12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

CURVATURE EFFECTS ON COMPRESSIBLE
TURBULENT SHEAR LAYERS

Kristen V. Matsuno
Department of Mechanical Engineering
Stanford University
Stanford, CA 94305
kmatsuno@stanford.edu

Sanjiva K. Lele
Department of Mechanical Engineering
Department of Aeronautics & Astronautics
Stanford University
Stanford, CA 94305
lele@stanford.edu

ABSTRACT

shear layer centerline. Experimental studies of simple mixing layer configurations with streamwise curvature at moderately compressible levels have not been conducted. Plesniak
et al. (1994) also conducted experiments of nearly incompressible, mildly curved mixing layers, and observed that shear layers grew approximately linearly and that vortical structuresthat developed were related to upstream disturbances rather
than the Taylor-Görtler instability. Stability analysis by Otto
et al. (1996) indicated that centrifugal instability modes in theform of longitudinal vorticies are supported in curved, incompressible mixing layers if the centerline curved into the fasterstream. Experiments of reacting, curved mixing layers with afocus on scalar mixing and hydrodynamic instabilities wereconducted by Karasso & Mungal (1997), where it was observed that mixing efficiency did not increase in unstable configurations. Later, curvature effects on turbulent channel flows
were studied by Nagata & Kasagi (2004) with a specific focus on increasing curvature intensity on turbulence intensity.
Several experimental studies of streamwise curvature effects
in turbulent boundary layers have been conducted (Jeans &Johnston (1982); Wang et al. (2017)), but similar experimentsof curved shear layers across a comparable parameter space arelacking. Aside from our prior work (Matsuno et al. (2022)),
numerical studies of curved mixing layers in the highly compressible regime have not been conducted.
The primary goal of this work is to improve the understanding of turbulent mixing layers influenced by compressibility and curvature. Large eddy simulations (LES) of turbulent shear layers at low and high Mc , and stable and unstable curvature configurations, are analyzed to improve thecurrent understanding of the fundamental physics behind turbulent mixing in flows with multiple physical effects. Metrics
for shear layer thicknesses and growth rates, and including a
discussion of their evolution for stable and unstable configurations, are presented. The effects of increasing curvature intensity on turbulent statistics are demonstrated. The analogy
between centrifugal and buoyancy effects, following work by
Bradshaw (1973) and Nagata & Kasagi (2004), is also evaluated. Finally, compressibility and curvature effects for thepresent parameter range are compared.

Fundamental data representing the simultaneous effectsof compressibility, density variation, and curvature as primary
turbulence generation mechanisms remains sparse. This workbuilds off our previous suite of LES studies on the combined
effects of these three variables, with a specific focus on curvature effects. Various measures of shear layer thicknesses
and growth rates are presented. Exact self-similarity is notachieved, although mean field and turbulent stress profiles
show reasonable collapse. Growth rates and turbulent stresses
for stably and unstably curved configurations are compared,
and the effects of increasing curvature intensity are presented.
Lastly, a comparison with compressibility effects suggests thatfor the given parameter range, curvature and compressibility
effects are comparable.

INTRODUCTION
Many aerospace applications involve high-speed jet
plumes with complex flow physics and multiple turbulencemechanisms. A comprehensive understanding of turbulence inthese flows is essential in developing and improving numericalmethods for such vehicle design and analysis, and is currently
lacking. In particular, the shear layers in these rocket-motor
exhaust plumes exhibit streamwise curvature, along with pressure gradients, shock waves, and other complexities. Prior
studies focused on turbulent mixing have been limited to compressibility and variable density effects, but studies of turbulence at combining all three turbulence mechanisms (compressibility, density variation, and curvature), has been sparse.
An area which has been particularly less studied is thejoint effect of streamwise curvature and compressibility in
mixing layers, although streamwise curvature effects havebeen considered in more detail in other flows. An analogy
between the phenomena in turbulent shear flow under buoyancy and under streamwise curvature was first drawn by Bradshaw (1973), which depended on the close correlation between temperature and streamwise velocity fluctuations. Later,
Castro & Bradshaw (1976) studied the non-constant curvature
in impinging flow and examined Reynolds stresses along the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1: Physical parameters and curvature levels during

specific heats, γ = 1.4, and gas constant Rg .

initial and turbulent ‘asymptotic’ growthMc

u2 /u1δΓ (0)/rc

0.2

0.5
0.5
1.9
1.9
0.2
0.2
0.2
4.5
4.6
4.8

0.001
0.010
0.001
0.010
0.001
0.005
0.010
0.001
0.005
0.010

0.8

Turb.
δΓ /rc0.005 - 0.008
0.075 - 0.108
0.004 - 0.011
0.061 - 0.110
0.006 - 0.009
0.022 - 0.027
0.049 - 0.108
0.006 - 0.009
0.025 - 0.040
0.056 - 0.085

Finalδ̇Γ /∆u(0)
0.071
0.035
0.069
0.140
0.036
0.018
0.018
0.025
0.061
0.079

Final
∆u/∆u(0)
0.98
0.97
1.00
1.01
0.99
0.98
0.96
1.01
1.02
1.04

The temporally developing curved shear layer configured
as two co-annular streams which are periodic in the streamwise
and spanwise directions. The outer and inner radial free stream
values are denoted 1 and 2 , respectively, in the remainder ofthis paper. The initial conditions for the curved shear layer
are set such that vorticity is zero everywhere except within theshear layer region.

(3)

(i)

( j)

(i)

( j)

(i)

( j)

u′θ = ∑ A(r)cos(kθ θ + φ1 )cos(kz z + φ2 )
i, j

u′r =

∑ A(r)sin(kθ θ + φ1 )sin(kz z + φ2 )i, ju′z = ∑A(r)( j)i, j rkz
(i)

(i)

( j)

sin(kθ θ + φ1 )[(1 − 2r(r − rc ))cos(kz z + φ2 )( j)+ kθ sin(kz z + φ2 )]NUMERICAL METHODSThe high-resolution LES have been generated with the
CFD solver OVERFLOW. Convective fluxes are computedwith WENO5 and 4th order Runge-Kutta time integration. In
the mixing layer region, the grid has uniform and isotropic grid
spacing with more than 30 points across the initial shear layer.
For the inner and outer radial free streams, upper and lower‘buffer’ regions with grid stretching were applied. The upperand lower boundaries of the domain are adiabatic slips walls
which enforce zero radial velocity at the wall. Buffer regionsin the form of grid stretching and numerical dissipation werealso implemented; further details on the LES calculations are
given in our previous documentation (Matsuno et al., 2022).

!
>0ξ (r) = ξ g + ξ (1 − g)1r − rcg(r) =
1 + tanh2δ0

are added at selected spatial modes kθ = i2π/Lθ and kz =j2π/Lz for i, j ∈ [4, 8] with amplitude A(r) = exp[σ (r − rc )2 ].
These perturbations decay exponentially with a rate of σ = 5
into the free streams. The initial perturbation amplitude is
0.05∆ū and phases φ were chosen randomly for each mode.
The chosen form for velocity perturbations is given below.

The physical parameters governing this flow are the curvature intensity, the convective Mach number, the velocity ratio, and the density ratio across the shear layer. The curvatureintensity, defined as the ratio of the (initial) shear layer thickness compared to the radius of curvature, is selected to range
from δ0 /rc = 0.001 to δ0 /rc = 0.01. The convective Mach
number, Mc = ∆u/(c1 + c2 ), a measure of the mean flow’s
compressibility, was selected to be Mc = 0.2 and 0.8 in order toprovide comparison of compressibility and curvature effects.
Nominally unity density ratio configurations ρ2 /ρ1 = 1 wereselected for analysis in order to focus on simple curvature effects due to kinematics, without large mean density gradienteffects. For the curvature intensities investigated, centrifugal
instability attributed to the velocity and compressibility of themean flow can be considered. Eckhoff & Storesletten (1978)
gave the necessary stability criterion for a compressible columnar vortex (with uz = 0) as
u2
1 dρ
− 2θ
ρ dr c r(2)

The streamwise velocity profile is set by the circulation asuθ = Γ/r, and the pressure profile is computed by integrating
d p/dr = ρu2θ /r from r = rc to the upper and lower boundaries,
using a reference pressure pre f at the centerline. This reference pressure is set such that the speed of sound at the centerline is a constant cre f for all cases. The initial conditions satisfy potential, inviscid flow. To induce turbulence, solenoidal
velocity perturbations were added to the shear layer. By selecting simple profiles for each u′θ and u′r , the remaining spanwise
component of the fluctuating velocity can be solved analytically from the continuity equation to maintain a divergencefree initialization. For u′θ and u′r , sinusoidal perturbations

FLOW CONFIGURATION

u2
Φ= θ
r

Pt (r) = Pt,1 g + Pt,2 (1 − g)(1)

Eckhoff & Storesletten also demonstrated that the derivationof this expression recovers the classic Rayleigh criterion for
centrifugal stability in the incompressible limit, d(Γ2 )/dr > 0.
The present unity density ratio shear layers are stably curved
for u2 /u1 < 1, and unstably curved for u2 /u1 > 1, as summarized in Table 1.

RESULTS
Growth rates
Notation and coordinates used throughout the remainderof this work are the centerline radius rc , the centerline streamwise position s = rc θ , and the normalized transverse coordinate η = (r − rc )/δ99 . The 99% thickness, δ99 , is based off
the mass-averaged circulation profile, Γ̃ = rũθ , and is similar
to the visual thickness reported by previous experimental studies of planar shear layers. Another thickness defined from themean circulation, which is constant in each of the free streams,
can be computed. Equation 4 defines this ‘circulation’ thickness, δΓ , which is analogous to the definition of momentum

Initial conditions
Each stream in the curved shear layer is setup to bevorticity-free in the mean flow. The shear layer region is centered about the centerline radius rc with an initial thickness of
δ0 . The initial values for the constant free stream total pressure, Pt , and entropy, ξi , for each free stream i = 1, 2 are set
according to Eqn. 2-3. Both streams have the same ratio of2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1: Thickness and growth rate measures for stable and unstable cases at Mc = 0.8 and δ0 /rc = 0.01.

Figure 2: Spanwise velocity fields of stable (left) and unstable (right) cases for Mc = 0.8 and δ0 /rc = 0.01. The domain

has been truncated in the radial extent for visualization purposes.

thickness for planar shear layers.

δΓ (t) =1
ρ0 (∆Γ)2

Z R1
R2

ρ̄(Γ1 − Γ̃)(Γ̃ − Γ2 )drto the plot of δ99 , the evolution of the circulation thickness isa much smoother due to its integral form. Both time historiesof the shear layer thickness definitions are normalized by the
initial value and predict the same trend in growth rates withrespect to curvature, namely that the destabilizing configuration results in a growth rate four times larger than the stableconfiguration.
The growth of the circulation thickness normalized by
the initial velocity difference, δ̇Γ /∆u, indicates approximately
three periods of adjustment in the shear layer evolution. First,
a period of transient perturbation growth occurs, corresponding to the peaks in growth rates at t ∗ = 80 for the unstable caseand t ∗ = 140 for the stable case. The second period is a transition to well-developed turbulence, between t ∗ ∈ [80, 130] forthe unstable case and t ∗ = [150, 270] for the stable case. A similar evolution can be seen for the shear stress integrated acrossthe shear layer, when normalized by the instantaneous thickness δ99 ∆u2 . The distinction between this second transitional
period and the asymptotic growth period is not clearly defined,
but have been estimated based off these plots. The corresponding start times with respect to the shear layer evolution havebeen marked with transparent dotted lines in Fig. 1. The curvature ratios δΓ /rc during the periods chosen for collectionof statistics are given in Table 1. In this work, cases are referred to by their initial curvature levels. However, the overallchange in curvature intensity seen by the turbulent shear layer
are more moderate. For example, while the approximately planar configuration has an initial curvature of 0.001, its final curvature intensity as measured by δΓ /rc approaches that of theintermediate curvature cases. Lastly, Table 1 also presents the
final growth rate normalized by the initial velocity difference
for each case. These tabulated rates are not exactly constant,but represent the final turbulent growth rate measured before
eddies begin to be constrained by the computational domain.

(4)

An expression for the growth rate based off the circulation
thickness can be defined by taking the time derivative of Eqn.
41 . This analysis follows a similar derivation of planar shear
layer growth rate by Vreman et al., where the integrand is
first expanded and rewritten in terms of the mean momentum
and kinetic energy equations. The explicit expression used tocompute an integral growth rate, neglecting viscous effects, is′′ u′′ .
given below, where K = 21 ũ2θ , and R12 = ug
θ r

dδΓ
2
=
dt
ρ0 (∆Γ)2

Z R1

"
r

R2

∂
(rρ̄ ũr K + rũθ ρ̄R12 )
∂r
#
∂ ũθ
2 ∂ p̄
− r ρ̄R12+ r ũrdr∂r∂r
2

Various measures of shear layer evolution are shown in Fig. 1.
The left-most figure shows the evolution of the 99% thickness
for two high speed shear layers at Mc = 0.8 with stabilizing
and destabilizing curvature. After a period of transient growth,
the flow becomes turbulent once the shear layer has growth
to approximately twice its initial δ99 thickness. This occurs
around a normalized convective time of t ∗ = 100 for the unstable case, and t ∗ = 240 for the stable layer. While the transient
growth phase is not a focus of this study, it has been consistently observed that the more stable configurations require alonger transitional period before turbulence begins. Compared

Stable vs. unstable configuration
Unstable and stable configurations at δ (0)/rc = 0.01 andMc = 0.8 will be compared in this section. The velocity ratio

1 Note: Our previous derivation of this growth rate erroneously ex-

cluded terms from this integrand; the equation for integrated growthrate is corrected here.

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3: Mean profiles of stable and unstable configurations at Mc = 0.8 and δ0 /rc = 0.01.

Effects of increasing curvature intensity

is reciprocal between the stable and unstable cases, while thevelocity difference ∆u is initially the same2 . Differences inshear layer evolution can clearly be seen in Fig. 1. Transition
to turbulence and development into an asymptotically growing
turbulent state occurs about twice as quickly for the unstablecase compared to the stable case. Even during the peak growth
period of the stable shear layer, the growth rate remains lessthan the growth rate of the planar shear layer.
Instantaneous spanwise velocity fields for the two configurations are shown in Fig. 2. In all configurations, this azimuthal flow travels in the clockwise direction. Two distinct
differences between the turbulence in the two configurations
can be easily observed by comparing these two visualizations.
The magnitude of turbulent fluctuations in the unstable caseis larger than that of the stable case (both plots use the samecolor scale). This difference in turbulent stresses can also beseen in the profiles of TKE in Fig. 4. The peak TKE in theunstable case is nearly twice that of the stable case. Note thatthe striation along the shear layer is opposite for the stable andunstable cases; this direction of shearing for the turbulent eddies is consistent with the streamwise velocity gradient in thetwo cases. In the stable configurations, the streamwise velocity uθ is higher along the outer free stream; and in the unstableconfigurations, uθ is higher along the inner free stream. The
corresponding evolution of mean velocity profiles during multiple snapshots during turbulent growth are plotted by radiusr in Fig. 3 (a) to show spreading of the shear layer. There isa small, negative velocity gradient in each of the free stream
regions in order to ensure that the free streams have constantcirculation and do not introduce vorticity to the mixing layer.
Note that this requires the velocity difference across the shear
layer to also change as the shear layer spreads, though thechange is relatively small (see Table 1). For the stable cases,∆u decreases in time, whereas it increases in time for the unstable cases. In planar shear layers, ∆u remains constant andthe mean shear decreases as the shear layer grows. In the stable configuration, because the velocity difference decreases as
the shear layer grows, the mean shear contributing to TKE production and eddy growth are further reduced.
The mean circulation for these same profiles are plottedby (r − rc )/δ99 in Fig. 3 (b) to show approximate similarity
during a subset of the turbulent period. Likewise, collapsed
profiles of TKE and turbulent shear stress R12 , normalized by
the instantaneous velocity difference, are also compared. For
both the stable and unstable cases, the peak R12 magnitude isconsistently one third of the TKE magnitude. The shear stressis positive in the unstable case and negative in the stable case,which is consistent considering the production of shear stressis dependent on the mean velocity gradient.

The effects of increasing curvature for both stabilizing
and destabilizing curvature are shown in Fig. 4. The growthrate for the stable case with intermediate curvature, δ0 /rc =0.005, is lower than previously anticipated, and the final
growth rate approaches the final growth rate of the higher curvature case. A similar trend of closeness between TKE andshear stress magnitude can be seen between theses two cases
as well. An explanation for this unexpected trend in the stably
curved cases remains to be determined.
The curvature intensity appears to influence the transientperiods of the stable and unstable growth rates, such that caseswith intermediate curvature levels of δ0 /rc = 0.005 have athickness growth which is mostly bounded by the lower andhigher curvature cases. The asymptotic growth periods between the stable and unstable configurations show some notable differences. The unstable configurations appear to reach
their asymptotic growth period more quickly compared to thestable configurations. These periods of growth correspondto the regions where the integrated shear stress also begin toplateau. In contrast, the stabilizing configurations did not exhibit any region of growth which was definitively linear. The
upper left two plots in Fig. 4 instead show that the δΓ in thestable cases have progressively slower growth rates which continue to drift until the computational domain is exceeded. Normalized TKE and shear stress profiles for the unstable cases
have distinct magnitudes, with the intermediate curvature casesitting evenly between the low and higher curvature cases. In
contrast, the difference in TKE and shear stress magnitudes forthe stabilizing curvature cases is less evenly distributed. Consistent with the observed growth rates, the cases with initiallyhigher curvature have similar magnitudes of TKE and shear
stress which are distinct from the low curvature case.
If the turbulence in curved free shear layers is consideredsimilar to that in boundary layer flow along a concave or convex surface, structural differences in the mixing layer may be
expected. In particular, the presence of Görtler vorticies orother spanwise-coherent structures may be expected to formin the turbulent region. While some qualitative differences canbe observed in the mixing layer’s spanwise organization while
perturbations grow into turbulence, a clear difference coherent structures was not observed when the flow becamee fully
turbulent. This observation remained unaffected by compressibility levels. At both low and high speeds, during periods ofwell-developed turbulent growth, large-scale spanwise vorticies were not seen in the mixing region.
Small differences can be observed in instantaneous snapshots of streamwise vorticity for unstable configurations, as
shown in Fig. 5. These snapshots are shown for comparable
times during the shear layer evolution (δ99 (t)/δ99 (0) = 1.6).
A low curvature levels, the vortical region is compact andnearly uniform in height across the span of the mixing layer.

2 In a temporally developing, planar shear layer the growth rate depends on the velocity difference, and not the free stream velocity ratio.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 4: Evolution of thicknesses, growth rates, TKE, and R12 for stable (top) and unstable (bottom) cases at Mc = 0.8.

Figure 5: Instantaneous normalized streamwise vorticity at contour levels ωθ δ99 /∆u ∈ [−5, 5] along a spanwise (r − z)plane for unstable cases and rofiles of streamwise vortex stretching. The domain has been truncated for visualization
purposes. Top: Mc = 0.2 at δ0 /rc = 0.001 (left) and δ0 /rc = 0.01 (right). Bottom: Mc = 0.8 at δ0 /rc = 0.001 (left) andδ0 /rc = 0.01 (right).
At higher curvature levels, a less compact region of vorticity isobserved in the mixing region. In addition this sparsity, slightstretching of streamwise vorticity can be observed from theanisotropy of contour levels. This effect is more obvious forthe low speed cases, where elongated iso-contours of vorticity
are seen. The right-most profiles in Fig. 5 show the turbulent
vortex stretching term of the mean streamwise vorticity transport equation, ω ′j u′i, j , normalized by the instantaneous scales.
Three profiles at times leading up to the visualization time areplotted to indicate the drift in value during this transient period. Statistical noise is evident in the profiles, but a generalform of the vortex stretching profile can be gleaned. As seen
qualitatively in the instantaneous visualizations, the normalized stretching of streamwise vorticity is slightly amplified forincreased curvature intensity. The intermittent vorticity fieldalso suggests that higher levels of curvature achieve equivalent
shear layer thicknesses over a shorter development period.
These visualizations are shown to highlight the differences in the vortical structures during early shear layer evolution. However, as the turbulence developed further, larger

vortical structures or roll cells were not observed. Profiles of
streamwise velocity fluctuations, u′′θ , during the time period offully developed turbulence (δ99 (t)/δ99 (0) = 3) are shown in
Fig. 7. Streaks of elongated streamwise coherence are visible, but definite differences in the organization of u′′θ were notobserved. The apparent lack of large-scale spanwise structurein the fully turbulent regime is similar to experimental observations by Jeans & Johnston (1982) for incompressible turbulent boundary layers under concave curvature. Experiments by
Wang et al. (2017) showed clusters of spanwise vorticity andswirl in supersonic boundary layers under convex curvature,
but the clusters’ spanwise extent was not shown.
Regions of sparse vorticity may be expected as the shear
layer ‘falls apart’ during the early evolution of unstable cases,
compared to stable cases with a restoring force of the freestream pressure gradient. In stable cases, due to the velocity
ratio u2 /u1 < 1, the free stream pressure gradient in the radial
direction which scales as d p/dr ∼ u2 /r is largest along theouter free stream. As a result, it is possible that vorticies which
may otherwise stray into the outer free stream flow from the5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
main mixing region are pushed back by the increasing meanpressure further from the mixing layer. The parameter governing this balance between shear and centrifugal production
can be written similarly to a flux Richardson number R f forstratified flow. As discussed in detail by Bradshaw (1973) andlater revisited by Nagata & Kasagi (2004), an analogous R f
for centrifugal flows can be written considering the productionof turbulent stresses Rθ θ and Rrr . Taking the first term of Pθ θ
as the shear production term and R12 ũθ /r as the centrifugal
production term of Pθ θ and Prr , the balance between shear andcentrifugal terms is given in Eqn. 5.

Pθ θ = −2R12



ũθ
r



ũ
ũ
− 4R12 θ
Prr = 4R12 θrr. ∂  ũ θR f = 2ũθ r2
(5)
∂r r

∂
∂r

Profiles of R f for stable and unstable flows, plotted withsolid lines in Fig. 6, indicate increasing curvature for unstable flows gives a more negative value of R f in the mixing
layer and increasing curvature for stable flows gives a morepositive value of R f , especially near the mixing layer edges.
For the stable cases, the velocity gradient includes two zerocrossings which cause the discontinuity in R f near the mixing
layer edges. Both the stable and unstable configurations giveR f ≈ −1 in the free streams, which implies the two terms in
the Rθ θ production cancel and the TKE production is zero.
Thus, this definition of R f confirms that the free streams arestable. The parameter for Eckhoff’s stability criterion, Φ as
defined in Eqn. 1, is also plotted with dashed lines in Fig. 6 todemonstrate the centrifugal stability of the curved shear layer
is dependent only on the shear layer region and not on the freestream behavior.

Figure 7:

Instantaneous visualizations of normalized streamwise velocity fluctuations at contour levelsu′′θ /∆u ∈ [−0.5, 0.5] along a streamwise (θ − z) plane
for centrifugally unstable cases. The streamwise extent
has been truncated by 50% for visualization purposes.

Comparison to compressibility effects
The effect of compressibility in planar shear layers is afour- to five-fold suppression of growth rate between incompressible and highly compressible shear layers (Papamoschou& Roshko (1988)). For the Mc range in this work, which
varies from quasi-incompressible at Mc = 0.2 to moderately
compressible at Mc = 0.8, the data shows ≥ 50% decrease in
growth rates. As given in Table 1, at low, stabilizing curvatures, the growth rate reduction is 0.036/0.071 = 0.507. At
higher, destabilizing curvatures, the growth rate reduction is0.079/0.14 = 0.564. In comparison, the difference in growthrate δΓ /∆u between the most stable and the most unstablecases gives a factor of four difference at both Mc = 0.8 and

Mc = 0.2. In the present data, the compressibility effect remains relevant across the ranges of curvature intensities explored. The trends observed suggest that, for the given rangeof curvature intensities, the effects of stabilizing and destabilizing curvature may be just as important as compressibility
effects.

CONCLUSIONS & FUTURE WORKThis work augments our prior investigation of turbulencein curved, compressible shear layers. Two shear layer thicknesses based on the mean circulation profile are used andthickness growth rates for stable and unstable cases are presented. While a period of turbulent stress collapse exists, theflow is not perfectly self similar. Increasing curvature intensity
increases turbulent kinetic energy and shear stress for unstably
curved cases, and slightly decreases the stress magnitudes forstably curved cases. An analogous ‘flux’ Richardson number
is evaluated and defines the stability of the flow consistent with
Eckhoff’s and Rayleigh’s centrifugal stability criteria. DNS
calculations of the spatially developing equivalent of this temporal shear layer are in progress.

Figure 6: Profiles of R f (solid lines) and Φ (dashed lines)

for high speed stable (left) and unstable (right) caseswith increasing curvature intensity.
6

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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