Detached-Eddy Simulation of Shock Wave/Boundary-Layer Interactions in a Planar Transonic Nozzle

Chapter 1
Introduction
This study aims at better understanding a benchmark calculation on a transonic nozzle analysed
both experimentally and numerically in a wide variety of papers available in literature, dating from
theseventiesuptonowadays. Thefinaltargetofallthedifferentapproacheshasbeentocharacterize
the low-frequency unsteadiness of SWBLI that takes place in the duct. A solid confidence with
numerical methodologies capable of capturing such unsteady phenomena in both qualitative and
quantitative ways is fundamental so as to have better knowledge in predicting possibly dangerous
situations, e.g. the catastrophic generation of dynamic side loads that limit considerably the design
of the expansion nozzles of liquid rocket engines, and so their performances.
In particular, separation flows induced by shocks and subsequent reattachment are associated
to many different kinds of flows, ranging from transonic airfoils, supersonic inlet, missile base flows
to, for the final purpose of the case in analysis, overexpanded nozzles. These complex interactions
of boundary layers with compression or expansion waves lead to flows that exhibit a low-frequency
unsteadiness which still needs a univoque explanation, but that can cause buffeting, instability,
thermal loadings, aerostructure fatigue with the coupling of the pressure oscillations with structures
resonant frequencies.
The fundamental problem is that a true knowledge of the physics of supersonic flows with a plethora
of phenomena like shock reflection at wall, shock/shock and shock/boundary-layers interaction,
with a deep understanding of the source or sources of low-frequency unsteadiness is still required.
A possible approach is developing improved computational capabilities, able to investigate complex
configurationsincluding3Dflows,andfosteringnon-standardsignalanalysisinordertoextractmore
information about the mechanism at the origin of the shock movement joined with the shedding of
vortical structures.
Talking about overexpanded nozzles, the experimental analysis of separated flows for full-scale
rocket nozzles is difficult and expensive, given the fact that the measurements of interest are relative
to just few seconds or less. However, an accurate estimate of this very short transient period is
crucial for the nozzle safe life since fluctuating pressure loads, consequence of shock wave trains or
interactions oscillating in time, are so severe that the operation of the engine and launch vehicle
are endangered irreparably (a series of practical examples of the consequences of uncontrolled and
usteady off-axis forces is available in literature).
1
1.1. Shock Wave/Bounday-Layer Interactions
In this chapter some notions about the physical phenomenon of SWBLI are proposed, followed
by an essential survey of the problematics encountered in real rocket nozzles, always with an eye
to the state of the art of the used approaches that can be found in literature. In the third section
a presentation of the studied case is provided, with attention to the peculiarities of the numerous
past studies and in the end an overview of the thesis organization is given to the reader.
1.1 Shock Wave/Bounday-Layer Interactions
The presence of an adverse pressure gradient in a supersonic flow alters the features of the turbulent
boundary layer, since the higher-pressure level is accounted thanks to a shock wave system. When
the strength of the adverse pressure gradient or equivalently of the generated shock is enough, the
kinetic energy of the fluid particles is transformed into potential energy at such a point that the
particles that have lower velocity are stopped or even forced to flow in the reverse direction: the
inviscid outer flow imposes a pressure gradient that cannot be negotiated leading to separation.
At the separation point the wall shear stress becomes zero, being the flow either subsonic or
supersonic. For a two-dimensional boundary layer where y is the direction normal to the wall and
u is the parallel to wall velocity, one has at the onset of separation
 w
=  @u
@y
 w
= 0 (1.1)
where  is the molecular viscosity as will be described better in ch.2 and the w subscript indicates
that the quantities are evaluated on the wall.
Considering the momentum equation in wall-parallel direction and the non-slip condition
1
for an
arbitrary point on the wall it is possible to demonstrate that
  @
2
u
@y
2
 w
=
@p
w
@x
(1.2)
and using (1.1), since
@
2
u
@y
2
has to be positive at the separation point, in order to have separation
there has to be an adverse pressure gradient.
Figure 1.1: Scheme of the formation of a separated flow over a body with curvilinear generatrices.
1
See ch.2 for further details.
Giacomo Della Posta 2
1.1. Shock Wave/Bounday-Layer Interactions
1.1.1 Unsteady Motion General Features: Basic Interactions Classification
Two- and three-dimensional cases of SWBLIs have been studied during the last fifty years, providing
a large amount of experimental data that is satisfactory for 2D interactions, but not so much for
3D ones. Many different canonical geometries experience SWBLIs and some basic configurations
can be identified, according to [67], [84] and [12]:
 the compression (swept and unswept) ramps flows (2D);
 the impinging-reflection shocks (2D);
 the normal shock, e.g. turbomachine cascades, air intakes, supersonic diffuser (2D);
 the pressure jump, e.g. overexpanded nozzles (3D);
 the blunt fin (3D).
(a) Ramp flow. (b) Shock reflection. (c) Blunt fin.
Figure 1.2: Schematic diagram of three canonical SWBLIs in supersonic flow. [67] [12]
Even if the causes that induce separation can be very different, these basic interactions exhibit
strong similarities in their unsteadinesses, showing the same features of the static wall pressure (see
fig.1.3). After a steep rise shortly after the onset of interaction I, separation starts fromS. Almost
constant pressure plateau is reached for an extent that is a measure of the dimensions of the closed
recirculation bubble. From the plateau pressure, another steep rise of the pressure is observable as
the R point of reattachment is approached. Corresponding to the first steep rise the coalescence
of compression waves brings the formation of a separation shock wave, whose lowest part is called
separation shock foot. The foot oscillates over a regionL
i
named intermittent region, meanwhile the
extent of the separated region is L
sep
.
All of these kinds of interactions show that SWBLI is an intermittent phenomenon and that
three-dimensionality has to be taken into account to properly describe the correct behaviour of the
present unsteadinesses.
About the origin of the dangerous low-frequency oscillations in the pressure signals and in the
separation shock foot, principally two different mechanisms have been proposed.
Some researchers suggest that the fluctuations in the upstream boundary layer drive the interaction,
meanwhile other results sustain the hypothesis of a dependence of the oscillation from some large-
scale instability that is intrinsic to the separated flow downstream of the beginning of separation. In
the review [12], it is concluded that probably the thruth is in the middle and both the dependences
are active for the general SWBLI. However for strongly separated flows, or for flows with large
Giacomo Della Posta 3
1.2. Towards Rocket Nozzles
Figure 1.3: Typical static wall pressure for a 2D basic configuration of SWBLI. [67]
enough recirculation bubble, the downstream mechanism dominates. For these situations, a global
instability forces the bubble to pulsate, leading the reattachment point to flap. This flapping causes
the synchronous movement of all the shock wave system, including the separation line and obviously
the separation shock itself. In particular, as exploited by [101], the shock-induced separated flow
can be seen as a low-order forced dynamical system in which the forcing source role is played by
the upstream boundary layer fluctuation and the downstream separated region fluctuations, with
the first having a reduced impact on the system with increasing size of the separated flow.
1.2 Towards Rocket Nozzles
The SWBLI is at the center of one of the basic fluid-dynamics phenomena in supersonic expansion
nozzles of rocket engines and its control or not is at the root of one of the possible limitations
of overall performances. At certain ratio of chamber to ambient pressure, Nozzle Pressure Ratio
(NPR), the jump imposed brings the formation of a shock system, sometimes very complex, and
hence shock/turbulent boundary-layer interaction inside the nozzle: a combination of gas-dynamic
phenomena takes place and mixes together and they include incident shocks, Mach reflections,
reflected shock, triple point, but also viscous phenomena such as induced separation, recirculating
bubbles and also shear layers.
In modern advanced launchers, the first stage of the rocket is responsible for providing thrust
through an altitude range that extends from an ambient pressure of 1 bar, at sea-level, to near
vacuum. During the transient phase of the beginning of the flight, the nozzle is so obliged to work
in overexpanded flow conditions, i.e. with an ambient pressure higher than the theoretical wall exit
pressure required for adapted attached flow. When the jump is strong enough, the viscous layer
separates owing to the fact that it is not able to sustain the adverse pressure gradient imposed by
the inviscid outer layer. The SWBLI that has origin in the nozzle shows strong unsteadiness and
moreover can cause asymmetrical flow separation that is highly undesirable, since lateral forces can
be generated, the so-called "side-loads", which may damage the nozzle.
For example, in the J-2S prototype engine, the successor of the J-2 Saturn V engine, a major
failure was experienced when the engine was torn violently from its gimbal structure. Likewise,
fatigue cracks and rupturing of the nozzle fuel coolant feed line surrounding the nozzle outer wall
[75] have been identified on the Space Shuttle Main Engine (SSME) as a result of extreme side loads.
Issues concerning excessive side loads have also been reported in Europe on the Vulcain engines [37]
and in Japan on the LE-7A engine [104].
Giacomo Della Posta 4
1.2. Towards Rocket Nozzles
Figure 1.4: Developed as part of ESA’s Ariane 5 Evolution Programme, Vulcain 2 provides 20% more
thrust compared to its predecessor. The nozzle extension is manufactured by Volvo Aero Corporation. [77]
In order to improve nozzle performance under overexpanded flow conditions the assessment of
side-loads is crucial, but is also a very difficult issue, since their experimental evaluation involves
the distinction of the actual aerodynamic side-loads from the complessive dynamic response of the
engine mechanical system. From a numerical point of view, since the oscillations have characteristic
low frequencies, the simulations have to be capable of describing turbulent unsteady fluctuations
and 3D flow structures for a sufficient physical time so as to describe properly the asymmetrical
forces, with a reasonable computational power.
In order to understand the problem of SWBLI in overexpanded nozzle, a description of increas-
ing difficulty is given starting from the quasi-one-dimensional model of de Laval nozzle and then
describing the different flow patterns that can take place in real nozzles with particular attention
to the side-loads problem and their possible origins.
1.2.1 Introduction to Nozzles Flow Patterns: Quasi-One-Dimensional de Laval
Nozzle
The nozzle is the component of the engine in which thermal energy is converted into kinetic energy
with the aim of producing thrust by the expulsion of high velocity fluid, thanks to an expansion
process.
To a first approximation, the nozzle flow can be studied through the quasi-one-dimensional flow
equations, under the assumption of isentropic and stationary flow, of ideal gas and considering a
duct with only a slow variation of area in axial direction. These equations are able to establish a
relation between the flow condition and the geometry of the simple nozzle considered, since the area
law along the longitudinal direction imposes the flow mass rate along the nozzle. The hypoteses
done bring to the following expressions
2
h
0
=h +
u
2
2
=cost)
T
0
T
= 1 +
  1
2
M
2
(1.3)
2
For a more complete dissertation about this topic, the interested reader can look for example at [32]. For a more
precise definition of the variables, a short summary of the fundamental equations of fluid dynamics is given in ch.2.
Giacomo Della Posta 5
1.2. Towards Rocket Nozzles
where h is the enthalpy, linked to the temperature T through the specific-heat coefficient for con-
stant pressure C
p
under the assumption of calorically perfect gas, with  a constant typical of the
considered gas. The 0 subscript indicates a total quantity and M is the Mach number. Under the
assumption of isentropic flow, it is possible to demonstrate that
  0
=
 1 +
  1
2
M
2
  1
 1
(1.4)
p
p
0
=
 1 +
  1
2
M
2
    1
(1.5)
Imposing the conservation of the mass flow rate and using (1.3) - (1.5), it is possible to relate
the ratio of two different areas relative to two different axial positions to the Mach value averaged
on the section of the duct at those longitudinal coordinate, thanks to the expression
A
2
A
1
=
M
1
M
2
 1 + (  1)=2M
2
2
1 + (  1)=2M
2
1
  +1
2( 1)
(1.6)
Considering the variation of the areaA(x), in order to have a continuous acceleration from subsonic
to supersonic flow, it is possible to demonstrate that it is necessary to join a convergent duct to a
divergent one with a minimum area, the throat, in which the transition takes place. The minimum
section withM = 1 between subsonic and supersonic flow is called choked throat. Depending on the
NPR, different situations can be observed according to the model and considering that only normal
shock can be taken into account in the duct wherever a non-isoentropic solution should be necessary
to force the pressure to reach ambient condition, since the model is quasi-one-dimensional. In the
proper-defined overexpanded interval of pressure ratio
3
, a shock system takes place outside of the
nozzle, whereas in underexpanded condition expansion waves are present at the end. These two
systems are responsible for adapting the pressure at the exit of the nozzle to the ambient pressure,
that can be higher (b to e), lower (g) or even equal (f) to project condition where the nozzle is
adapted (exit pressure equal to ambient pressure).
The procedure to determine the position of the normal shock wave position and therefore, the
strength of the pressure jump imposed on the flow, when the ambient pressure is in the range b -
f
+
, is composed of iterative procedure in which at the beginning the area of the duct where the
shock takes place is hypothesized. Through the relation (1.6) in which the second area is the throat
area, assumed in choking, the Mach number in front of the normal shock can be derived and so the
flow features on either side of the discontinuities. Through the knowledge of the exit area and the
critical areas ratio across the shock, a value of the NPR =p
0
=p
a
can be inferred. The comparison
with the real chamber to ambient pressure ratio is the measure that steers the choise of a new area
value and so on until convergence.
Itisevidentasthemodelisreallysimplicisticandinordertounderstandthecomplexphenomena
thatinvolvealsotheeffectsofboundarylayer,turbulence,unsteadinessandthree-dimensionaleffects
is not enough. However, it represents the simplest approach to the nozzle problem, able to give a
first idea of nozzle flows.
3
In fig.1.5 the overexpanded region goes from d to f, meanwhile in the rest of the work it goes from b to f,
including also shocks in the divergent section.
Giacomo Della Posta 6
1.2. Towards Rocket Nozzles
Figure 1.5: Pressure distribution along the de Laval nozzle, according to the quasi-monodimensional model
and schematic of corresponding flow patterns. [18]
1.2.2 Flow Patterns in Supersonic Rocket Nozzles
Depending on the geometry and on the pressure ratio, very different flow patterns are observed
in real supersonic rocket nozzles, in which the interaction between the shock wave system and the
turbulent boundary layer with the separation region is central and requires particular attention.
During the years, flow separation in nozzles and in general flow patterns that develope under
overexpanded conditions have been the subjects of many studies. These have demonstrated through
experiments on either subscale or full-scale nozzles, that two different separation processes can be
observed. They are briefly described in the following paragraph.
Giacomo Della Posta 7
1.2. Towards Rocket Nozzles
Free Shock Separation (FSS)
The boundary layer separates at a certain point on the nozzle wall and never reattaches. It was
the first mechanism observed and the pressure profile along the wall is dominated by the SWBLI
present where a supersonic flow separates. In fig.1.6 it can be seen how the recirculating region
caused by separation, in which the external fluid is sucked into a subsonic region, is at the origin
of an oblique shock wave near the wall, the separation shock. If an internal shock is present, e.g.
TOC nozzles, this or the separation shock, interacts with the Mach disk/stem at the triple point,
where a reflection shock arises. The separated flow then quits the nozzle as a supersonic free jet.
Because of the oblique shock, the wall pressure first rises and then reaches a plateau at a value that
is slightly lower than the ambient level, due to loss at the lip associated to small counter rotating
vortex. The annular supersonic plume is limited by two mixing layer, an internal and external one.
Figure 1.6: Internal shock structure under FSS regime in a TOP nozzle with example of pressure evolution
along the wall. [1]
Restricted Shock Separation (RSS)
Hypothesized and described for the first time in the pioneering work of Nave and Coffey [62] during
cold-flow subscale tests of the J-2S engine’s project in the early 70s, this regime is observed only
for highly overexpanded conditions. A completely different and peculiar flow pattern develops that
includes reattachment of the flow, initially deduced by the evolution of pressure along the wall,
which showed an unsteady behavior downstream of the separation point with strong irregularity
finallyreachingapressurehigherthanambientvaluecausedbythealternatingshocksandexpansion
waves induced by reattachment.
A special shock structure, the cap-shock, and a low-speed trapped vortex downstream of the Mach
disk/stem are characteristics of this situation. In some kinds of nozzle geometries an internal shock
is present in order to match the change in the direction imposed by the shape of the wall; this
interacts with the Mach disk far downstream to the throat and joins the transmitted cone-shaped
oblique strong reflected shock into a triple point. The separation shock and the strong reflected
Giacomo Della Posta 8
1.2. Towards Rocket Nozzles
The problem is that it is difficult to understand wheter upstream or downstream perturbations are
responsible of the asymmetry. One possible explaination is that the largest turbulent structures
that develop upstream of separation after the throat can trigger asymmetrical behaviour. Another
answer to the problem is that the pulsations in the separated region originated by turbulent shear
layer, result in a irregular pressure distribution that acting through the subsonic recirculation region
contributes to determining the separation point location. In brief, according to [84], the possible
origins for side-loads suggested in literature are:
Tilted pressure line Models that assume a tilted separation line in FSS regime correctly estimate
side-loads only for special nozzle families.
Side-loads due to random pressure pulsation A combination of random oscillations of the
separation line and random pressure pulsations in the separated flow region is supposed.
Side-loads due to aero-elastic coupling Closed loop fluid-structure effects can be considered
practically just with simple decoupled models, given the complexity in generating accurate
asymmetric models of the nozzle-engine support system. However small pressure fluctuations
can modify significantly the nozzle contour.
Side-loads due to a change of separated flow structure AssumingthatthetransitionFSS!
RSS is not instantaneous, a transitory phase exists during which FSS and RSS cohexist in
different spatial regions. The pressure behaviour is totally different and the separation point
of RSS regime is further downstream: strong lateral forces are generated.
Figure 1.10: A NASA picture of the three space shuttle main engines during the transient start-up, with
the typical teepee like separation line. [84]
Giacomo Della Posta 11