sharma and Murugan [24] to locate
the dominant turbulence mixing noise source of a Mach 0.8 jet from an elliptical nozzle.
4
Chapter 2
Supersonic Jet Noise and Noise Source Location
2.1 Introduction
Supersonic jets used for practical applications are mostly imperfectly expanded with a quasi-
periodic shock cell structure existing in their plumes. So, unlike subsonic jets which radiate
turbulent mixing noise alone, supersonic jets radiate additional noise apart from noise due to
turbulent mixing, which are the two shock-associated noise components. One of which has
discrete frequencies and is commonly referred to as the "screech tone" and the other
component is broadband like the turbulent mixing noise, and is known as the "broadband
shock-associated noise". These make supersonic jet noise distinctly different from that of
subsonic jets. In the present investigation, all the supersonic jets under study are ideally
expanded and hence, do not involve the two shock-associated noise components, but only the
turbulent mixing noise.
2.2 Turbulent Mixing Noise
Jet flows contain turbulent structures ranging from fine to large scale. Both the fine and large
scale turbulent structures contribute to the turbulent mixing noise. This is evident from the
works of Tam [25], Goldstein [26] and several others. However, the magnitude of the noise
generated by these scales depends primarily on two factors, viz. the Mach number of the jet
and its temperature. Subsonic jets usually have low turbulence convection Mach numbers
(relative to the ambient speed of sound) unless the jet temperature is very high, due to which
the large turbulent structures are incapable of efficient noise generation and the dominant part
of subsonic jet noise is produced by the fine scale turbulence.
On the contrary , supersonic jets have large turbulence structures propagating downstream at
high convective Mach numbers (sometimes even supersonic with respect to the ambient
sound speed) which makes the large scales efficient noise generators, responsible for the
dominant part of the turbulent mixing noise, while the fine scale turbulence produces the
background noise. And when the convection Mach number of the large turbulent structures
becomes supersonic relative to the ambient speed of sound (as in the case of high Mach
number and/or high temperature supersonic jets), they radiate Mach waves which easily
predominates over the noise from the fine scale structures [12]. Thus, the large turbulence
structures are the dominant noise source of supersonic jets and experimental evidence
available from the works of Tam [27] and others has proved this. Later, Tam and Chen [28]
proposed the successful stochastic instability wave model to predict the noise from
supersonic jets, generated by the large turbulence structures which was widely accepted and
further extended.
5
2.3 Near-Field Pressure Model
It is possible but not efficient to carry the Large-Eddy Simulation to the acoustic far-field to
compute the noise directly. Hence, most computational strategies for noise estimations or
locating noise source use LES to simulate the jet near-field and employ acoustic analogies
like the Lighthill's integral solution or the Kirchhoff's formulation to compute the radiated
acoustic far-field. But, such strategies involve a two stage process for locating noise sources
and can be tedious.
The near-field pressure model proposed by Arndt et. al. [15] provides a simpler way of
locating the dominant noise source in free shear flows. The model was developed with the
objective of displaying the essential features of spatial decay and spectral variations. The
spherical wave equation's point-source solution which is consistent with turbulent free shear
flows has been used. Lighthill [29] proved that the solution of the spherical wave equation
must have a quadrupole character in turbulent free shear flows as there can be no sources of
mass and no unbalanced forces. Since, only a single noise source is considered, this solution
does not model the magnitude of the pressure fluctuations of a turbulent jet.
The relationship between pressure and velocity in an irrotational flow is given by the
unsteady Bernoulli equation .
𝑃 − 𝑃∞
𝜌=
𝛿𝜑
𝛿𝑡−∇𝜑.∇𝜑
2 (2.1)
where, ϕ is the velocity potential and P∞ is the pressure far from the flow. Arndt et. al. [15]
further stated that the pressure fluctuations governed by equation 2.1 can be divided into two
parts: The propagating or acoustic fluctuations that occur far away from a source in the far-
field and the non-propagating or hydrodynamic fluctuations that occur near the source in the
near-field. Arndt et. al. also proposed that, while the convective term in the Bernoulli
equation is usually neglected in the acoustic approximation, this approximation which is valid
in the far-field also appears to be valid in the near-field, provided one does not get too close
to the noise source. It is also assumed that the turbulent shear flow is composed of a finite
number of individual sources.
The reader is advised to refer [15] for a detailed description of the near-field pressure model,
where the appropriate velocity potential solution provided by Morse and Ingard [30] is used
to arrive at the solution for the mean-square pressure (intensity), I using the boundary
condition for an axial quadrupole. The final expression for the intensity I is
𝐼 = (𝑃 − 𝑃∞)2
𝜌0 𝑎0= 𝜌0𝑎0𝑈0
2(𝑘𝑅0)2[𝑅0
𝑟]6{
2 + 2𝑖𝑘𝑟 + (𝑖𝑘𝑟)2
𝐵}2 (2.2)
where, 𝐵 = 6 − 3(𝑘𝑅0)2 + 𝑖 6𝑘𝑅0 − 𝑘𝑅0 3 (2.3)
6
k is the wavenumber, 𝜌0𝑎0 is the acoustic impedance, 𝑈0 is the acoustic source velocity, 𝑅0 is
the source size and 𝑟 is the distance from the source.
The product (𝑘𝑅0) is a constant as it is assumed that long wavelength disturbances are
associated with large sources and short wavelength disturbances are associated with small
sources. The value of this constant is unknown and since, we are interested in the variations
in the intensity rather than the absolute magnitude, its precise value is not required. Figure 2.1
is an example of a near-field pressure spectrum which shows four distinct regions, viz. Low
wavenumber, Energy containing, Inertial subrange and Far-field.
Figure 2.1. An example of near-field pressure spectrum measured by Arndt et. al. showing all
four spectral regions [15].
In equation (2.2), if the product of wavenumber and distance becomes large (𝑘𝑟 ≫ 1), then
the intensity shows far-field behavior and spatially decays as 𝑟−2. This means
𝐼 ∝ 𝜌0𝑎0𝑈02(𝑘𝑟)−2 (2.4)
and the wavenumber can be converted into frequency using the relation
𝜔 = 𝑎0𝑘 (2.5)
which implies that the intensity wil have a spectral decay of 𝜔−2.
7
If the product of wavenumber and distance becomes small (𝑘𝑟 ≪ 1), provided (𝑟 > 𝑅0) then
the intensity shows near-field behaviour and from equation (2.2)
𝐼 ∝ 𝜌0𝑎0𝑈02(𝑘𝑟)−6 (2.6)
Upon inspection of preliminary velocity data by Arndt et. al. [15], they observed that in the
energy containing region where the spectral level is relatively flat, at a constant wavenumber,
the pressure intensity has a spatial decay of
𝐼 ∝ 𝑟−6 (𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) (2.7)
while in the inertial sub-range, for a constant distant 𝑟, the intensity has a spectral variation of
𝐼 ∝ 𝑘−6.67 (𝑟 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) (2.8)
Figure 2.2 shows the experimental arrangement used by Arndt et. al. [15] to verify the near-
field pressure model. In their measurements, a microphone was traversed radially outwards
from the edge of a jet, at a certain stream-wise location from the nozzle exit. The normalized
pressure spectra of the measured pressure signal was then obtained for each radial location
with the frequencies converted into acoustic wavenumbers using equation (2.10). Figure 2.1
represents the normalized spectrum of pressure with the product of acoustic wavenumber and
distance (𝑦) on the abscissa. Theoretically, y should be the distance between the face of the
microphone and center of the mixing region. But, in their measurements, it was found that the
center of the mixing region was approximately equal to the nozzle lip line. In the present
work too, y has been defined as the distance between the nozzle lip line and the point of
interest outside the jet edge.
Figure 2.2. Experimental setup used by Arndt et. al. to test the near-field pressure model [15].
8
When the normalized pressure spectra for different radial points at a given stream-wise
location are plotted together, it enables one to look at the spatial decay of the intensity (I) at a
constant wavenumber or constant Strouhal number (StD), where
𝑆𝑡𝐷 =𝑓𝐷
𝑈𝑒 (2.9)
where, f is the frequency, D is the nozzle exit diameter and Ue is the jet velocity
and, 𝑘 =2𝜋𝑓
𝑎0=
2𝜋𝑆𝑡𝐷𝑈𝑒
𝐷𝑎0 (2.10)
In the studies conducted by Arndt et. al. [15], two axial location were chosen, x/D = 1.5 and
x/D = 2.25 for the subsonic jets that they studied. Their normalised spectra plot for x/D = 1.5
is shown as an example in figure 2.3 below.
Figure 2.3. Near-field spectra plots for a low Mach number jet at x/D = 1.5 obtained by Arndt
et. al. [15]
Each solid curve in figure 2.3 represents the normalized pressure spectrum at a different
radial location outside the mixing region. The topmost curve is for the radial location nearest
to the nozzle lip line, while the subsequent curves are for radial locations corresponding to
increasing distance between the microphone face and the nozzle lip line. It is clearly evident
from figure 2.3 that, as the microphone is traversed radially outwards, the magnitude of the
peak intensity decreases and the spectra become more broadband. Each symbol on these
curves represents a constant Strouhal number (St) whose values are also provided in figure
9
2.3. In the energy containing region, a best fit line through the data points or constant
Strouhal number symbol corresponding to StD = 0.5 (which also represents the peak intensity
of each spectrum), exhibits a (ky)-6
decay which is consistent with the (y)-6
behaviour for
constant (k) (same as equation 2.7). Also, it can be seen from figure 2.3 that, in the low
wavenumber region, the slope through any of the data points (constant StD symbol) on each
curve is less than the (y)-6
spatial decay rate as expected and in this region, for lower and
lower wavenumbers, the spatial decay rate in terms of (y) becomes flatter and flatter.
In these experiments, the exact location of the noise source was unknown. Arndt et. al. [15]
postulated that, had they known the exact noise source location, the correct value of (y) could
have been used to obey the (r)-6
law. To this, a reviewer pointed out that, the reverse
statement is also true. With the assumption that the (r)-6
law is valid, the exact downstream
location of the noise source can be predicted. This gives us a unique and simple method to
determine the stream-wise location of the noise source. In the present work, the same concept
has been employed to predict computationally, the stream-wise location of the dominant
noise source in three ideally expanded supersonic jets.
10
Chapter 3
Large-Eddy Simulation
3.1 Filtering
In Large-Eddy Simulations the large scale turbulent fluctuations are resolved directly, while
the small scale turbulent fluctuations remain unresolved and have to be modeled. The
separation of the large and small scales is done by filtering the equations of conservation of
mass, momentum and energy. For this purpose a spatial filter G with a filter width ∆ is used.
𝑓 = 𝐺 𝑥 − 𝜉 𝑓(𝜉)𝑑𝜉∞
−∞
(3.1)
The overbar in equation 3.1 represents the resolved, filtered or large scale portion of the
function. In the present work, the solution is not explicitly filtered and it is assumed that the
numerically computed solution is a filtered representation of the exact solution. This
assumption has been justified by DeBonis [9] based on the results of his Fourier analysis,
where the numerical schemes and solution filtering behaved like a spectral cut-off filter of
the exact solution.
For convenient recovery of the terms corresponding to the unfiltered equations, Favre
weighting (density based weighting) is used in the filtering process. The Favre weighted term
is represented by the tilde as shown in equation 3.2.
𝑓 = 𝜌𝑓
𝜌 (3.2)
3.2 Axisymmetric form of Governing Equations
Jets from circular nozzles have been simulated in the present work and since, the geometry of
circular nozzles is symmetric about their centerlines, the axisymmetric form of the Navier-
Stokes equations have been used to reduce the amount of computer memory, storage and
CPU time. The axisymmetric form of Navier-Stokes equations assumes that there are no
velocity components or gradients in the circumferential direction of the cylindrical co-
ordinate system.
11
The filtered form of the continuity, momentum and energy equations comprise of resolved
(large scale) and unresolved (small scale) terms. These resolved terms correspond directly to
the unfiltered equations in form and the unresolved or sub-grid terms are modeled. The final
form of the modeled governing equations (in vector notation) as per cylindrical co-ordinates
(x,r) is
𝑈𝑡 + 𝐹𝑥 + 𝐺𝑟 + 𝑄 = 𝐹𝑥𝑣 + 𝐺𝑟
𝑣 + 𝑄𝑣 (3.3)
where
𝑈 =
𝜌 𝜌 𝑢 𝜌 𝑣 𝜌 𝑒𝑡
(3.4)
the inviscid fluxes in axial (x) and radial (r) directions are
𝐹 =
𝜌 𝑢
𝜌 𝑢 2 + 𝑝 𝜌 𝑢 𝑣
(𝜌 𝑒𝑡 + 𝑝 )𝑢
, 𝐺 =
𝜌 𝑣 𝜌 𝑢 𝑣
𝜌 𝑣 2 + 𝑝 (𝜌 𝑒𝑡 + 𝑝 )𝑣
(3.5)
and viscous fluxes in axial and radial directions are
𝐹𝑣 =
0𝜏 𝑥𝑥𝜏 𝑥𝑟
𝑢 𝜏 𝑥𝑥 + 𝑣 𝜏 𝑥𝑟 − 𝑞 𝑥
, 𝐺𝑣 =
0𝜏 𝑥𝑟𝜏 𝑟𝑟
𝑢 𝜏 𝑥𝑟 + 𝑣 𝜏 𝑟𝑟 − 𝑞 𝑟
(3.6)
where “ ̃ ” and “ ̄ ” represent Favre filtered and spatially filtered quantities, respectively. The
axisymmetric source terms Q and Qv are
𝑄 =
𝐺1
𝑟𝐺2
𝑟𝐺3
𝑟−
𝑝
𝑟
𝐺4
𝑟
, 𝑄𝑣 =
𝐺1𝑣
𝑟
𝐺2𝑣
𝑟
1
𝑟
2
3 𝜇 +𝜇 𝑡 ∇.𝑈 +2 𝜇 +𝜇 𝑡
𝑣
𝑟+ 𝜑 +
𝐺3𝑣
𝑟
𝜖 + 𝐺4𝑣
𝑟
(3.7)
12
where the different terms appearing in the above equations (3.5 - 3.7) are given as follows
∇.𝑈 = 𝛿𝑢
𝛿𝑥+
𝛿𝑣
𝛿𝑟+
𝑣
𝑟 (3.8)
𝜏 𝑥𝑥 = −2
3 𝜇 + 𝜇𝑡 ∇.𝑈 + 2 𝜇 + 𝜇𝑡
𝛿𝑢
𝛿𝑥 − 𝜑 (3.9)
𝜏 𝑟𝑟 = −2
3 𝜇 + 𝜇𝑡 ∇.𝑈 + 2 𝜇 + 𝜇𝑡
𝛿𝑣
𝛿𝑟 − 𝜑 (3.10)
𝜏 𝑥𝑟 = 𝜇 + 𝜇𝑡 𝛿𝑣
𝛿𝑥+
𝛿𝑢
𝛿𝑟 (3.11)
𝜑 = 2
3𝜌 𝑘𝑠𝑔𝑠 (3.12)
𝑞 𝑥 = −𝐶𝑝 𝜇
𝑃𝑟+
𝜇𝑡𝑃𝑟𝑡 𝛿𝑇
𝛿𝑥 (3.13)
𝑞 𝑟 = −𝐶𝑝 𝜇
𝑃𝑟+
𝜇𝑡𝑃𝑟𝑡 𝛿𝑇
𝛿𝑟 (3.14)
The total filtered energy is given by
𝑒 𝑡 = 𝑒 + 1
2𝑢𝑘𝑢𝑘 , 𝜌𝑒 =
𝑝
𝛾 − 1 (3.15)
and the filtered pressure term is
𝑝 = 𝛾 − 1 𝜌 𝑒 𝑡 − 1
2𝜌 𝑢 𝑘𝑢 𝑘 −
1
2𝑇𝑘𝑘 (3.16)
3.3 Sub-Grid Scale Modeling
The contributions of the unresolved or sub-grid scale terms in the filtered form of the
governing equations must be modeled. For this purpose, a model based on the incompressible
sub-grid scale model of Smagorinsky [5] with additional terms to account for compressibility
as prescribed by Moin et. al. [6] and Vreman et. al. [7] has been employed. The popular
Smagorinsky model is an eddy viscosity model where the sub-grid scale stress tensor is
modeled as an eddy viscosity multiplying the resolved stress tensor. The compressible form
13
of the sub-grid scale stress tensor (Tij) based on the work of Moin et. al. [6] is used here and
is given by
𝑇𝑖𝑗 = 2
3𝜌 𝑘𝑠𝑔𝑠𝛿𝑖𝑗 − 2𝜇𝑡 𝑆 𝑖𝑗 −
1
3∇.𝑈 𝛿𝑖𝑗 , 𝑆 𝑖𝑗 =
1
2 𝛿𝑢 𝑗
𝛿𝑥𝑖+
𝛿𝑢 𝑖𝛿𝑥𝑗
(3.17)
where δij is the Kronecker delta function and 𝑆 𝑖𝑗 is the strain rate. The sub-grid scale turbulent
kinetic energy (ksgs
) and the sub-grid scale eddy viscosity (μt) are
𝑘𝑠𝑔𝑠 = 𝐶𝐼∆2 2𝑆 𝑖𝑗𝑆 𝑖𝑗 , 𝜇𝑡 = 𝐶𝜌 ∆2(2𝑆 𝑖𝑗𝑆 𝑖𝑗 )1/2 (3.18)
where the filter width (∆), is given by
∆ = (𝛿𝑥𝛿𝑦)1/2 (3.19)
and the coefficients C and CI are user defined constants. Rogallo and Moin [31] gave a range
of values for C in the range 0.01≤ C ≤0.0576 and for CI in the range 0.0025≤ CI ≤0.009.
The sub-grid scale heat flux (Qi) is modeled based on the work by Moin [6], the sub-grid
scale turbulent dissipation rate (ε) is modeled based on Vreman's work [7] and these are given
by
𝑄𝑖 = 𝜇𝑡𝑃𝑟𝑡
𝛿𝑇
𝛿𝑥𝑖 , 𝜖 =
𝐶3
2∆𝑇𝑘𝑘 (3.20)
Vreman determined the value of C3 to be 0.6 and the same value has been used in the present
work for all the three Mach numbers. The turbulent Prandtl number (Prt) used in the present
computations is, Prt = 0.4 (for all the three Mach numbers). Table 3.1 gives the values of the
constants C and CI used in the present work for the different Mach number jets.
M C CI
1.3 0.012 0.00575
1.4 0.012 0.00575
2 0.02 0.00575
Table 3.1. Values of the sub-grid model constants used for different Mach number jets
14
3.4 Numerical Method
3.4.1 Numerical Scheme
The inviscid fluxes are computed using a fifth-order in space, low dissipation Weighted
Essentially Non-Oscillatory (WENO) finite difference scheme. The algorithm used for this
WENO approximation is the WENO reconstruction procedure prescribed by Shu [21]. The
temporal discretization is done using an optimal second-order Total Variation Diminishing
(TVD) Runge-Kutta method given by Gottlieb and Shu [23].
3.4.2 Computational Grid and Boundary Conditions
The computational domain used for the simulation of all the three ideally expanded jets is
shown in figure 3.1. The computational domain extends 20.25 times the nozzle exit diameter
in the stream-wise direction and 6 times the nozzle exit diameter in the radial direction. The
uniform rectangular computational grid used for each of the jets is fine and contains 1000
points in the stream-wise direction and 400 points in the radial direction. The nozzle lip
length is 1.25D and its thickness is 0.18D.
Figure 3.1. Computational domain of the jets.
The boundary conditions used in the simulations are shown in figure 3.1. These boundary
conditions are similar to the ones implemented by Rona and Zhang [32] in their study of
axisymmetric supersonic jets. Along the jet axis, the axisymmetric boundary condition has
been imposed. At the open flow boundaries, quantities such as, the axial and radial
components of velocity, pressure and density have been extrapolated linearly from the
freestream. The supersonic jet inlet has fixed values of the conservative variables. At the
solid boundaries, i.e. the nozzle lip surfaces, the inviscid wall condition has been imposed.
15
Chapter 4
Results and Discussion
4.1 Mean Flow Properties
Three ideally expanded jets (M = 1.3, 1.4 and 2) have been simulated. The LES for each jet
was initially run for about four acoustic times to remove the start up transients and to develop
and establish the flow field. During this period, a CFL value of 0.35 was used for each jet to
develop the flow field. This required about 50000 - 60000 iterations depending upon the jet
under study. Once the flow field was established, the LES solution was run for several
iterations with constant time stepping for each jet to obtain a steady time averaged solution
(details of which are given in Table 4.1) and to generate pressure data for the spectra
computations, which is eventually used for locating the noise source using the Near-Field
Pressure Model. The mean flow properties are used to validate the computational model
employed to simulate the three, ideally expanded, axisymmetric supersonic turbulent jets.
M No. of iterations Time Step (∆t) Corresponding
CFL
Simulation
Time (s)
1.3 116000 115 ns 0.268 0.01334
1.4 123456 108 ns 0.263 0.01333
2 84800 236 ns 0.33 0.02
Table 4.1. Simulation details for the three jets in terms of no. of iterations, CFL, time step and
simulation time.
Figure 4.1. Pressure History at a point on the nozzle lip line and at a stream-wise location of
x/D = 9 for the jets with Mach numbers 1.4 and 2.
The pressure history at a point on the nozzle lip line, at a stream-wise location of x = 9D is
shown in figure 4.1 for two of the jets (M = 1.4 and 2). The pressure has been normalised by
the ambient pressure, P∞ and time has been normalised by the ratio D/Ue (where, D is the
16
nozzle diameter and Ue is the jet velocity at nozzle exit). Such pressure data has been
generated at various stream-wise and corresponding radial locations for spectral computations
which is required for the analysis using the Near-Field Pressure Model.
The flow conditions for the ideally expanded jets are tabulated in table 4.2. The time
averaged properties obtained from the present LES solution are compared with experimental
data. Figure 4.2 shows the time averaged normalised centerline velocities of M =1.4 and M
=2 jets. For the M =1.3 jet, the time averaged centerline Mach number has been compared
with the measurements of Hileman and Samimy [13]. In case of the M = 1.4 jet, the present
mean centerline velocity (normalised by Ue) is compared with the experimental data of Witze
[33] as well as the time averaged axisymmetric LES solution of DeBonis [9]. The mean
centerline velocity (normalised by Ue) of the M = 2 jet has been compared with the
measurements of Seiner et. al. [34] as well as the data obtained by Rona and Zhang [32]
using their 'short' Time Dependent Reynolds Averaged Navier-Stokes equations (TRANS)
with the two equation, k-ω turbulence model.
Nozzle
exit
Mach
number
(M)
Nozzle
exit
Diameter
(D)
mm
Jet
velocity
at nozzle
exit (Ue)
m/s
Nozzle
exit
Density
(ρe)
kg/m3
Nozzle exit
Temperature
(Te)
K
Reynolds
number
(Re)
Ambient
Pressure
(P∞)
Pa
Ambient
Temperature
(T∞)
K
Ambient
Density
(ρ∞)
kg/m3
1.3 25.4 382.31 1.64 215.26 1.0 × 106
101320 288 1.225
1.4 25.4 411 1.59516 215.945 1.2 × 106 98862 297 1.15982
2 49.89 511.5 2.168 162.81 5 × 106 101320 288 1.225
Table 4.2. Flow conditions for the ideally expanded jets.
For the M = 1.3 and M = 2 jets, the predicted time averaged centerline Mach number and
normalised centerline velocity exhibit minor fluctuations close to the nozzle exit owing to a
series of weak shock waves in the jet plume. Such weak shock cells are also visible in the
experimental data as well as the prediction of Rona and Zhang [32] for the M = 2 jet (figure
4.2). Also, in case of the M = 1.4 jet, the experimental result and the prediction by DeBonis
[9] show the existence of weak shocks in the jet plume. In experiments, this could happen due
to slight imperfections in the nozzle geometry or boundary layer growth that prevents parallel
exit flow, whereas, in simulations, these predictions are very sensitive to the nozzle inlet
conditions. Such weak shocks are not present in the prediction of the time averaged centerline
velocity for the M = 1.4 jet, as the nozzle inlet parameters were defined by values with
sufficient significant digits.
Other important observations from figure 4.2 are the predicted potential core lengths and the
decay of the centerline velocity/Mach number. For all the three jets, the potential core lengths
have been consistently under-predicted in the present LES. These comparisons are
summarized in table 4.3. For the M = 2 jet, the current prediction of the potential core length
falls short of the measured value by 1D, whereas Rona and Zhang over-predicted the
17
potential core length for this jet by 1.5D. The present LES predictions of the potential core
lengths of all the three jets are therefore, in fair agreement with the measured values.
M Potential Core
Length (Present)
Potential Core
Length (Expt.)
Potential Core Length
(Numerical)
Difference between present
LES & Expt. result
1.3 5.5D 6D [13] − -0.5D
1.4 6D 7.5D [33] 6D [11] -1.5D
2 10D 11D [34] 12.5D [32] -1D
Table 4.3. Potential core lengths of the three jets compared with measured data.
From figure 4.2, it becomes clear that, beyond the potential core, the decay rate of the
normalised centerline velocity (centerline Mach number in case of M =1.3 jet) in the present
LES prediction is less, compared to the measured decay rate. Till the end of the potential
core, the LES predictions have good agreement with experiments, but, beyond that, the
prediction deviates from the experiments as the decay rate of the centerline velocity is lesser.
Similarly, the time averaged radial profiles of normalised axial velocity for the M = 1.4 jet, at
stream-wise locations of x/D = 2, 4, 6, 8, 10 and 12 are shown in figure 4.3. The present
prediction has been compared with the measurements of Witze [33] and the axisymmetric
LES solution of DeBonis [9]. The present LES solution for the mean radial profile of axial
velocity shows good agreement with experiment up to 8 jet diameters, but beyond that, the
prediction deviates from the experiment. The spreading of the jet for stream-wise locations
beyond x/D = 8 is greater for the experiment than the present axisymmetric LES prediction.
This is to be expected, as the predicted decay of the centerline velocity was lesser than the
experiment (figure 4.2).
The first and the major reason for such departures could be the invalidity of the axisymmetric
assumption beyond the potential core of the jet, due to the existence of three-dimensional
turbulent structures (similar observation was made by DeBonis [9]). Good agreement is
found with measurements, upstream of the end of the potential core in terms of the mean
centerline velocity and the mean radial profiles of axial velocity, indicating that the
axisymmetric assumption is valid in this region. Beyond the potential core, the prediction
departs from experiment, with lesser decay rate in the centerline velocity and lesser spreading
of the jet.
The second reason for the estimated shorter potential cores, reduced decay rate of the
centerline velocity/Mach number and the reduced spreading of the jet than the experimental
data beyond the potential core, could be because, the sub-grid scale model is unable to
produce adequate dissipation of the large-scale eddies. Another factor contributing to the
deviations of the flow-field averages from experiments, in the region beyond the potential
core, could be the poorer averaging in that region. This happens because, the flow structures
in the region downstream of the potential core are of much lower frequency than those
upstream of the end of the potential core.
18
(a)
(b)
(c)
Figure 4.2. Time averaged normalised centerline velocities of M = 1.4 (b) and M = 2 (c) jets
and the time averaged centerline Mach number of M = 1.3 (a) jet, vs. stream-wise distance.
19
Figure 4.3. Time averaged radial profiles of normalised axial velocity at various stream-wise
locations for the M = 1.4 jet.
However, for the M = 1.4 jet, the present prediction shows better reduction in the peak values
of the radial profiles of axial velocity, a better decay of the time averaged normalized
centerline velocity and a better spreading of the jet for stream-wise locations beyond the jet's
potential core compared to the axisymmetric LES prediction of DeBonis. This is because, the
20
present prediction's grid resolution is very fine compared to the coarse grid used by DeBonis
[9].
Figure 4.4 (a) shows the half velocity points (R0.5) normalised by the nozzle exit diameter, of
the M = 2 jet. The half velocity points are compared with the experimental measurements of
Seiner and Ponton [35] and the TRANS solution of Rona and Zhang [32]. A half velocity
point indicates the radial distance at which the velocity is 50 % of the velocity on the
centerline of the jet at a given stream-wise location.
Figure 4.4. (a) Time averaged half velocity points and (b) Similarity profiles for the M = 2
jet.
The predicted half velocity points in figure 4.4 (a) clearly exhibit the high speed potential
core region (x/D ≤ 10) and the fully developed region beyond the potential core, which is
21
evident from the appreciable change in the slope of the half velocity points outside the
potential core. In the potential core region, the present prediction of half velocity points
indicates a diminished spreading rate of the jet shear layer compared to the experimental data
and the prediction of Rona and Zhang [32]. Although the potential core length has been
overestimated in the TRANS simulation of Rona and Zhang, their half velocity points are
slightly more conformal to the experimental data within the potential core region. But,
beyond the end of the potential core, the present LES prediction of half velocity points
conforms very well to the experimental data of Seiner and Ponton [35], even better than the
TRANS prediction [32]. In the region beyond the potential core, termed as the mixing/fully
developed region, the time averaged radial profile of axial velocity is characterized by
geometric self similarity as depicted in figure 4.4 (b). In this figure, the time averaged radial
profiles of axial velocity of the M = 2 jet at stream-wise locations of x/D = 13, 14, 16 and 18
are presented, where the axial velocity is normalised by the local jet centerline velocity (Uc)
and the radial coordinate is normalised by the jet half-width or the half-velocity point (R0.5) at
that stream-wise location.
The instantaneous contours of vorticity, density, axial and radial velocity for the M = 1.4 jet
are shown in figure 4.5 (a-d). The instantaneous vorticity contours in figure 4.5 (a), obtained
from the LES solution show weakly resolved vortices near the nozzle lip (the nozzle lip is the
black rectangular patch on the right hand side of the contour plots). Further downstream, the
vortices evolve and grow in size forming large scale turbulent structures. In turbulent
supersonic jet flows, these large-scale structures are primarily responsible for the dominant
part of the turbulent mixing noise, with the fine scale turbulence producing the background
noise.
Another interesting observation from the vorticity contours (figure 4.5(a)), is the two
structures that seem to be merging together. The structure upstream of the jet plume being
closer to the jet centerline could be moving faster and trying to overtake the downstream
structure which is away from the jet center and moving at lower speed and in the process is
merging with the downstream structure. It is evident from these instantaneous contours, that,
LES resolves many of the flow features in turbulent jet flows and is capable of providing
much more data about the flow physics via. the time history of turbulent fluctuations that is
required for unsteady analysis.
22
(a) Instantaneous Vorticity Contours
(b) Instantaneous Density Contours
(c) Instantaneous Axial Velocity Contours
(d) Instantaneous Radial Velocity Contours
Figure 4.5. Instantaneous contours of vorticity, density, axial and radial velocity for the M =
1.4 jet.
23
4.2 Dominant Noise Source Location
The near-field pressure model discussed in chapter 2 is now utilised to locate the dominant
noise source of the three ideally expanded supersonic jets. Figure 4.6, 4.7 and 4.8 show the
near-field normalized pressure spectra at different stream-wise locations for the M = 1.3, M =
2 and M = 1.4 jets, respectively. At a given stream-wise location, multiple radial locations
outside the jet's shear region are used to record the pressure histories and the distance
between adjacent radial locations is 0.5D. Discrete Fourier Transform (DFT) is performed on
these pressure signals to convert them from time domain to frequency domain. The pressure
is then normalised in the same way, as suggested by Arndt et. al. [15], so that the quantity on
the ordinate of each plot represents the normalised mean-square pressure or intensity, I
(similar to equation 2.2). On the abscissa, the frequency f has been converted into
wavenumber k using equation 2.10 and then k is non-dimensionalised by the radial distance
Y, where Y is the distance between a radial location and the nozzle lip-line (similar to the
distance of the microphone from the nozzle lip-line, as used by Arndt et. al). Both the
abscissa and the ordinate are expressed in log-scale. In each of these plots, different symbols
have been used. These symbols or data points represent the intensities of individual spectra at
constant values of Strouhal number (St) or k.
As discussed previously in section 2.2, that, if at a particular stream-wise location, the best-fit
lines through these constant St data points in the energy containing region exhibit the (Y)-6
spatial decay of intensity for constant k, then that stream-wise distance is the location of the
dominant noise source. Consider the M = 1.3 jet (figure 4.6). The pressure spectrum for
progressively increasing (nearest to the nozzle lip-line to farthest from the nozzle lip-line)
radial locations outside the shear region is plotted at each stream-wise location. In these
spectra plots, the intensity of the fluctuations decreases and the spectra become more
broadband as the distance between the radial location and the nozzle lip-line (or the shear
region) increases.
At the stream-wise location of x/D = 2, the near-field pressure spectra do not exhibit the four
distinct regions, viz. Low wavenumber, Energy containing, Inertial subrange and Far-field
clearly. The different symbols in these plots represent data points corresponding to a constant
St or k. The best fit line through these data points neither shows the near-field behaviour
(hydrodynamic fluctuations) nor the far-field behaviour (acoustic fluctuations). At x/D = 6,
the three regions, viz. Low wavenumber, Energy containing and Far-field begin to emerge in
the pressure spectra plots. The data points corresponding to St = 0.12 and 0.14 in the energy
containing region do not exhibit the (kY)-6
spatial decay, but the data points for St = 0.2
nearly show the (kY)-6
spatial decay (near-field behaviour). Also the data points
corresponding to St = 1.5 in the higher wavenumber region, nearly obey the (kY)-2
spatial
decay (far-field behaviour).
24
St: ( ) 0.12, ( ) 0.14, ( ) 0.2, ( ) 1.5
kY
(P/
Ue2)2
(kD
)-1
100
10110
-2
10-1
100
101
102
103
104
105
106
(a) x/D = 2, M =1.3
(kY)-6
(kY)-2
kY
(P/
Ue2)2
(kD
)-1
100
10110
0
101
102
103
104
105
106
107
(b) x/D = 6, M = 1.3
(kY)-6
(kY)-6
(kY)-2
25
St: ( ) 0.12, ( ) 0.14, ( ) 0.2, ( ) 1.5
kY
(P/
Ue2)2
(kD
)-1
100
10110
0
101
102
103
104
105
106
107
(c) x/D = 8, M =1.3
(kY)-6 (kY)
-6
(kY)-2
kY
(P/
Ue2)2
(kD
)-1
100
10110
0
101
102
103
104
105
106
107
(d) x/D = 9, M = 1.3
(kY)-6
(kY)-6
(kY)-2
26
St: ( ) 0.12, ( ) 0.14, ( ) 0.2, ( ) 1.5
Figure 4.6. Near-field normalized pressure spectra plots for M = 1.3 jet, at various stream-
wise distances for locating the dominant noise source (digital bin-width, ∆f = 150 Hz).
kY
(P/
Ue2)2
(kD
)-1
100
10110
0
101
102
103
104
105
106
107
(e) x/D = 10, M = 1.3
(kY)-6
(kY)-6
(kY)-2
kY
(P/
Ue2)2
(kD
)-1
100
10110
0
101
102
103
104
105
106
107
(f) x/D = 14, M = 1.3
(kY)-6
(kY)-2
27
Further downstream, the three regions (Low wavenumber, Energy containing and Far-field)
and their features become more prominent. For x/D = 8, in the energy containing region, the
slopes through the data points for St = 0.12 and 0.14 exhibit spatial decays slightly less than
(kY)-6
, while St = 0.2 data points follow the (kY)-6
law. The far-field behaviour is also shown
by the St = 0.2 data points, as the best fit line through them obeys the (kY)-2
spatial decay.
For the M = 1.3 jet, the dominant noise source location is x/D = 9, since, the best fit lines
through data points in the energy containing region, corresponding to St = 0.12, 0.14,and 0.2
obey the (kY)-6
spatial decay of intensity I. In the high wavenumber region, the influence of
the radiated acoustic field can also be seen, with the intensity decaying spatially as (kY)-2
(St
= 1.5 data points). For x/D>9, i.e. for the stream-wise locations x/D = 10 and 14, the constant
St data points in the energy containing region show spatial decay rates flatter than (kY)-6
, but
the influence of the radiated acoustic field is still prevalent in these stream-wise locations
with the St = 1.5 data points still exhibiting the (kY)-2
spatial decay. Another important
observation from these plots, which was also mentioned in the studies by Arndt et. al. [15], is
that the energy-containing scales have moved to lower wavenumber as the stream-wise
distance increases, which is to be expected.
Similar analysis has been carried out for the M = 2 and M = 1.4 jets. Figure 4.7 shows the
near field pressure spectra for M = 2 jet at increasing stream-wise distances. The spectra at
x/D = 4 (figure 4.7(a)) do not show the Low wavenumber, Energy containing and Far-field
regions and lack the near and far field characteristics. Instead, these spectra plots resemble
the background noise that would be generated by fine scale turbulence. As in the case of the
M = 1.3 jet, here too, with increasing stream-wise distance, the Low wavenumber, Energy
containing and Far-field regions begin to emerge in the pressure spectra plots and become
distinctly visible for x/D ≥ 8 . For the M = 2 jet, the dominant noise source lies at x/D = 12
(figure 4.7(e)), since, the best fit lines through data points in the energy containing region,
corresponding to St = 0.15, 0.22,and 0.27 obey the (kY)-6
spatial decay. Also, for x/D ≥ 8, the
(kY)-2
law (far-field behaviour) is valid in the high wavenumber region (St = 2 data points).
Here also, the energy-containing scales move to lower wavenumber as the stream-wise
distance increases.
28
St: ( ) 0.22, ( ) 0.5, ( ) 2
kY
(P/
Ue2)2
(kD
)-1
100
10110
-3
10-2
10-1
100
101
102
103
104
105
(a) x/D = 4, M = 2
(kY)-6
(kY)-2
kY
(P/
Ue2)2
(kD
)-1
100
101
10210
-3
10-2
10-1
100
101
102
103
104
105
(b) x/D = 6, M = 2
(kY)-6
(kY)-6
(kY)-2
29
St: ( ) 0.22, ( ) 0.27, ( ) 0.5, ( ) 2
kY
(P/
Ue2)2
(kD
)-1
100
101
10210
-2
10-1
100
101
102
103
104
105
106
(c) x/D = 8, M = 2
(kY)-6 (kY)
-6
(kY)-2
kY
(P/
Ue2)2
(kD
)-1
100
101
10210
-2
10-1
100
101
102
103
104
105
106
(d) x/D = 10, M = 2
(kY)-6 (kY)
-6
(kY)-2
30
St: ( ) 0.12, ( ) 0.15, ( ) 0.22, ( ) 0.27, ( ) 2
Figure 4.7. Near-field normalized pressure spectra plots for M = 2 jet, at various stream-wise
distances for locating the dominant noise source (digital bin-width, ∆f = 100 Hz).
kY
(P/
Ue2)2
(kD
)-1
100
101
10210
-2
10-1
100
101
102
103
104
105
106
(kY)-6
(kY)-6
(kY)-2
(e) x/D = 12, M = 2
kY
(P/
Ue2)2
(kD
)-1
100
101
10210
-1
100
101
102
103
104
105
106
107
(f) x/D = 16, M = 2
(kY)-2
(kY)-6 (kY)
-6
31
The spatial variations of the intensity given by the near-field pressure model in the different
regions of the near-field pressure spectra are now used to locate the dominant noise source of
the M = 1.4 jet. The normalised pressure spectra at increasing stream-wise distances for this
jet are shown in figure 4.8. Starting with the stream-wise location x/D = 6, the three different
regions of the spectra (Low wavenumber, Energy containing and Far-field) start to emerge,
similar to the M = 1.3 jet. For the M = 1.4 jet, the dominant noise source is located at x/D =
9 (figure 4.8(c)), with the data points in the energy containing region corresponding to St =
0.1, 0.13,and 0.16 decaying spatially as (kY)-6
and showing the near-field behaviour. For the
M = 1.4 jet, the influence of the radiated acoustic field in the high wavenumber region, is
apparent for all the stream-wise locations (x/D = 6, 8, 9, 10, 12 and 14) given in figure 4.8.
This is because, the St = 1.5 data points, show the (kY)-2
spatial decay of the intensity (far-
field behaviour).
For all the three jets, the constant St or constant k data points in the energy containing region
at the stream-wise location of the dominant noise source show the (kY)-6
spatial decay of
intensity. At stream-wise locations other than that of the noise source (upstream as well as
downstream), the constant St data points in the energy containing region (if the energy
containing region exists at those locations) do not show the near-field characteristics ((kY)-6
spatial decay). Also, at all the stream-wise locations greater than or equal to the jets' potential
core lengths, the far-field characteristics ((kY)-2
spatial decay) are shown by the data points in
the high wavenumber zone, indicating the existence of the radiated acoustic field originating
from the fine-scale turbulence. The results for all the three jets are summarised in table 4.4.
32
St: ( ) 0.13, ( ) 0.16, ( ) 0.22, ( ) 1.5
kY
(P/(U
e2))
2(
kD
)-1
100
10110
-2
100
102
104
106
108
(a) x/D = 6, M = 1.4
(kY)-6
(kY)-2
kY
(P/(U
e2))
2(
kD
)-1
100
10110
-2
100
102
104
106
108
(b) x/D = 8, M = 1.4
(kY)-6 (kY)
-6
(kY)-2
33
St: ( ) 0.1, ( ) 0.13, ( ) 0.16, ( ) 1.5
kY
(P/(U
e2))
2(
kD
)-1
100
10110
-2
100
102
104
106
108
(c) x/D = 9, M = 1.4
(kY)-6 (kY)
-6
(kY)-2
kY
(P/(U
e2))
2(
kD
)-1
100
10110
-2
100
102
104
106
108
(d) x/D = 10, M = 1.4
(kY)-6
(kY)-2
(kY)-6
34
St: ( ) 0.1, ( ) 0.16, ( ) 1.5
Figure 4.8. Near-field normalized pressure spectra plots for M = 1.4 jet, at various stream-
wise distances for locating the dominant noise source (digital bin-width, ∆f = 150 Hz).
kY
(P/(U
e2))
2(
kD
)-1
100
10110
-2
100
102
104
106
108
(e) x/D = 12, M = 1.4
(kY)-6
(kY)-2
kY
(P/(U
e2))
2(
kD
)-1
100
10110
-2
100
102
104
106
108
(kY)-6 (kY)
-6
(kY)-2
(f) x/D = 14, M = 1.4
35
Table 4.4 compares various parameters predicted in the present study with experimental
results. The parameters for the M =1.3 and M = 2 jets have been compared with the
experimental measurements of Hileman and Samimy [36], [13] and Kastner et. al. [14].
Parameters like the dominant noise source location and potential core length show good
agreement with the experimental data. For the M = 1.4 jet, the parameters are compared with
the measurements of Witze [33]. Due to the lack of experimental data regarding the
convective velocity and the dominant noise source location of the M = 1.4 jet in current
literature, the present predictions of these parameters could not be compared for this jet.
The predicted dominant noise source location for the M = 1.3 jet shows good agreement with
the measurement of Hileman and Samimy [36]. In case of the M =2 jet, the diameter of the
nozzle used in the LES is almost twice the nozzle exit diameter used for the experiment by
Hileman and Samimy, hence, the Reynolds no. for the present prediction is also, nearly twice
the Reynolds no. of the experiment. Nevertheless, both the Reynolds number values (in
simulation as well as experiment) for the M = 2 jet are of the same order of magnitude and
the predicted dominant noise source location has good agreement with the experiment. So, it
appears that, for a jet of a given Mach number, the noise source location does not vary
significantly unless, there's a drastic change in the Reynolds no. (like an order of magnitude
variation).
Also, the dominant noise source location of the M = 1.4 jet is the same as that of the M = 1.3
jet. It seems that the noise source location has not changed for the M = 1.4 jet, since, the
change in Mach number from 1.3 to 1.4 is not that significant and the difference in the
Reynolds number for these two jets (M = 1.3 and M = 1.4) is also insignificant. The
dominant noise source for ideally expanded supersonic jets originates from the large-scale
structures within the mixing layer. For the M = 1.3 and M = 1.4 jets, the predicted average
convective velocities (table 4.4) of the large-scale turbulent structures in the jets' shear layer
are subsonic (Mach 0.72 and 0.75, respectively) relative to the ambient. Hence, for these jets,
the dominant jet noise component is the turbulent mixing noise.
However, in case of the M = 2 jet, the average convective velocity (both predicted and
measured) of the large-scale structures is supersonic (predicted Mach number relative to the
ambient = 1.07) with respect to the ambient. Hence, the large-scale turbulent structures in this
jet's mixing layer will radiate Mach waves and these Mach wave radiations will be the
dominant jet noise component for the M = 2 jet.
36
Mach no. (M) 1.3 1.4 2
Expt. Nozzle exit diameter, mm 25.4 25.4 25.4
Present Nozzle exit diameter (D), mm 25.4 25.4 49.89
Expt. Reynolds no. (Re) 1.0 × 106 1.2 × 10
6 2.6 × 10
6
Present Reynolds no. (Re) 1.0 × 106 1.2 × 10
6 5.0 × 10
6
Expt. Convective velocity (Uc), m/s 231 − 302
Predicted Convective velocity (Uc), m/s 252 261 364
Potential core length (expt.), x/D 6 7.5 11
Potential core length (predicted), x/D 5.5 6 10
Dominant noise source location (expt.), x/D 9.6 − 11.3
Dominant noise source location (prediction), x/D 9 9 12
Table 4.4. Comparison of parameters for the three jets with experimental data.
37
Chapter 5
Conclusions and Future Scope
Large-Eddy Simulation has been used to simulate three ideally expanded axisymmetric
supersonic jets. The simulations are first validated with experimental data, in terms of the
time averaged flow properties. Good agreement with measurements has been found in the
potential core region of the jets, in terms of the time averaged centerline velocity/Mach
number and the time averaged radial profiles of axial velocity. Downstream of the end of the
potential core, the estimates depart from measurements and these discrepancies have been
attributed to factors like, invalidity of the axisymmetric assumption beyond the potential core
of the jet, inadequate dissipation of the large-scale eddies produced by the sub-grid scale
model and poorer averaging of the flow parameters in that region. These estimates can be
improved by having a three dimensional approach to the simulations with better sub-grid
scale modelling. The use of a dynamic sub-grid scale eddy viscosity model, like the one
proposed by Germano et.al. [8] could help improve the predictions, even for axisymmetric
LES of turbulent supersonic jets.
In the present study, the methodology employed to estimate the dominant noise source
locations in the three jets is based on the near-field pressure model proposed by Arndt et. al.
[15]. As discussed previously, this model is based on the point-source solution of the
spherical wave equation and does not model the magnitude of the pressure fluctuations due to
a turbulent jet. Instead the methodology for locating the dominant noise source in jet is a
qualitative approach, as it relies on features like the spatial decay and spectral variations of
the intensity in the different regions (Energy containing, Low wavenumber and Far-field) of
the near-field pressure spectra. The predicted stream-wise locations of the noise sources of
the M = 1.3 and M = 2 jets show good agreement with experimental measurements. The
noise source location of a third jet (M = 1.4) was also predicted, but could not be validated
due to lack of experimental data.
The present study using LES has proved to be very promising for jet noise simulations and
predictions. The approach used in the present study for locating the dominant noise source
can also be extended to imperfectly expanded jets, theoretically. For imperfectly expanded
jets with shock cells in their jet plumes, in addition to the turbulent mixing noise, the two
shock-associated noise components viz. the broadband shock-associated noise and the
screech tones are present. Studies by Tam [12] suggests that the relative intensity of these
three noise components is a strong function of the direction of observation. The turbulent
mixing noise is the most dominant noise component in the downstream direction of the jet,
the broadband shock-associated noise is more intense in the upstream direction and screech
tones too are most dominant in the upstream direction.
38
However, the broadband shock-associated noise dominates over the turbulent mixing noise
over a wide range of observation angles (30o - 100
o) in the upstream directions. The
broadband shock-associated noise also predominates over the screech tones over a
considerable observation angle range (60o - 100
o) in the upstream direction. Hence, for
imperfectly expanded jets, the broadband shock associated noise is an important noise
component.
Recently, Berland, Bogey and Bailly [37] used LES to simulate an underexpanded jet. Upon
extrapolation of the near-field data obtained from the LES to the far-field, they observed that
the radiated noise included the three characteristic noise components (broadband shock-
associated noise, screech tones and turbulent mixing noise) of imperfectly expanded jets.
Numerical flow visualizations of the shock-vortex interactions in the vicinity of the third
shock-cell provided evidences of screech noise sources being related to the shock-leakage
theory proposed by Suzuki and Lele [38]. They showed that the third compression shock
within the jet can leak outside through low vorticity regions (saddle points of the vortices) of
the shear layer and generates upstream propagating acoustic waves which are then
responsible for producing screech tones.
The sound production mechanism via shock leakage, was proposed by Suzuki and Lele [38]
in order to explain the acoustic radiation that occurs when a weak shock interacts with the
shear layer. This could possibly be the broadband shock-associated noise source which
ultimately leads to the creation of screech tones. The technique used to determine the
dominant noise source location in the present work can be used to predict the stream-wise
location of the broadband shock-associated noise source, in theory. It could perhaps be used
to verify the hypothesis that, the location of the shock leakage and the location of the
broadband shock-associated noise source are the same. If proven correct, this hypothesis can
be used for developing quantitative prediction models for screech tones.
39
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