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The document presents a study on the computational analysis of over-expanded flow in convergent-divergent nozzles using the Fluent CFD code. It highlights the limitations of classical inviscid theory in accurately predicting complex flow features and discusses the effectiveness of various turbulence models in simulating viscous flows. The findings indicate that the shock structures and flow characteristics differ significantly from inviscid predictions, particularly at higher nozzle pressure ratios (NPR).
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0% found this document useful (0 votes)
12 views14 pagesPaper CFD
The document presents a study on the computational analysis of over-expanded flow in convergent-divergent nozzles using the Fluent CFD code. It highlights the limitations of classical inviscid theory in accurately predicting complex flow features and discusses the effectiveness of various turbulence models in simulating viscous flows. The findings indicate that the shock structures and flow characteristics differ significantly from inviscid predictions, particularly at higher nozzle pressure ratios (NPR).
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ResearchGate
CED Analysis of Over-Expanded Flow in Convergent Divergent Nozzles
1 240s
Bi rare
-comentooning page was pleased ab vibaansnan 040M 207 CCSIR-National Aerospace Laboratories, Belur
3TH PAL
CED ANALYSIS OF OVER-EXPANDED FLOW IN CONVERGENT
DIVERGENT NOZZLES
MT. Thanusha; B.V, Rahul; Swapneel Roy AA. Khan
Student - Aeronautical Society of India Senior Scientist
Propulsion Division Propulsion Division
Bangalore-560 037, India Bangalore-S60 037, India
Email: shfaque@nal.es.in
Abstract,
The classical one-dimensional inviscid theory does not reveal the complex low features in an
cover-expanded nocile accurately. The code Fluent hasbeen used to simulate the viscous uid
(air) low passing through thee different 2-D Convergent Divergent (C-D) nozzles for nozzle
pressure ratios (NPR) corresponding to over expanded flow. The steady state RANS equation
‘with five diferent turbulence models has been simulated for turbulence model stdy. Two
different ypes of boundary layer grids: grid (y"=1) and grid 2(y"=50) have been simulated.
Shear Stress Transport ka (SSTKW) turbulence model with grid 1 has been chosen for
‘urbulence modeling. Both inviscid and viscous flows have been simulated. The converged
solutions captured asymmetric lambda shockinailthe three nacles ut higher NPRs or viscous
‘flows. Italo predicted aftershock and low separation depending upon NPR. The shock wave
‘angle ofthe incidentand reflected shock generally reduces with increase in NER or NPRSL.92.
The trust co-efficient of ll the theoretical, predicted inviscid and predicted viscous solutions
reduce with increase in NPR upto a certain value of NPR and then start increasing. The
predicted viscous resulis (shock structure, shock location, sie of normal shock. aftershockand
asymmetric lambda shocks) are close tothe experiments in mast ofthe cases. Te aftershock
phenomenon provides dual mode flow at the exit ofthe nozcle.
CCSIR-National Aerospace Laboratories, Belur
‘Nomenclature = Shock
| { =Throat
A = Nozzle are ut = Upper, Lower wal
F = Actual Nozzle thrust 1,2. Just before, after the shock
Fi Ideally expanded nozzle theust
M_— =Local Mach number
P,P). = Static, Total pressure ‘Acronyms
T,Tp, = Static, Total Temperature FSS =Free Shock Separation
a@ © =Half nozzle wall angle NPR. = Nozzle pressure ratio (PyP,)
8 = Flow deflection angie RSS _= Restricted Shock Separation
B = Shock wave angle
Introduction
Subscript ‘The one-dimensional inviscid isentropic flow in aC-D
nozzle isa classical text-book problem, which has differ-
© Exit ‘ent flow regimes depending upon NPR. The inviscid the-
ir =Incident, Reflected shock ‘ory predicts a simple shock structure consisting of a
n_=Normal component normal shock followed by a smooth recovery to exit
Paper Code : V6T NURDS-2015, Manuscript received on 05 Jul 2013, Reviewed, revised and accepled as a Fall Length Contibuted
Paper on 07 ct 2014 pressure in the divergence part of a choked nozzle for
PRs corresponding to the over-expanded flow regime,
Bui, in reality, mull-dimensionality and viscous effects
Tike wall boundary layer and flow separation drastically
alter the flow in a C-D nozzle. Viscous effects thicken the
‘boundary layer before the shock and the base ofthe shock
also becomes thick or bifureate the shock in the form of
lambda shape depending upon the nature and thickness of
boundary layer [1]. So, the shock contains normal shock
‘known as Mach stem inthe central region while the shocks
‘lose to both the boundaries become thicker or bifurcate
as lambda shock,
‘The frst le of the lambda shock (known as incident
shock) turns the low away from the wall while the second
leg (known as reflected shock) tres to turn back the flow
to the original direction. The coincidence of Mach stem,
incident shock and reflected shockiisknown as triple poi.
At higher NPR, the flow separation in C-D Nozzle is
asymmetric, where one lambda foot is larger than the
‘other. The asymmetric flow separation has been predicted.
for higher NPR (1.5 < NPR < 2.4) depending upon the
inital low field [2]. The whole flow behind the lambda
shock is divided into two region separated by slip stream,
‘The slp steam originates from the triple point [3-4] The
flow behind the Mach stem (away from the boundary)
becomes subsonic because of the normal shock in the
‘central region. However, the flow behind the reflected
shock is sill supersonic. Since the area between the slip
stream and shear layer behind the reflected shock is in-
creasing for a short distance, hence, the flow accelerates
for a certain distance with expansion wave. The central
region of the flow becomes subsonic behind the Mach
sem, But it also accelerates because of convergent-diver-
‘gent region made by wavy slip stream, The slip stream
becomes supersonic at the position where itis intercepted
by the expansion wave [4]. Ultimately, the flow may
‘experience another shock in the downstream known as
aflershock to match the ambient pressure, For higher NPR
(NPR22), the separation is not the result of a stronger
shock/boundary layer interaction, butiteomes through the
natural tendeney of an over-expanded nozzle flow [5-6]
‘Two different patterns of flow separation may occur in
‘over-expanded nozzle (Free shock separation and re-
stricted shock separation). In free shock separation (FSS),
the flow separates from the wall and separation continues,
tll the exit of the nozzle. However, in restricted shock
separation (RSS), the separated flow reattaches tothe wall,
and becomes supersonic in the downstream of the reat-
JOURNAL OF AEROSPACE SCIENCES & TECHNOLOGIES
VOL.67, No.
tachment point {7-10}. The peak value of wall static pres
sure is associated with reattachment of flow {11}. Two
«different vortical regions have been found during nozzle
start-up [12]. One vortex is due to the boundary layer
separation from wall, whereas the second vortex spreads
in the divergent section and has inviscid origin, The un-
steady nature of shock boundary interaction is explained
in (13-14}. The over-expanded nozzles ate characterized
by unsteady shock induced separation.
Since, over-expanded flow results in total pressure
losses, its important to predict the thrust loss and rust
performance of the nozzle, Papamoschou [15] suggested
‘thatthe thrust predictions (compared with inviscid predic
tions) in the over-expanded nozzles would be inaccurate
because of the non-uniformity ofthe flow a the exit, The
off-design nozzle thrust efficiency could be improved by
‘encouraging stable separation (witha passive porous cav
ity) and controlling the location and extent ofthe separa-
tion [16]. ‘The over-expanded flow in the nozzle is
unsteady by nature and the shock oscillates with a maxi
rum displacement of about 10-30 mm depending on the
area ratio [17]. In the beginning, the lambda shock was
symmetric bu it becomes asymmetric after a certain time
for higher NPRs [18]. The asymmetry in lambda shock is
a key factor for mixing enhancement [19]. Investigation
[19} contains detailed experimental study of flow in rec-
‘angular over-expanded supersonic nozzles exploring the
‘complex nature of such flows. The prediction of such
flows also presents great challenge toany CFD code, The
‘resent computational work was aimed at simulating three
different configurations of nozzles [19] forthe given range
‘of NPRs with the help of a commercial CFD code Fluent,
with the objective of understanding complex flow struc-
ture in a C-D nozzle
Objective
In supersonic combustion RAMIET (SCRAMIET)
facility, combustion usually takes place at supersonic /
‘dual speed (supersonic and subsonic). For hypersonic air
‘breathing engine, there is a possibility to have dual mode
‘combustion in the experimental facility (20), For dual
‘mode combustion in the combustor, the flow atthe inlet
‘of the combustor should be in dual mode. The aim of this
work is to explore the situation to obtain the Now at the
‘exit ofthe nozzle in the state of dual mode. This flow can
‘be passed through the combustor to obtain the combustion
in dual mode. In order to meet the objective, these models
have been chosen forthe detailed CFD analysis {19} FEBRUARY 2015
Computational Details
‘The computational domains forthe three diferent C-D
nozzles have been generated based on the information
given in the reference (19]. GAMBIT has been used 0
generate geometry and grid i the computational domain.
‘The code Fluent has been used to solve 2-D steady state
Reynolds-averaged Navier-Stokes (RANS) equations in
three different C-D nozzles - (1. AYA=L, h=3.03°,
Symmettic about centerline 2. AVA=1.5,<4=0.00°, Sym.
metric about centertine,3. AJA, oy=4.33, A=Ay,
asymmetic about centeriine) for different NPRS, The
turbulence study has been made by simulating the flow
with five different turbulence models (Standard ke
(SKE), Reaizable ke (RKE), Renormalization Group k-e
(RNGKE), Standard ko (SKW) and Shear Sess Teans-
port ka (SSTKW)) with gridi (*=1) for NPR=141
(aozalel). All the three kee models (SKE, RNGKE and
KE) have also been tied with grid2 (y"=50) and with
standard wall funetion based oa log law fr the same NPR.
From accuracy and convergence consideration, the second
‘order upwind scheme was used forall the conservation
‘equations after geting convergence with first order up-
wind scheme, Grd adaptation based on pressure gradient
criterion was also invoked in order to capture the flow
iscontinties accurately. With grid adaptation, the final
solutions in most ofthe cases were obtained on grids that
had 2 to 3 times the original numberof cells. The compu
tation has been made for iavisid flow through C-D nozzle
too, The let boundary conditions consisted of total pres-
sure, Pp=3.5x10° Nim? and total temperature, Ty=300K.
‘The exit static pressure was varied to obtain different
IPRs (NPR=1,20, 1.25, 1.32, 141, 1.65, 192, 2.03 and
2,36 in nozzlet, NPR=I.22, 1.28, 1.36, 140, 150, 1.77,
1.89 and 2.16in nozale2 and NPR=IL19, 1.28, 1.34, 1.4,
181, 2.21, 232 and 239 in nozzle)
Results and Discussion
Inviscid Analysis
The inviscid prediction matches well with I-D inviscid
‘theory in regard of shock location and shock structure (one
‘normal shock followed by smooth recovery of pressure)
for lower NPRs. According to 1-D inviscid theory, area
ratio (AYA, atthe shock location is higher than the exit
area ratio for higher NPRs (i.e. Shock location is outside
the nozzle). But, inviscid solution predicted shock atthe
‘exit itself for higher NPRS to match static pressure speci-
fied at the outlet boundary condition. In some cases, shock
isnot predicted atthe exit ofthe nozzle for higher NPRS
(NPR22.32) in Nozzle3. This may be because of the
CFD ANALYSIS OF OVER-EXPANDED FLOW 3
insufficient value of the specified static pressure at the
‘outlet to get the shock.
Turbulence Model Study
‘The viscous predictions with 5 different turbulence
models (SKE, RKE, RNGKE, SKW, SSTKW) and two
different grids have been made by simulating Nozzlel at
NPR =1.41 with 2" order upwind discretization and with
pressure based grid adaptation, The simulations with
RNGKE turbulence models posed convergence problem,
“The predictions of SSTKW with grid] shows better agree
ment with the experiment in egard to the shock location
in terms of area ratio (AY/A), aftershock and sharpness of
the shock as compared to the other models as shown in
Fig.|. So, SSTKW turbulence model with gril has been
‘chosen for all the simulations whose results are diseussed,
below.
Viscous Analysis
‘The viscous prediction with 1" order upwind discret
zation could not eaptue the actual shook structure (ie.
aftershock) for any NPR. However, the 2 order upwind
dliseretized solations predicted lambda shock near walls,
Mach stem (normal shock inthe ental region) and afte.
shock forhigher NPRs, The pressure based grid adaptation
‘with 2" order upwind discretization further improved the
results in regard to sharpness of shock and number of
afteshock as shown in Fig.2. I also indicates th signi
‘ant diference inthe shock location with viscous predic-
tion as compared to inviscid prediction. Generally, the
mass flow rate reduces insignificantly from inviscid to
‘iseous low. The reduction in mass flow ate with viscous
simulation is due tothe boundary layer development near
the boundaries, It further reduces from lower order dis-
cretization to higher order discretization as weil as with
pressure based grid adaptation. Tis indicates that bound
ary layer prediction is captured and improved with higher
‘order discretization as well as with grid adaptation. In
Some cases, second oder viscous simulations converged
with Muctuations in residuals which indicat that flow is
unsteady in ature
Figure 3 shows Mach number contours in the nozzles
for different NPRs with pressure based grid adaptation
"The curvature ofthe Mach number contours indicates that
the flow is not totally uniform across the height of the
nozzle due to presence of boundary layers on the walls. A
‘more complex shock structure namely, a lambda shock is
clearly visible in the nozzle a higher NPRs (NPR21.25) 4 JOURNAL OF AEROSPACE SCIENCES & TECHNOLOGIES
In addition, at higher NPRs (NPR2I.32), the lambda
shock is nt symmetric about the centerline in all the three
norales. The reason for the asymmetry in lambda shock
for nozzle} may be the asymmetry in geometry. But
nozzle1 and nozzle2 do not contain any asymmetry in
geometry or non-uniformity in inlet boundary condition,
In spite ofthat, asymmetric lambda shock has been pre
dicted. The unsteady nature of the over-expanded flow
makes the lambda shock to oscillate within a certain
distance and finally convert the symmetric lambda shock
into asymmetric for higher NPRs. The flow has been found
lobe stable with oscillation afler conversion to asymmmet-
rie lambda shock in unsteady similations too [18]. This
asymmetry has been attributed to the unsteadiness in the
nozzle flow and subsequently to the oscillations in the
lambda shock. The cause of this nature can be related to
the shock oscillation and state of boundary layer separa-
tion downstream of the main shock at both the walls. The
similar type of asymmetric lambda shock has been ob-
served in the experiments too [19]. The computed and
measured flow features are nesrly symmetric for lower
INPRs as the flow unsteadiness was seen to be minimal
Furthermore, in general, the length ofthe central normal
shock (Mach stem) reduces with the increase in NPR. The
Tambda shock structure (symmetric and asymmettic) and
reduction in the size of Mach stem with increase in NPR
‘generally agrees withthe experiment. The shock location
in terms of area ratio is significantly different from that
predicted by the one-dimensional inviscid theory. Figt
shows shock location in terms of area ratio obtained from
viscous CED prediction, Experimental results, inviscid
CED prediction and 1-D inviscid theory. Both Viscous
predictions and experimentally measured shocks sit at
smaller axial distance (ie. smaller area ratios) than the
corresponding inviscid values. However, the inviscid pre-
dictions of shock locations are generally close to the
‘corresponding values obtained from inviscid theory for
lower NPRS. But, inviscid shock location could not be
predicted correctly for higher NPRs because ofthe insuf-
ficient exit area In those cases the shock generally sits at
the exitof the nozzle. The viscous predicted values gener-
ally compae well withthe experimental result,
Figure 5 schematically shows the locations of the
shock structure for different values of NPRs both for
viscous predictions and experimental results shown in
[19}, For NPR<1.25, the shock is almost normal with
slight lt near the Boundary wall. However, for NPR21.25,
the shock structure consists of a Mach stem (Normal
shock) atthe centre and two oblique shocks (incident and
reflected shock) at each side of the nozzle walls. The size
VOL.67, No.
‘of Mach stem reduces wit increase in NPR. The incident
shock turns the flow away from the wall while the reflected
shock ties to turn it back towards the axial direction, The
flow separates atthe incident shock location and may or
may not reattach in the downstream depending upon the
shock structure. Fig.6 shows the plot of wall shear stress
for three different nozzles at different NPRs, The sepa-
rated flow, generally, reattaches for lower NPR. The value
‘of NPR, for which separated flow reattaches, varies for
different nozzles (NPRS1.Al for Nozzle1, NPRSI.R9 for
Norale2, and NPRSI.44 for Nozzle3). The smaller leg
side of lambda shock reattaches forall values of NPR
However, for larger leg side of the lambda shock, the
separated flow continues til the exit ofthe nozzle without
reattachment at higher NPRs. The former separation is
‘known as restricied shock separation (RSS), where the
flow reataches downstream and the later is known as free
shock separation (FSS), where flow continues to be sepa
rated till the exit
In the centerline region also, very interesting phe-
nomenon takes place for NPR2140. Usually, lambda
shock does not sit alone but is followed by aftershocks,
‘The appearance of aftershocks is illustrated in Fig.8,
Which shows the computed Mach number at the Mach
stem forthe three different nozzles at different NPRs. The
Mach number distribution exhibits sharp oscillations,
‘which indicates thatthe flow decelerates through anormal
shock, re-expands to higher speed, shocks down, re-ex
pands, and so on, giving rise to aftershocks. The re-expan-
sion may lead to local subsonic or supersonic velocities
depending upon NPR values as shown in Fig 8. The numn-
ber and strength of aftershock increase with inerease in
NPR. These aftershocks (more number of aftershocks for
higher NPRs) have been observed in the experiments too
[19}. This aftershock phenomenon provides the dual mode
flow at the exit of the nozzle that can be sent through the
‘combustor to get the dual combustion inthe combusr.
‘Boundary layer shock interaction convert the normal
shock into (wo oblique shocks (incident and reflected
shock) in boundary layer region. The flow compresses
Ubrough the incident shock and turn the flow away from
the wall. Because ofthe turning of flow towards the center,
the flow separates and the effective area of the flow
reduces as shown in Fi.7. The reflected shock ties to turn
the flow towards original flow direction. The flow behind
the reflected shock is still supersonic for a small region
just above the shear layer. The whole flow behind the
lambda shock is divided into two region separated by slip.
stream as explained earlier. Fig.9 shows isoline of M=1 FEBRUARY 2015
(oni line) near the main shock. It indicates the shear layer
downstream of the shock near the boundary. Between
shear layer and slip line, the flow is supersonic down-
stream ofthe reflected shock, Inthe central region, there
is closed loop of sonic line behind the Mach ster. Inside
the closed loop, the flow is subsonic. But the flow becomes.
supersonic dowastream of the closed loop experiencing
again a shock known as aftershock. The schematic dia-
_gram of C-D Nozzle in Mach stem region has been shown
in Fig.10, It indicates that the acceleration of flow to
supersonic speed behind the lambda shock in the central
region of the nozzle is due to the turning of flow.
Table-1 shows the angles (flow deflection angle @,
shock angle B) for the incident and reflected shock ob-
lained from the perfect gas oblique shock theory. For
NPRSI.92, Band B, reduce with increase in NPR at both
the top and bottom walls. But, for higher NPRs, it in-