[1]Omar Es-Sahli

[1]Department of Aerospace Engineering, Mississippi State University, 75 B. S.
Hood Rd, Mississippi State, 39762, MS, USA

2]Institute of Fluid Science, Tohoku University, 2 Chome-1-1 Katahira, Sendai,
980-8577, Japan

# Gauging Centrifugal Instabilities in Compressible Free-Shear Layers Via
Nonlinear Boundary Region Equations

Adrian Sescu  Yuji Hattori  *  [

###### Abstract

Curved free shear layers emerge in many engineering problems involving complex
flow geometries, such as the flow over a backward facing step, flows with wall
injection in a boundary layer, the flow inside side-dump combustors, or wakes
generated by vertical axis wind turbines, among others. Previous studies
involving centrifugal instabilities have mainly focused on wall-flows where
Taylor instabilities between two rotating concentric cylinders or Görtler
vortices in boundary layers, resulting from the imbalance between the
centrifugal forces and the radial pressure gradients, are generated. Curved
free shear layer flows, however, have not received sufficient attention,
especially in the nonlinear regime. The present work investigates the
development of centrifugal instabilities evolving in a curved free shear layer
flow in the nonlinear compressible regime. The compressible Navier-Stokes
equations are reduced to the nonlinear boundary region equations (BRE) in a
high Reynolds number asymptotic framework wherein the streamwise wavelengths
of the disturbances are assumed to be much larger than the spanwise and wall-
normal counterparts. We study the effect of the freestream Mach number
$\boldsymbol{M_{\infty}}$, the shear layer thickness $\boldsymbol{\delta}$,
the amplitude of the incoming disturbance $\boldsymbol{A}$, and the relative
velocity difference across the shear layer $\boldsymbol{\Delta V}$ on the
development of these centrifugal instabilities. Our parametric study shows,
among other things, that the kinetic energy of the curved shear layer flow is
directly proportional to $\boldsymbol{\Delta V}$ and $\boldsymbol{A}$ while it
is inversely proportional to $\boldsymbol{\delta}$.

##  1 Introduction

The stability of curved shear layer flows depends on the velocity difference
across the shear layer and the radius of the curvature. For a free shear layer
with no curvature, as known as a plane shear layer, the Kelvin-Helmholtz
instability is the dominant turbulent mechanism [8]. In this case, the two-
dimensional disturbances are more unstable than their three-dimensional
counterpart [15], and the Kelvin-Helmholtz instability introduces
predominantly spanwise oriented vortices. Rayleigh [12] proved that the
presence of an inflection point in the basic velocity profile is a necessary
condition for the Kelvin-Helmholtz instability. On the other hand, for a
curved mixing layer flow with an inflectional velocity profile, the Kelvin-
Helmholtz instability mechanism is still present, along with centrifugal
instabilities in the form of streamwise oriented Görtler type vortices.

Görtler vortices are mostly known to appear inside a boundary layer flow along
a concave surface due to the imbalance between radial pressure gradients and
centrifugal forces (e.g., Gortler [4], Hall [5], Swearingen & Blackwelder
[16]). For highly curved walls, for example, vortex formation occurs more
rapidly and can significantly alter the mean flow causing the laminar flow to
transition into turbulence. Under certain conditions, Görtler vortices can be
efficient precursors to transition. The growth rate of these counter-rotating
streamwise vortical structures depends on the surface curvature and the
receptivity of the boundary layer to freestream disturbances and surface
imperfections. This type of instability is predominant in many engineering
applications. Thus, understanding the processes leading to their development
and predicting their occurrence using efficient and tractable methods could
potentially advance the overall understanding of turbulence.

Numerous theoretical and numerical studies covered centrifugal instabilities
in incompressible curved free shear layer flows. Plesniak et al. [10, 11]
conducted extensive experimental measurements investigating curved two-stream
mixing layers to show how centrifugal effects yield streamwise vortices. The
untripped case within this suite of experiments exhibited organized streamwise
vorticity, while the tripped case did not. The researchers explain that this
is due to the spatially stationary streamwise vortices that provide extra
entrainment to the flow in the tripped case (Bell & Mehta [1] showed this for
a plane mixing layer). Hu et al. [6] and Liou [7] focused on the effect of the
curvature on the inflectional Rayleigh modes, which they found to be minimal,
although the curvature excites an unstable three-dimensional disturbance with
the amplitude increasing as the streamwise wavenumber decreases. The
analytical and numerical study of Otto et al. [9] showed that the unstable
modes largely depend on surface curvature. They also employed numerical
simulations to solve the parabolic equations, assuming that the wavenumber and
Görtler number are both of order one. They found that as the difference
between the freestream speeds increased, the layer became more susceptible to
centrifugal instabilities.

In the present work, we analyze the development of centrifugal instabilities
in high-speed compressible curved free shear layer flows via an efficient
numerical algorithm based on the nonlinear boundary region equations (NBREs) –
a parabolized version of the Navier-Stokes equations under the assumption that
the streamwise wavenumber associated with the disturbances is much smaller
than the cross-flow wavenumbers. The study considers the effects of a wide
range of Mach numbers, the amplitude of the freestream disturbance, $A$, the
shear layer thickness, $\delta$, and the velocity difference across the shear
layer, $\Delta V$, on the development and growth of these centrifugal
instabilities. The study shows, among other things, that the kinetic energy
level of the curved shear layer flow is directly proportional to $\Delta V$
and $A$ while it is inversely proportional to $\delta$. Increasing $A$ induces
larger instability structures, as expected, which may be beneficial for
enhancing mixing. The location of the maximum energy moves farther downstream
as the freestream Mach number increases.

##  2 Problem formulation and numerical algorithm

###  2.1 Scalings

All dimensional spatial coordinates $(x^{*},y^{*},z^{*})$ are normalized by
the spanwise separation $\lambda^{*}$, while the dependent variables by their
respective freestream values, except the pressure, which is normalized by the
dynamic pressure:

$\displaystyle\bar{t}=\frac{t^{*}}{\lambda^{*}/V_{\infty}^{*}};\hskip
11.38109pt\bar{x}=\frac{x^{*}}{\lambda^{*}};\hskip
11.38109pt\bar{y}=\frac{y^{*}}{\lambda^{*}};\hskip
11.38109pt\bar{z}=\frac{z^{*}}{\lambda^{*}}$ (1)

$\displaystyle\bar{u}=\frac{u^{*}}{V_{\infty}^{*}};\hskip
11.38109pt\bar{v}=\frac{v^{*}}{V_{\infty}^{*}};\hskip
11.38109pt\bar{w}=\frac{w^{*}}{V_{\infty}^{*}};\hskip
11.38109pt\bar{\rho}=\frac{\rho^{*}}{\rho_{\infty}^{*}}$ (2)

$\displaystyle\bar{p}=\frac{p^{*}-p_{\infty}^{*}}{\rho_{\infty}^{*}V_{\infty}^{*2}};\hskip
11.38109pt\bar{T}=\frac{T^{*}}{T_{\infty}^{*}};\hskip
11.38109pt\bar{\mu}=\frac{\mu^{*}}{\mu_{\infty}^{*}};\hskip
11.38109pt\bar{k}=\frac{k^{*}}{k_{\infty}^{*}}$ (3)

where $\lambda^{*}$ is the spanwise wavelength of the disturbances,
$(u^{*},v^{*},w^{*})$ are the velocity components, $\rho^{*}$ the density,
$p^{*}$ is pressure, $T^{*}$ temperature, $\mu^{*}$ dynamic viscosity, $k^{*}$
thermal conductivity, and all quantities with $\infty$ at the subscript
represent conditions at infinity.

Reynolds number based on the spanwise separation, Mach number and Prandtl
number are defined as

$\displaystyle
R_{\lambda}=\frac{\rho_{\infty}^{*}V_{\infty}^{*}\lambda^{*}}{\mu_{\infty}^{*}},\hskip
14.22636ptM_{\infty}=\frac{V_{\infty}^{*}}{a_{\infty}^{*}},\hskip
14.22636ptPr=\frac{\mu_{\infty}^{*}C_{p}}{k_{\infty}^{*}}$ (4)

where $\mu_{\infty}^{*}$, $a_{\infty}^{*}$ and $k_{\infty}^{*}$ are freestream
dynamic viscosity, speed of sound and thermal conductivity, respectively, and
$C_{p}$ is the specific heat at constant pressure. As for boundary layer flows
over curved surfaces, we here define the equivalent global Görtler number as

$\displaystyle G_{\lambda}=\frac{R_{\lambda}^{2}\lambda^{*}}{r^{*}}$ (5)

where $r^{*}$ is the radius of the curvature.

###  2.2 Boundary region equations: a parabolized form of the Navier-Stokes
equations

If the streamwise wavenumber of the disturbances evolving inside the shear
layer are much larger that the wavenumbers corresponding to the crossflow
directions, then the Navier-Stokes equations can be transformed into a
parabolic set of equations in the framework of high Reynolds number
asymptotics.

For a full compressible, Newtonian flow, the primitive form of the Navier-
Stokes equations with non-dimensional variables are considered here in the
form

$\displaystyle\frac{D\bar{\rho}}{Dt}+\rho\left(\frac{\partial\bar{u}}{\partial\bar{x}}+\frac{\partial\bar{v}}{\partial\bar{y}}+\frac{\partial\bar{w}}{\partial\bar{z}}\right)=0$
(6)

$\displaystyle\bar{\rho}\frac{D\bar{u}}{D\bar{t}}=-\frac{\partial\bar{p}}{\partial\bar{x}}+\frac{1}{Re_{\lambda}}\frac{\partial}{\partial\bar{x}}\left[\frac{2}{3}\mu\left(2\frac{\partial\bar{u}}{\partial\bar{x}}-\frac{\partial\bar{v}}{\partial\bar{y}}-\frac{\partial\bar{w}}{\partial\bar{z}}\right)\right]+\frac{\partial}{\partial\bar{y}}\left[\mu\left(\frac{\partial\bar{u}}{\partial\bar{y}}+\frac{\partial\bar{v}}{\partial\bar{x}}\right)\right]+\frac{\partial}{\partial\bar{z}}\left[\mu\left(\frac{\partial\bar{w}}{\partial\bar{x}}+\frac{\partial\bar{u}}{\partial\bar{z}}\right)\right]$
(7)

$\displaystyle\bar{\rho}\frac{D\bar{v}}{D\bar{t}}=-\frac{\partial\bar{p}}{\partial\bar{y}}+\frac{1}{Re_{\lambda}}\frac{\partial}{\partial\bar{y}}\left[\frac{2}{3}\mu\left(2\frac{\partial\bar{v}}{\partial\bar{y}}-\frac{\partial\bar{u}}{\partial\bar{x}}-\frac{\partial\bar{w}}{\partial\bar{z}}\right)\right]+\frac{\partial}{\partial\bar{x}}\left[\mu\left(\frac{\partial\bar{v}}{\partial\bar{x}}+\frac{\partial\bar{u}}{\partial\bar{y}}\right)\right]+\frac{\partial}{\partial\bar{z}}\left[\mu\left(\frac{\partial\bar{v}}{\partial\bar{z}}+\frac{\partial\bar{w}}{\partial\bar{y}}\right)\right]$
(8)

$\displaystyle\bar{\rho}\frac{D\bar{w}}{D\bar{t}}=-\frac{\partial\bar{p}}{\partial\bar{z}}+\frac{1}{Re_{\lambda}}\frac{\partial}{\partial\bar{z}}\left[\frac{2}{3}\mu\left(2\frac{\partial\bar{w}}{\partial\bar{z}}-\frac{\partial\bar{u}}{\partial\bar{x}}-\frac{\partial\bar{v}}{\partial\bar{y}}\right)\right]+\frac{\partial}{\partial\bar{x}}\left[\mu\left(\frac{\partial\bar{w}}{\partial\bar{x}}+\frac{\partial\bar{u}}{\partial\bar{z}}\right)\right]+\frac{\partial}{\partial\bar{y}}\left[\mu\left(\frac{\partial\bar{v}}{\partial\bar{z}}+\frac{\partial\bar{w}}{\partial\bar{y}}\right)\right]$
(9)

$\displaystyle\bar{\rho}\frac{D\bar{T}}{D\bar{t}}=\frac{1}{PrRe_{\lambda}}\left[\frac{\partial}{\partial\bar{x}}\left(k\frac{\partial\bar{T}}{\partial\bar{x}}\right)+\frac{\partial}{\partial\bar{y}}\left(k\frac{\partial\bar{T}}{\partial\bar{y}}\right)+\frac{\partial}{\partial\bar{z}}\left(k\frac{\partial\bar{T}}{\partial\bar{z}}\right)\right]$
$\displaystyle-$
$\displaystyle(\gamma-1)M_{\infty}^{2}\left[p\left(\frac{\partial\bar{u}}{\partial\bar{x}}+\frac{\partial\bar{v}}{\partial\bar{y}}+\frac{\partial\bar{w}}{\partial\bar{z}}\right)-\frac{2}{3}\mu\left(\frac{\partial\bar{u}}{\partial\bar{x}}+\frac{\partial\bar{v}}{\partial\bar{y}}+\frac{\partial\bar{w}}{\partial\bar{z}}\right)^{2}\right]$
$\displaystyle+$
$\displaystyle(\gamma-1)M_{\infty}^{2}\frac{\mu}{Re_{\lambda}}\left[2\left(\frac{\partial\bar{u}}{\partial\bar{x}}\right)^{2}+2\left(\frac{\partial\bar{v}}{\partial\bar{y}}\right)^{2}+2\left(\frac{\partial\bar{w}}{\partial\bar{z}}\right)^{2}+\left(\frac{\partial\bar{u}}{\partial\bar{y}}+\frac{\partial\bar{v}}{\partial\bar{x}}\right)^{2}+\left(\frac{\partial\bar{w}}{\partial\bar{x}}+\frac{\partial\bar{u}}{\partial\bar{z}}\right)^{2}+\left(\frac{\partial\bar{v}}{\partial\bar{z}}+\frac{\partial\bar{w}}{\partial\bar{y}}\right)^{2}\right]$

where

$\displaystyle\frac{D}{D\bar{t}}=\frac{\partial}{\partial\bar{t}}+\bar{u}\frac{\partial}{\partial\bar{x}}+\bar{v}\frac{\partial}{\partial\bar{y}}+\bar{w}\frac{\partial}{\partial\bar{z}}$
(11)

is the substantial derivative. The pressure $\bar{p}$, the temperature
$\bar{T}$ and the density $\bar{\rho}$ of the fluid are combined in the
equation of state in non-dimensional form, $\bar{p}=\bar{\rho}\bar{T}/\gamma
M_{\infty}^{2}$, assuming that non-chemically-reacting flows are considered.
Other notations include the dynamic viscosity $\mu$, and the free-stream Mach
number $M_{\infty}=V_{\infty}^{*}/a_{\infty}^{*}$. The dynamic viscosity $\mu$
and thermal conductivity $k$ are linked to the temperature using a power law
in dimensionless form,

$\displaystyle\mu=T^{b};\hskip 17.07164ptk=\frac{C_{p}\mu}{Pr}$ (12)

where $b=0.76$ (Ricco & Wu [13]), $C_{p}=\gamma R/(\gamma-1)$, $\gamma=1.4$,
and $Pr=0.72$ for air.

We re-scale the streamwise distance and time co-ordinate at which the vortex
system forms by the following $\mathcal{O}(1)$ variables:
$x=\bar{x}/R_{\lambda}$, and the time as $t=\bar{t}/R_{\lambda}$. Note that
the distance in the wall-normal and spanwise directions are the same,
$y=\bar{y}$, $z=\bar{z}$. Another thing to mention is that, in this region,
the crossflow velocity component is small compared to the streamwise velocity
component, and pressure variations are negligible. Appropriate dominant
balance considerations suggest that the dependent variables in this region
must also re-scale as follows:

$\displaystyle u=\bar{u};\hskip 11.38109ptv=\bar{v}/R_{\lambda};\hskip
11.38109ptw=\bar{w}/R_{\lambda};\hskip 11.38109pt\rho=\bar{\rho};$
$\displaystyle p=\bar{p}/R_{\lambda}^{2};\hskip 11.38109ptT=\bar{T};\hskip
11.38109pt\mu=\bar{\mu};\hskip 11.38109ptk=\bar{k};$ (13)

Working out the order-of-magnitude analysis of the Navier-Stokes equations, we
obtain the parabolic set of equations, which we refer to as the nonlinear
compressible boundary region equations (NCBRE)

$\vec{V}\cdot\nabla\rho+\rho\nabla\cdot\vec{V}=0$ (14)

$\rho\vec{V}\cdot\nabla u=\nabla_{c}\cdot\left(\mu\nabla_{c}u\right)$ (15)

$\rho\vec{V}\cdot\nabla v+G_{\lambda}u^{2}=-\frac{\partial p}{\partial
y}+\frac{\partial}{\partial y}\left[\frac{2}{3}\mu\left(3\frac{\partial
v}{\partial y}-\nabla\cdot\vec{V}\right)\right]+\frac{\partial}{\partial
x}\left(\mu\frac{\partial u}{\partial y}\right)+\frac{\partial}{\partial
z}\left[\mu\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial
y}\right)\right]$ (16)

$\rho\vec{V}\cdot\nabla w=-\frac{\partial p}{\partial
z}+\frac{\partial}{\partial z}\left[\frac{2}{3}\mu\left(3\frac{\partial
w}{\partial z}-\nabla\cdot\vec{V}\right)\right]+\frac{\partial}{\partial
x}\left(\mu\frac{\partial u}{\partial z}\right)+\frac{\partial}{\partial
y}\left[\mu\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial
y}\right)\right]$ (17)

$\rho\vec{V}\cdot\nabla
T=\frac{1}{Pr}\nabla_{c}\cdot\left(k\nabla_{c}T\right)+(\gamma-1)M_{\infty}^{2}\mu\left[\left(\frac{\partial
u}{\partial y}\right)^{2}+\left(\frac{\partial u}{\partial
z}\right)^{2}\right]$ (18)

where $\vec{V}$ is the velocity vector and $\nabla_{c}$ is the crossflow nabla
operator:

$\vec{V}=u\vec{i}+v\vec{j}+w\vec{k};\hskip
17.07164pt\nabla_{c}=\frac{\partial}{\partial
y}\vec{j}+\frac{\partial}{\partial z}\vec{k}$ (19)

The effect of the wall curvature is contained in the term involving the global
Görtler number $G_{\lambda}$ in the second momentum equation.

A small artificial disturbance is imposed at the inflow boundary in the form:

$v^{\prime}=A\cos\left(\frac{\pi
z}{\lambda}\right)\exp\left[-\frac{(y-y_{0})^{2}}{\sigma^{2}}\right]$ (20)

where $A$ is a small amplitude (in this study $A=0.04$). $\lambda^{*}$ is the
spanwise wavenumber (dictating the spanwise separation of the centrifugal
instabilities), and $\sigma$ represents the extent of the disturbance in the
$y$ direction. In the present work, $\lambda$ is kept constant at 0.8 cm. Es-
Sahli et al. [2] elaborately studied the effect of $\lambda^{*}$ on the
development of centrifugal instabilities in curved free shear layers. The
NCBREs are solved using the algorithm developed in Es-Sahli et al. [3] and
previously in Sescu and Thompson [14]. We slightly adjusted the algorithm to
accommodate the free shear layer setting.

##  3 Results

In this section, we present and discuss a set of results from the curved free
shear layer numerical simulations. The flow domain is split into ‘fast’ and
‘slow’ streams, both having velocities $V_{f}$ and $V_{s}$, respectively,
which we define as; $V_{f}=V_{\infty}$ and $V_{s}=(1-\Delta V)V_{\infty}$,
where $\Delta V$ is the relative velocity difference (here, we set this
difference at four levels, $20\%$, $30\%$, $40\%$, and $50\%$). The
temperature in the fast stream is set to $T_{\infty}$, while in the slow
stream is set to $0.9T_{\infty}$. In our parametric study, we consider three
Mach numbers in the fast stream of $2.0$, $4.0$, and $6.0$, respectively (the
lowest velocity in the slow stream will correspond to a Mach number of $1.0$).
We also consider three values for the shear layer thickness $\delta$ at the
inflow boundary at $0.2$, $0.4$, and $0.6$, where the velocity variation
between the fast and slow streams is modeled via a hyperbolic tangent function
$0.5(1-\tanh{(y-y_{0})/\delta})$, whith $y_{0}$ representing the location of
the shear layer. A similar function is used to model the variation of the
temperature in the shear layer, with the same thickness, although in reality
the thickness of the thermal layer may be slightly different.

The Reynolds number $R_{\lambda}$ based on the faster freestream and the
spanwise separation of the disturbance and the global Görtler number
$G_{\lambda}$ are kept the same for all cases at $10^{6}$ and $2\times
10^{5}$, respectively (the kinematic viscosity and the curvature of the wall
were varied to achieve constant $R_{\lambda}$ and $G_{\lambda}$ for all
simulations). The grid is uniform in the spanwise direction taking into
account that the flow is periodic in this direction, while in the radial
direction the grid is stretched towards the top and bottom far-field
boundaries. The marching in the streamwise direction is achieved by means of
an explixit Euler method, with equally-spaced discretization. At the inflow
boundary, centrifugal instabilities are excited by the non-dimensional
artificial disturbance in equation 20 imposed on the base flow at the inflow
boundary, with the amplitude set at $0.04$.

Figure 1: Contour plots of the streamwise velocity $\boldsymbol{u}$ at
different streamwise locations for $M=2$ and $\delta=0.2$: top -
$V_{s}=80\%V_{f}$; bottom - $V_{s}=50\%V_{f}$.

Figure 2: Contour plots of the streamwise velocity $\boldsymbol{u}$ at
different streamwise locations for $M=4$ and $\delta=0.2$: top -
$V_{s}=80\%V_{f}$; bottom - $V_{s}=50\%V_{f}$.

Figure 3: Contour plots of the streamwise velocity $\boldsymbol{u}$ at
different streamwise locations for $M=6$ and $\delta=0.2$: top -
$V_{s}=80\%V_{f}$; bottom - $V_{s}=50\%V_{f}$.

Figures 1-3 present consecutive contour plots depicting the magnitude of
crossflow velocity for different Mach numbers and velocity differences $\Delta
V$, with the shear layer thickness $\delta$ set to $0.2$. The color scheme
represents the flow velocity, with white at the top indicating the fast stream
and black at the bottom indicating the slow stream. These contour plots
illustrate the progression of the centrifugal instabilites in the streamwise
direction; we hypothesize that the incoming disturbance is sufficiently high
to allow the centrifugal instabilities to take over the Kelvin-Helmholtz type
instabilities (this mathematical model is not capable of predicting Kelvin-
Helmholtz type instabilities because the problem is steady and the system of
equations is parabolic). To characterize the shape of the centrifugal
instabilities, we identify two types of structures: "primary" and "secondary."
The primary flow structure refers to the mushroom-like formation that evolves
as the main instability, as exemplified by the fourth panel of the first row
in figure 1. On the other hand, the secondary structures are elongated
features that emerge from the edges of the primary flow structure, as seen,
for example, in the fourth panel of the second row in figure 2.

By comparing the first to the second row of each figure 1, 2 or 3, we notice
that by increasing the velocity difference across the shear layer, $\Delta V$,
accelerates the development of the mushroom-like primary structures and makes
the secondary structures more prominent. The color transition from black to
white seen in the bottom rows of each of these figures suggest that the higher
the velocity difference across the shear layer, the more intense the mixing in
the shear layer is. Comparing the centrifugal instabilities at the same
streamwise coordinate across various Mach numbers, we observe a delay in the
growth of the mushroom-like structures as $M_{\infty}$ increases. We also
observe that the flow structures become thinner as the Mach number increases.
For instance, in the third panel of the first row in figure 1, for which
$M_{\infty}=2$, the mushroom structure has already begun to form, whereas in
the corresponding panel of figure 3, where $M_{\infty}=6$, the mushroom shape
has not yetr developed. This finding suggests that, for high Mach number free
shear layer flows, the same mixing efficiency is achieved further downstream.

To quantify the thermal effects of these instabilities, vertical profiles of
temperature disturbance $T_{d}(x,y,z)=T(x,y,z)-T_{m}(x,y)$, where $T_{m}(x,y)$
is the spanwise mean component of temperature $T(x,y,z)$, through the center
of the mushroom shape are included in figures 4-6. The profiles are compared
to each other for different shear layer thicknesses and different velocity
difference levels across the shear layer. They all show that increasing the
thickness of the shear layer increases the temperature disturbance, and that
this increase is slightly less significant at high Mach numbers. As expected,
by increasing the velocity difference across the shear layer increases the
amplitude levels of the temperature disturbance at all Mach number. Also, the
vertical extent of these disturbances seem to increase with increasing the
velocity difference, especially towards the slow stream (for example, in
figure 4, the top boundary of the disturbance is roughly in $y=1$ for all
velocity differences, while the bottom boundary is roughly in $y=0.9$ for
$\Delta V=80\%$ and in $y=1.2$ for $\Delta V=50\%$).

Figure 4: Profiles of temperature disturbance $T_{d}(y)$ for the
$\boldsymbol{M=2}$ case. Figure 5: Profiles of temperature disturbance
$T_{d}(y)$ for the $\boldsymbol{M=4}$ case. Figure 6: Profiles of temperature
disturbance $T_{d}(y)$ for the $\boldsymbol{M=6}$ case.

Figure 7: Vortex energy distribution of different parametric settings for the
$\boldsymbol{M=2}$ case.

Figure 8:  Vortex energy distribution of different parametric settings for the
$\boldsymbol{M=4}$ case.

Figure 9: Vortex energy distribution of different parametric settings for the
$\boldsymbol{M=6}$ case.

The kinetic energy distribution is calculated as

$E(x)=\frac{1}{\Delta
V^{2}}\intop_{z_{1}}^{z_{2}}\intop_{0}^{\infty}\left[\left|u(x,y,z)-u_{m}(x,y)\right|^{2}+\left|v(x,y,z)-v_{m}(x,y)\right|^{2}+\left|w(x,y,z)-w_{m}(x,y)\right|^{2}\right]dzdy,$
(21)

where $u_{m}(x,y)$, $v_{m}(x,y)$, and $w_{m}(x,y)$ are the spanwise mean
components of velocity, and $z_{1}$ and $z_{2}$ are the coordinates of the
boundaries in the spanwise direction (note that the energy here is also scaled
by $\Delta V^{2}$, which implies that the energy variation is the result of
disturbances developing in the shear layer). In figures 7-9, we plot $E(x)$
for different values of $\Delta V$ and in figure 10 we plot the same energy
for different values of $\delta$. The disturbance energy $E(x)$ seems to be
directly proportional to $\Delta V$ as it is highest for $\Delta V=50\%$ and
decreases as $\Delta V$ is reduced for all considered cases. Moreover, the
streamwise location of the energy saturation (the point at which the energy
starts to level off) moves farther downstream as $\Delta V$ decreases,
especially for higher Mach numbers (see figure 9). On the other hand, as seen
in figure 10, $E(x)$ is inversely proportional to $\delta$, which may be
because viscous effects are more predominant in thicker shear layers; in this
figure, we superposed the smallest (solid line) and the highest (dashed line)
velocity difference levels, indicating that the the trend is the same for both
(the other two velocity difference levels fall in between). We must point out
that the energy reduction due to the increase in the shear layer thickness is
rather significant, perhaps because the variation of the shear layer thickness
is substantial (th e largest $\delta$ is three times greater that the smallest
$\delta$).

(a) (b) (c)

Figure 10: Effect of the shear layer thickness $\delta$ variation on the
vortex energy distribution for $\Delta V=80\%$ (solid lines) and $\Delta
V=50\%$ (dashed lines): a) $M=2$; b) $M=4$; c) $M=6$.

##  4 Conclusions

In this paper, we investigate the nonlinear development of centrifugal
instabilities in a compressible curved free shear layer flow using a numerical
solution to the boundary region equations, specifically, a parabolized form of
the Navier-Stokes equations. Our focus lies on understanding the
characteristics of these centrifugal instabilities, which exhibit similarities
with the development of G"ortler vortices in boundary layer flows over concave
surfaces. The study encompasses variations in the free stream Mach number
($M_{\infty}$), the relative velocity difference between the two streams of
the shear layer ($\Delta V$), the shear layer thickness ($\delta$), and the
amplitude of the inflow disturbance ($A$).

Upon closer examination of the kinetic energy plots for the $M_{\infty}=2$
case in figure 7, we observe that $E(x)$ is directly proportional to $\Delta
V$ and inversely proportional to $\delta$ across all the considered Mach
numbers. However, the increase in shear layer thickness has an insignificant
effect on energy reduction, with a mere $1\%$ drop in $E$ resulting from a
$100\%$ increase in $\delta$, as evident in figure 10. Furthermore, increasing
the amplitude of the inflow disturbance ($A$) slightly boosts the kinetic
energy, with less than $1\%$ increase in $E$ observed for a $100\%$ increase
in $A$. Interestingly, a larger magnitude of $A$ hampers the influence of the
relative velocity difference. The energy curves corresponding to different
$\Delta V$ values exhibit a considerably reduced gap when comparing the
$A=0.02$ and $A=0.04$ cases. Similar trends were observed in the parametric
study of centrifugal instability development for the $M_{\infty}=4$ and
$M_{\infty}=6$ cases.

Examining the contour plots of crossflow velocity magnitude in figures 1-3 for
$\Delta V=30\%$, we find that increasing the disturbance amplitude leads to
significant growth in the mushroom-like structure’s amplitude and renders the
secondary structures more visible, indicating increased mixing for all Mach
numbers under consideration. When comparing different Mach numbers at the same
crossflow plane location, we observe a slower development of mushroom-like
structures as $M_{\infty}$ increases, suggesting that achieving the same
mixing efficiency would require traveling further downstream for higher Mach
numbers. Consequently, numerical simulations for higher Mach number cases
would incur higher computational costs due to the need for larger grid sizes
to maintain comparable mixing efficiency.

## References

  * [1] Bell, J. H. & Mehta, R. D. (1990) Development of a two-stream mixing layer within tripped and untripped boundary layers, AIAA J., Vol. 28, pp. 2034-2042.
  * [2] Es-Sahli, O., and Sescu, A. and Afsar, M. (2020) Nonlinear centrifugal instabilities in curved free shear layers, AIAA Scitech, AIAA 2020-2239.
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