J. Fluid Mech. (2023), vol. 973, A36, doi:10.1017/jfm.2023.768
Study of the linear models in estimating coherent
velocity and temperature structures for
compressible turbulent channel flows
Xianliang Chen1 , Cheng Cheng2 , Jianping Gan1 and Lin Fu1,2,3, †
1 Department
of Mathematics and Center for Ocean Research in Hong Kong and Macau (CORE), The
Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PR China
2 Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and
Technology, Clear Water Bay, Kowloon, Hong Kong, PR China
3 HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute, Futian, Shenzhen, PR China
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
(Received 6 June 2023; revised 6 August 2023; accepted 11 September 2023)
Linear models, based on stochastically forced linearized equations, are deployed for
spectral linear stochastic estimation (SLSE) of the velocity and temperature fluctuations
in compressible turbulent channel flows with a bulk Mach number of 1.5. Through
comparison with the direct numerical simulation (DNS) data, an eddy-viscosity-enhanced
model (eLNS) outperforms the one not enhanced (LNS) in computing the coherence and
amplitude ratio of streamwise velocity at different wall-normal heights, but they both
largely deviate from DNS regarding the temperature prediction. For further investigation,
the eigenspectra and pseudospectra of the linear operators are scrutinized. The eddy
viscosity is shown to stabilize the eigenmodes and decrease the non-normality of the
vortical modes. Consequently, the relative importance of acoustic and entropy modes
increases, and they can contribute 20 % to 55 % of the response growth, which is not
supported by DNS. Hence, it is an intrinsic defect of the eLNS model introduced by
turbulence modelling. After a procedure of cospectrum decomposition, the contributions
of acoustic and entropy components are filtered out. The resulting SLSE quantities for
velocity, temperature and their coupling are basically agreeable with DNS, demonstrating
that the coherent temperature fluctuation is dominated by advection and other vortical
motions, instead of the compressibility effects. Moreover, a parameter study of Reynolds
and Mach numbers (from 0.3 to 4) is conducted. The semi-local units are shown to well
collapse the velocity SLSE quantities to the incompressible case for streamwise-elongated
structures of high coherence.
† Email address for correspondence: linfu@ust.hk
© The Author(s), 2023. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/
licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original
article is properly cited.
973 A36-1
X. Chen, C. Cheng, J. Gan and L. Fu
Key words: turbulent boundary layers, turbulence modelling, compressible turbulence
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
1. Introduction
Understanding and modelling compressible turbulent flows are fundamentally significant
for aerodynamic applications (Bradshaw 1977). Turbulent boundary layers strongly
affect the surface drag and heat transfer, so accurate predictive models are important
for reliable vehicle design and flow control (Gatski & Bonnet 2013). Compared with
the incompressible one, additional thermodynamic processes are present in high-speed
turbulent flows, such as heat transfer, acoustic fluctuations, dilatational work and
high-enthalpy effects (Di Renzo, Fu & Urzay 2020; Di Renzo & Urzay 2021; Fu et al.
2021; Fu, Bose & Moin 2022). Due to the theoretical complexity and practical significance,
compressible wall-bounded turbulence has become a hot topic.
The mean flow properties of compressible turbulence have been extensively studied.
The hypothesis of Morkovin (1962) is widely verified, stating that at moderate free
stream Mach numbers (Ma∞ 5), the dilatation effect is small, and any differences from
incompressible turbulence can be accounted for by mean variations of fluid properties
(Coleman, Kim & Moser 1995; Pirozzoli, Grasso & Gatski 2004; Duan, Beekman &
Martin 2010). Lagha et al. (2011) even show that the hypothesis is applicable with Ma∞
up to 20 for flat-plate boundary layers. Thereby, velocity transformation can be designed
using only the mean flow. The transformed streamwise velocity profile can match the
incompressible one within and below the logarithmic region with very high accuracy
(van Driest 1951; Trettel & Larsson 2016; Griffin, Fu & Moin 2021; Bai, Griffin & Fu
2022). Meanwhile, the mean temperature is shown to be nearly a quadratic function of
the mean velocity (Walz 1969; Duan & Martín 2011; Zhang et al. 2014). Strong Reynolds
analogy (SRA) and its extensions are proposed, connecting the temperature and velocity
fluctuations (Morkovin 1962; Huang, Coleman & Bradshaw 1995; Zhang et al. 2014). In a
recent work, Cheng & Fu (2023a) note that the profile of turbulent Prandtl number Prt in
supersonic channels is very close to the incompressible case with heat transfer (Antonia,
Abe & Kawamura 2009). They suggest that the coherent temperature fluctuations behave
like a passive scalar, and the coupling between the velocity and temperature fields results
largely from the advection effect.
In addition to the mean flow, much progress has been made in understanding
the structures of velocity and temperature fluctuations, owing to the advancement of
experimental and numerical techniques, especially the direct numerical simulation (DNS,
Moin & Mahesh 1998). In the near-wall region, small-scale temperature streaks are
observed elongated in the streamwise direction, analogous to the streamwise velocity
streaks (Coleman et al. 1995; Duan et al. 2010). The spanwise spacing of streaks is
shown to decrease with rising Ma on adiabatic walls, and increase by wall cooling; it
experiences much less variation if expressed in semi-local units (Morinishi, Tamano &
Nakabayashi 2004). Within and above the logarithmic layer, energy-containing large-scale
and very-large-scale motions (LSMs and VLSMs), as first reported in incompressible
flows (Kim & Adrian 1999), are identified in supersonic boundary layers experimentally
(Ganapathisubramani, Clemens & Dolling 2006; Bross, Scharnowski & Kähler 2021)
and numerically (Ringuette, Wu & Martín 2008; Huang, Duan & Choudhari 2022).
Unlike the near-wall motions, the characteristic scales of LSMs and VLSMs are not very
sensitive to compressibility effects (Pirozzoli, Bernardini & Grasso 2008; Williams et al.
2018; Cheng & Fu 2022b). The LSMs populating the logarithmic and outer regions are
973 A36-2
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Estimating coherent velocity and temperature structures
known to exert footprints, i.e. have effects of velocity modulation and superposition,
on the near-wall motions (Abe, Kawamura & Choi 2004; Hutchins & Marusic 2007),
which is the basis of the inner–outer interaction model (IOIM, Marusic, Mathis &
Hutchins 2010). Using the DNS of supersonic boundary layers, Bernardini & Pirozzoli
(2011) and Pirozzoli & Bernardini (2011) demonstrate that the temperature fluctuations
also experience inner–outer interaction, though the modulation effect seems weaker
than the velocity. Yu & Xu (2022) extend IOIM to realize a predictive model for both
velocity and temperature fields in compressible boundary layers, considering the modified
SRA and the superposition and modulation from LSMs. Meanwhile, the inner–outer
interaction of temperature fluctuations indicates that they can also be regarded as a
wall-attached quantity, hence described by the well-known attached-eddy model (AEM,
Townsend 1976). This is supported by accumulating evidence concerning the geometrical
self-similarity of temperature in the logarithmic region (Cheng & Fu 2022b; Yu et al.
2022; Chen et al. 2023). Moreover, Cheng & Fu (2023a) point out that in the logarithmic
region, only the scales corresponding to the attached eddies and VLSMs are firmly
coupled; the Reynolds number Re acts as the crucial similarity parameter in constructing
the coupling, rather than the Mach number.
To estimate and model coherent large-scale structures, the spectral linear stochastic
estimation (SLSE) approach is helpful. SLSE originates from the extensively used
stochastic estimation (Adrian 1979; Adrian & Moin 1988), which can provide an estimated
velocity signal given a measurement at another point from experiments or simulations.
Considering the multi-scale feature of turbulence, SLSE is performed in the Fourier
space, which takes advantage of the coherence of large-scale structures and suppresses
small-scale random noise (Tinney et al. 2006; Baars, Hutchins & Marusic 2016). A kernel
function HL is designed to relate the Fourier components at different wall-normal heights,
based on the linear coherence spectrum (LCS). Over the years, SLSE has been widely
deployed to study the multi-scale structures in incompressible flows and to help understand
and improve AEM and IOIM (Baars, Hutchins & Marusic 2017; Encinar & Jiménez 2019;
Cheng & Fu 2022a, 2023b; Cheng, Shyy & Fu 2022). Nevertheless, its application in
compressible flows is still limited. Very recently, Cheng & Fu (2022b, 2023a) extend the
SLSE framework for compressible flows. The self-similarity of thermodynamic quantities
and the coupling effects between velocity and temperature are thus investigated. This
extended SLSE framework is considered in this work for compressible channel flows.
By definition, the LCS and HL in SLSE involve ensemble-averaged cospectra,
so their calculation requires a series of instantaneous measurements or numerical
data, which are more challenging to obtain than the mean flow. In an alternative
method, Madhusudanan, Illingworth & Marusic (2019) obtain HL using the linearized
incompressible Navier–Stokes (NS) equations, which requires only the mean flow as the
input. Their stochastically forced, eddy-viscosity-enhanced linear model gives agreeable
LCS with the DNS data, except in the near-wall region. For improvement, Gupta et al.
(2021) model the amplitude distribution of the stochastic forcing using mean flow
quantities. The resulting LCS and HL are quite close to the DNS data, though the prediction
can still struggle in the near-wall region. Notably, the linearized NS equations have long
been utilized in turbulence research, as will be introduced at length later. Thereby, it is
a natural thought to develop a linear model for compressible turbulent flows, so that the
LCS and HL can be obtained using only mean flow quantities. In this way, the coherence
of velocity and temperature fluctuations, as well as their coupling effects, can be further
investigated. Such a linear model is of particular significance for compressible flows,
because the instantaneous experimental and DNS data are much more limited than the
incompressible counterparts due to larger parameter space and higher facility requirements
973 A36-3
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
X. Chen, C. Cheng, J. Gan and L. Fu
(Gatski & Bonnet 2013). It is the objective of this work to develop a linear model for
SLSE in compressible turbulent channel flows. We find that a direct extension of the
incompressible model to the compressible case does not provide satisfactory SLSE results,
especially for temperature fluctuations. Thereby, we will scrutinize the mathematical
properties and physical relevance of different linear models, and assess carefully their
behaviours.
To set more grounds for the present work, the approaches based on the linearized
NS equations for turbulent flows are introduced in more detail. Early works find that
the incompressible turbulent mean flow (one dimensional) is globally stable, i.e. there
are no unstable modes through linear stability analysis (Malkus 1956; Reynolds &
Hussain 1972). For this globally stable system, non-modal instability theory and the
resolvent-based input–output analysis, first applied for laminar flows (Trefethen et al.
1993; Schmid & Henningson 2001), are extended to study the transient behaviour of
turbulent mean flows. These linear approaches are successful in studying the multi-scale
characteristic motions (del Álamo & Jiménez 2006; Hwang & Cossu 2010; Abe, Antonia
& Toh 2018), constructing low-rank predictive models (McKeon & Sharma 2010; Moarref
et al. 2013; Illingworth, Monty & Marusic 2018), designing flow control strategies
(Moarref & Jovanović 2012; Ran, Zare & Jovanović 2020) and so on. Many of these
works adopt the eddy-viscosity-enhanced models, where the Reynolds stress fluctuation
is linearized using the eddy viscosity µt , to partly model the colour of the forcing. The
eddy-viscosity-enhanced models are shown to perform better than those without using
µt , especially for estimating fluctuations between different heights (Reynolds & Hussain
1972; Illingworth et al. 2018; Madhusudanan et al. 2019; Morra et al. 2021; Symon,
Illingworth & Marusic 2021). The linearized-equation-based approaches have also been
deployed for compressible turbulent flows. Alizard et al. (2015) perform the transient
growth analysis on turbulent boundary layers with Ma∞ up to 4. The inner and outer peak
modes are identified from the energy growth curves with different spanwise wavenumbers,
analogous to the incompressible cases. The former mode is related to the near-wall streaky
motions, and the latter represents the outer-layer LSMs (and VLSMs). Bae, Dawson &
McKeon (2020) extend the resolvent analysis to supersonic boundary layers and highlight
the distinct features of the relatively supersonic region in the wavenumber space, which is
not present in incompressible flows. Also, Chen et al. (2023) develop the linear response
analyses subject to both optimal harmonic and stochastic forcing for supersonic channel
flows, and analyse the response characteristics over wide ranges of Ma and Re. To model
the linearized Reynolds stress and turbulent heat flux, µt and SRA are introduced in some
of these works (Alizard et al. 2015; Pickering et al. 2021; Chen et al. 2023). Nevertheless,
no quantitative comparisons have been made with DNS or experiments in these works,
so the accuracy of these linear models remains unclarified. It is also a task of the present
work to make such a quantitative assessment regarding SLSE using the DNS database
constructed in the present authors’ group.
In addition, there are some theoretical issues that require to be addressed regarding
the linearized equations for compressible turbulent flows. For example, the singular value
of the resolvent by Bae et al. (2020) (their figure 1) has relatively strong oscillations
when the response mode is relatively supersonic. Madhusudanan & McKeon (2022) (and
also our own experience) note that relatively supersonic modes have amplified acoustic
components in the free stream, and the results can be dependent on the height of the
computational domain (which is artificially selected) for boundary layers. These facts
suggest that there can be some fundamental differences in the linear models between
incompressible and compressible flows, in addition to the widely addressed similarities.
However, the differences have not been fully resolved. Also, in deriving the compressible
973 A36-4
Estimating coherent velocity and temperature structures
y
v
x
z
u
ρˉ(y)
Tˉ( y)
Uˉ ( y)
w
Figure 1. Schematic and coordinate set-up for the compressible channel flow.
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
eddy-viscosity-enhanced models, multiple options are available to linearize different
terms, leading to numerous combinations. They deserve a careful discussion through
inter-model comparisons.
The primary attention of this work is on the following three aspects, which are also
its value. First, we carefully discuss the derivation and mathematical properties of the
linearized equations for compressible turbulent flows, which is instructive for interpreting
subsequent results (§§ 2–4). Second, we utilize the linear model to perform SLSE for
streamwise velocity, temperature, and their coupling, aiming to predict the coherent
velocity and temperature fluctuations based on the measurement at outer locations. Direct
comparison with DNS is conducted to assess their behaviours (§ 5). Third, we discuss the
physics inferred from SLSE results, especially for temperature, and conduct a parameter
study to see the effects of Reynolds and Mach numbers (§§ 5 and 6). Some remarks
are provided on the scope of this work. For incompressible flows, many linear models
have been developed to provide continuously improved results approaching the DNS and
experimental data (see the references above). These models have different treatments in
linearizing the Reynolds stress, modelling the forcing, utilizing or not instantaneous DNS
data and so on. Consequently, it is nearly impossible for the present work to extend and
compare among all these models, especially considering the still larger parameter space
in compressible flows. Thereby, we hope that the linear model in this work is both simple
and instructive. Some possible extensions and future works are discussed in §§ 7 and 8.
2. Governing equations and dataset
2.1. Compressible Navier–Stokes equations
The canonical compressible turbulent channel flow is considered, as illustrated in figure 1.
Assuming a calorically perfect gas, the governing NS equations are
∂ρ
+ ∇ · (ρu) = 0,
(2.1a)
∂t
2
∂u
ρ
+ u · ∇u = −∇p + ∇ · µ(∇u + ∇uT ) − ∇(µ∇ · u),
(2.1b)
∂t
3
2
∂T
∂p
T
2
ρcp
+ u · ∇T −
+ u · ∇p = µ ∇u : (∇u + ∇u ) − (∇ · u)
∂t
∂t
3
+∇ · (κ∇T),
(2.1c)
where ρ, u = [u, v, w]T , T and p = ρRT are the fluid density, velocity, temperature and
pressure, respectively; R and cp are the gas constant and isobaric specific heat; and µ and
973 A36-5
X. Chen, C. Cheng, J. Gan and L. Fu
κ are the molecular viscosity and thermal conductivity. Characteristic non-dimensional
parameters are the Mach, Reynolds, Prandtl and Eckert numbers,
Mab =
Ub
,
aw
Reb =
ρb Ub h
,
µw
Pr =
µcp
= 0.72,
κ
Ec =
Ub2
= (γ0 − 1)Ma2b ,
c p Tw
(2.2a–d)
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
where the subscript w denotes quantities at the wall (y = 0, 2h) and h is the channel
half-height. The speed of sound is a = (γ0 RT)1/2 with a constant specific heat ratio
γ0 = cp /cv = 1.4. The subscript 0 here is to distinguish from the LCS notation in
2h
§ 2.3. The bulk density ρb and bulk velocity Ub are defined as ρb = 0 ρ̄ dy/(2h)
2h
and ρb Ub = 0 ρu
dy/(2h), where the overbar denotes mean variables. The viscosity is
calculated through Sutherland’s law, where the fitting constant is 110.4 K and the reference
temperature is 293.15 K. No-slip and isothermal walls are set on both sides as the boundary
condition.
The basic variable set in (2.1) is q = [ρ, u, v, w, T]T . Both the Reynolds average (ϕ̄
/ ρ) are deployed.
with ϕ a time-dependent variable) and Favre average (ϕ̃ = ρϕ/ρ̄, ϕ =
The resulting two fluctuations are denoted as ϕ ′ and ϕ ′′ , respectively. For wall-bounded
turbulence, it is common to use wall viscous units with a superscript +, as x+ = x/δν ,
ρ + = ρ/ρw , u+ = u/uτ and T + = T/Tτ . Here the viscous length unit is δν = µw /(ρw uτ ),
the friction velocity and temperature are uτ = (τw /ρw )1/2 and Tτ = Qw /(ρw cp uτ ), where
τw and Qw are the wall mean shear and heat flux. The friction Reynolds number is Reτ =
h/δv . Furthermore, semi-local units are adopted, expressed with a superscript ∗ as u∗τ =
(τw /ρ̄)1/2 , δν∗ = µ̄/(ρ̄u∗τ ), so y∗ = y/δv∗ and Re∗τ = h/δv∗ .
2.2. DNS dataset and mean flow calculation
A series of DNSs for compressible turbulent channel flows has been conducted by the
present authors, as reported at length (Cheng & Fu 2022b). The calculations adopt two
Mab of 0.8 and 1.5, and cover a Reb range of 7667–20 020 (Reτ = 436–1150). The domains
are all rectangular, in the same sizes Lx × Lz × Ly = 4πh × 2πh × 2h. Previous studies
have verified that this set-up can capture most of the large-scale motions in the outer region
(Agostini & Leschziner 2014). Readers can refer to Cheng & Fu (2022b, 2023a) for more
analyses on the DNS statistics. The case Mab = 1.5, Reb = 20 020 (Reτ = 1150, Re∗τ =
780) is selected as the benchmark case for this study, which is of the highest Mab and Re
in our dataset. The valuable DNS data allow a quantitative evaluation of the linear model
results.
In addition to DNS, well-established universal relations can be used to obtain efficiently
the turbulent mean flows, by solving a set of ordinary differential equations (ODEs)
(Griffin, Fu & Moin 2022, 2023; Chen et al. 2023; Song, Zhang & Xia 2023). The
ODE-based method of Chen et al. (2023) is used here, which requires only the values
of Mab and Reb as the input. We will demonstrate in § 6 that the ODE-based mean flow
can be used in the linear models for SLSE, which will serve the parameter study.
2.3. Spectral linear stochastic estimation
As motivated in § 1, SLSE predicts the coherent portion of fluctuations at different heights
as a reflection of the superposition effects. For consistency with the equations in § 3.1, the
SLSE quantities are defined based on the Favre average. As the mean flow is homogeneous
973 A36-6
Estimating coherent velocity and temperature structures
in the wall-parallel directions, Fourier decomposition is applied on q′′ as
q′′ (x, y, z, t) =
∞
−∞
q̂′′ ( y, t; kx , kz ) exp[i(kx x + kz z)] dkx dkz ,
(2.3)
where kx and kz are the streamwise and spanwise wavenumbers, and q̂′′ is the shape
function. In SLSE, the spectral signals û′′ between the locations of measurement ym and
prediction yp are connected through a complex-valued kernel function HL (Tinney et al.
2006),
⎫
û′′p ( yp , t; kx , kz ) = HL,uu ( yp , ym ; kx , kz )û′′ ( ym , t; kx , kz ), ⎬
û′′ ( yp , t; kx , kz )û′′† ( ym , t; kx , kz )
(2.4a)
HL,uu ( yp , ym ; kx , kz ) = ′′
,⎭
′′†
û ( ym , t; kx , kz )û ( ym , t; kx , kz )
where · stands for ensemble average and superscript † denotes the complex conjugate.
If the time-independent HL,uu is obtained, then û′′ ( yp , t; kx , kz ) and u′′ (x, yp , z, t) at yp
can be estimated (u′′p or û′′p ) given an instantaneous field (u′′ or û′′ ) at ym considering all
coherent motions (Baars et al. 2016). In the following, kx and kz in the brackets will be
omitted if there is no ambiguity. It is worth mentioning that HL,uu for compressible flows
is defined differently among the literature. The u′′ in (2.4a) can be replaced with u′ or
other density-weighting forms. Cheng & Fu (2022b) show that different density-weighting
approaches do not alter the energy distribution among scales for the benchmark case here.
Also, with or without density weighting negligibly affects the streamwise and spanwise
length scales of the pre-multiplied spectra peaks. Therefore, the definition in (2.4a) is
used throughout. Similar to that for u′′ , SLSE can be deployed to estimate T ′′ ( yp ) given
T ′′ ( ym ) or u′′ ( ym ). The latter combination concerns the coupling effects between u′′ and
T ′′ (Cheng & Fu 2023a). The two kernel functions are defined as
T̂ ′′ ( yp , t)û′′† ( ym , t)
.
û′′ ( ym , t)û′′† ( ym , t)
T̂ ′′ ( ym , t)T̂ ′′† ( ym , t)
(2.4b)
Following Tinney et al. (2006), the amplitude of HL,uu (also for HL,TT , HL,uT ) is further
decomposed into two parts, as
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
HL,TT ( yp , ym ) =
T̂ ′′ ( yp , t)T̂ ′′† ( ym , t)
,
HL,uT ( yp , ym ) =
2
|HL,uu ( yp , ym )|2 = A2pm,uu ( yp , ym )γuu
( yp , ym ),
(2.5)
where Apm ( yp , ym ; kx , kz ) measures the fluctuation amplitude ratio at a specific length
scale, and γ 2 ( yp , ym ; kx , kz ) is the 2-D LCS; their expressions are
Apm,uu ( yp , ym ) =
|û′′ ( yp )|2
,
|û′′ ( ym )|2
2
γuu
( yp , ym ) =
|û′′ ( yp , t)û′′† ( ym , t)|2
. (2.6a)
|û′′ ( yp , t)|2 |û′′ ( ym , t)|2
2 comprises two individual 2-D energy spectra of u′′ at y and y ,
The denominator of γuu
p
m
while the numerator is the absolute value of the u′′ cospectrum between the two heights.
2 reflects the maximum correlation coefficient at each Fourier scale, hence
Therefore, γuu
instructive for understanding structural coherence. By definition, 0 ≤ γ 2 ≤ 1; γ 2 = 1
represents perfect coherence while γ 2 = 0 means no coherence. The phase of HL,uu is
not reflected in (2.5), which also matters as a measure of the structural angle difference in
973 A36-7
X. Chen, C. Cheng, J. Gan and L. Fu
2 and γ 2 are defined as
the x–z plane (Baars et al. 2016). Similar to (2.6a), γTT
uT
2
γTT
( yp , ym ) =
|T̂ ′′ ( yp , t)T̂ ′′† ( ym , t)|2
|T̂ ′′ ( yp , t)|2 |T̂ ′′ ( ym , t)|2
,
2
( yp , ym ) =
γuT
|T̂ ′′ ( yp , t)û′′† ( ym , t)|2
|T̂ ′′ ( yp , t)|2 |û′′ ( ym , t)|2
.
(2.6b)
Calculating HL , γ 2 and A2pm requires the 2-D cospectrum of q̂′′ ,
Φ( y, y′ ; kx , kz ) = q̂′′ ( y, t; kx , kz )q̂′′H ( y′ , t; kx , kz ),
(2.7)
where H denotes the Hermite transpose. The cospectrum is computed using a time series
of instantaneous measurements or numerical data. As an alternative, it can be obtained
through a linear model based on the linearized (2.1), as introduced in § 1. The construction
of the linear model is discussed below.
3. Linear models for compressible flows
Although the linear models in incompressible flows have been extensively addressed, their
compressible counterparts deserve a careful discussion.
3.1. Mean flow and fluctuation equations
To establish a linear model, the governing equation for q′′ is derived first. Following
standard procedures, the time-averaged (2.1) is
ρ̄
∂ ũi
∂ ũi
+ ũj
∂t
∂xj
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
ρ̄cv
∂ T̃
∂ T̃
+ ũj
∂t
∂xj
∂ ũj
∂ ρ̄
∂ ρ̄
+ ρ̄
= 0,
+ ũj
∂t
∂xj
∂xj
=−
∂ τ̄ij
∂ p̄
∂
′′ u′′ ) ,
+
+
(−ρ̄ u
∂xi
∂xj
∂xj i j
(3.1a)
i = 1, 2, 3,
(3.1b)
Reynolds stress
∂
=
∂xj
∂ T̃
κ̃
∂xj
− p̄
∂ ũj
∂
∂ ũi
′′ T ′′ )
+ τ̄ij
+
(−ρ̄cv u
j
∂xj
∂xj
∂xj
∂u′′j
∂u′′
+ τij′ i .
− p′
∂xj
∂xj
pressure
dilatation
turbulent
heat flux
(3.1c)
viscous
dissipation
Here the mean pressure is p̄ = ρ̄RT̃, and the viscous stress (and also for κ̃) is
approximated, following the suggestion by Gatski & Bonnet (2013), as
⎫
∂ ũi ∂ ũj 2 ∂ ũk
⎪
⎪
τ̄ij ≈ µ̃
+
−
δij ,
⎪
⎬
∂x
∂x
3
∂x
j
i
k
′′
(3.2)
∂uj
∂u′′i
∂ ũi ∂ ũj 2 ∂ ũk
2 ∂u′′k
⎪
′
′′
⎪
τij ≈ µ̃
+
−
δij + µ
+
−
δij ,⎪
⎭
∂xj
∂xi
3 ∂xk
∂xj
∂xi
3 ∂xk
where µ̃ ≈ µ(T̃), µ′′ ≈ (∂µ/∂T)T ′′ , and δij is the Kronecker delta. Equation (3.1) uses
the Favre-averaged quantities, so it can be regarded as a governing equation for the mean
973 A36-8
Estimating coherent velocity and temperature structures
flow q̃ = [ρ̄, ũi , T̃]T , though there are unclosed terms regarding q′′ = [ρ ′ , u′′i , T ′′ ]T . The
physical meanings of these unclosed terms, as underbraced, have been elaborated before
(e.g. Huang et al. 1995). In particular, the last two terms in (3.1c) are crucial for the energy
′′ u′′ /2 (Lele 1994).
transfer between the internal energy and turbulent kinetic energy k ≡ u
j j
For reference, the governing equation for k is provided as
′′
∂u′′j
∂k
∂k
′′ k′′ + p′ u′′ − τ ′ u′′ ) + p′
′′ u′′ ∂ ũi − ∂ (ρ̄ u
′ ∂ui . (3.3)
= −ρ̄ u
ρ̄
−
+ ũj
τ
j
i j
j
ij i
ij
∂t
∂xj
∂xj
∂xj
∂xj
∂xj
The governing equations for q′′ are obtained after a subtraction between (2.1) and (3.1).
The fluctuating continuity equation is
′
∂u′′j
∂ρ ′ u′′j
∂ ũj ′ ∂ ρ̄ ′′
∂ρ
∂ρ ′
+ ρ̄
.
(3.4a)
+
ρ +
uj = −
+ ũj
∂t
∂xj
∂xj
∂xj
∂xj
∂xj
Nρ′′
The equation is rearranged so that the terms on the left-hand side are linear terms of q′′ ,
while the one on the right-hand side is nonlinear, denoted as Nρ′′ . Similarly, the fluctuating
momentum equation (i = 1, 2, 3) takes the form of
′′
′
∂ui
∂u′′
∂ ũi ∂p′ ∂τij
ρ̄
+
−
+ ũj i + (ρ ′ ũj + ρ̄u′′j )
∂t
∂xj
∂xj
∂xi
∂xj
′ ′′
∂ρ ′ u′′i ũj
∂ρ ui
∂
′′ ′′
′ ′′ ∂ ũi
′′
′′
.
(3.4b)
=−
[ρ̄(u u − u u )] −
+ ρ uj
+
∂xj i j i j
∂t
∂xj
∂xj
Reynolds stress fluctuation
Nu′′
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
i
Same as (3.4a), the linear and nonlinear terms are placed on the two sides. The nonlinear
terms are classified into two groups. The first group is the fluctuation of the Reynolds
stress tensor, and the second group collects all the second- and higher-order terms related
to ρ ′ . The classification in (3.4b) is not unique. For example, one can expand Nu′′i using
(3.4a). Different possibilities will be discussed in § 7. Notably, when DNS is conducted,
there is a temporarily-varying body force term in the streamwise momentum equation to
fix the mass flux (e.g. Yao & Hussain 2020). This term is spatially uniform in the current
DNS data, so it appears only at the scale kx = kz = 0, hence not included in (3.4b). The
fluctuating internal energy equation is
′′
∂T ′′
∂
∂T ′′
∂T
′′ ∂ T̃
′
′′ ∂ T̃
κ̃
−
+κ
+ ũj
+ cv (ρ ũj + ρ̄uj )
ρ̄cv
∂t
∂xj
∂xj ∂xj
∂xj
∂xj
∂u′′j
∂u′′i
∂
′ ∂ ũj
′ ∂ ũi
′′ T ′′ )]
= − [ρ̄cv (u′′j T ′′ − u
+p
+ τij
+ p̄
− τ̄ij
j
∂xj
∂xj
∂xj
∂xj
∂xj
−
′
p
∂u′′j
∂xj
− p′
∂u′′j
∂xj
pressure dilatation
fluctuation
+
turbulent heat flux
fluctuation
∂u′′
τij′ i
∂xj
′′
′ T ′′ ũ
′ T ′′
∂u
∂ρ
∂ρ
∂
T̃
j
i
′
′′
−cv
.
− τij′
+ ρ uj
+
∂xj
∂t
∂xj
∂xj
dissipation rate
fluctuation
NT′′
(3.4c)
973 A36-9
X. Chen, C. Cheng, J. Gan and L. Fu
Likewise, NT′′ collects all the terms related to ρ ′ . There are additional terms of fluctuating
turbulent heat flux, pressure dilatation and dissipation rate, as underbraced. The governing
equation for k′′ ≡ (u′′j u′′j /2 − k) (Huang et al. 1995) can be similarly derived, whose full
expression is listed in Appendix A.
As seen, (3.4) is far more complicated than its incompressible counterpart for deriving
a linear model. The nonlinear terms Nρ′′ , Nu′′i and NT′′ all result from ρ ′ , not present
in incompressible cases. The remaining four nonlinear terms are classic ones in the
turbulence modelling theory.
3.2. Modelling issues and linearization
In general, there are two types of strategies to linearize (3.4). The first one, as adopted
by Bae et al. (2020) and Dawson & McKeon (2020), is to collect all the second- and
higher-order terms of q′′ , i.e. those underbraced in (3.4), into the nonlinear forcing term.
The resulting form is termed the LNS equation here, written in a standard operator form
as
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
∂q′′
= LLNS q′′ + f ′′LNS .
∂t
(3.5)
Here, the linear operator L is only mean flow related, and f ′′ collects all the nonlinear
terms. For later use, the temporal derivative term is not included in L. Notably, the
expression of LLNS is exactly the same as that in the linear stability theory for laminar
flows (Mack 1984).
The second strategy is to utilize turbulence modelling relations for possible linearization
of the nonlinear terms, which is the counterpart of the eddy-viscosity-enhanced model
for incompressible flows. However, the extension to compressible flows is much more
complicated with multiple combinations. More terms require linearization besides the
Reynolds stress term, which, as will be shown later, can lead to difficulties in closing
the equations. Our principle here is to avoid ad hoc fittings by ourselves and use only the
relations from classic modelling theory, which have been widely verified.
Analogous to the incompressible case, the fluctuation of the Reynolds stress term is
modelled using the linearized Boussinesq assumption,
− ρ̄(u′′i u′′j
′′ u′′ )
− u
i j
= µt
∂u′′j
∂u′′i
2
2 ∂u′′m
+
−
δij − ρ̄k′′ δij .
∂xj
∂xi
3 ∂xm
3
(3.6)
The turbulent heat flux is usually modelled using the SRA. The modified version by Huang
et al. (1995) takes the form of
T ′′ =
1 ∂ T̃ ′′
u .
Prt ∂ ũ
(3.7)
It assumes that T ′′ also satisfies the mixing-length relation as u′′ , and hints that T ′′ is
advection dominated since it is determined by u′′ and the mean-flow gradients. Based
on (3.7), there are two means to linearize the turbulent heat flux term. The first one
is to follow the Reynolds-averaged NS (RANS) modelling, e.g. the k–ω model by
973 A36-10
Estimating coherent velocity and temperature structures
Wilcox (2006), which extends (3.7) to all spatial directions. The resulting form is
′′
′′ T ′′ ) = µt ∂T .
′′ T ′′ = µt ∂ T̃ → −ρ̄(u′′ T ′′ − u
− ρ̄ u
j
j
j
Prt ∂xj
Prt ∂xj
(3.8a)
The second approach is a direct application of (3.7), as
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
′′ T ′′ ) = − 1 ∂ T̃ ρ̄(u′′ u′′ − u
′′ u′′ ),
− ρ̄(u′′j T ′′ − u
j
j
j
Prt ∂ ũ
(3.8b)
which is evaluated using (3.6). Consequently, the heat flux fluctuation is independent of
T ′′ but related to u′′i instead. Equation (3.8a) is deployed as we prefer to treat T ′′ as an
independent variable. The model using (3.8b) will be discussed in § 7.
Afterward, we focus on the pressure dilatation and dissipation terms in (3.4c). These
two also appear in the k′′ equation. Little can we learn from the modelling theory on
the linearization of pressure dilatation. The budget analysis for k shows that the pressure
dilatation and dilatational dissipation in (3.3) are negligible in the supersonic channel cases
(Huang et al. 1995). So intuitively, their fluctuations are also negligible. The remaining
part is the solenoidal dissipation, denoted as ρ̄ǫ. It can be evaluated from, e.g. the k–ǫ
model (Jones & Launder 1972), as ρ̄ǫ = ρ̄ 2 Cµ k2 /µt with Cµ a constant, thus linearized
in terms of k′′ . Note that k′′ cannot be expressed as a linear function of q̂′′ , so it needs
to be treated as an independent variable, which means that the k′′ equation needs to be
included in (3.4) to solve the system of six equations. We prefer not to introduce k′′ as an
independent variable for three reasons. First, there are difficulties in deriving a physically
reasonable energy norm for the fluctuation, including k′′ (see § 3.3). Second, more terms
(e.g. the diffusion term of k′′ ) appear in the k′′ equation, and their linearization adds to
the uncertainty and complexity of the linear model before a careful assessment. Third,
including the k′′ equation requires the profile of k as the input, which is harder to obtain
than q̃. As an alternative to exclude k′′ , the turbulent production and dissipation terms can
be combined in the spirit of algebraic models (see Appendix A), and the former can be
calculated using (3.6). The effects of including k′′ will be discussed in § 7.
Finally, there are nonlinear terms Nρ′′ , Nu′′i and NT′′ in (3.4), with no counterparts in the
modelling theory. These three terms are all related to ρ ′ , so in the spirit of Morkovin’s
hypothesis, they are of secondary importance. Therefore, they are not linearized but
included in the nonlinear forcing. Some supporting evidence will be provided in § 7.
Following the nomenclature for incompressible flows, the linear model utilizing
turbulence modelling is termed the eLNS (‘e’ for eddy-viscosity-enhanced) model. By
collecting the residual nonlinearity into the forcing term, the operator form is
∂q′′
= LeLNS q′′ + f ′′eLNS .
∂t
(3.9)
If using (3.6) and (3.8a) and combining the turbulent production and dissipation terms,
then LeLNS is in the same form as LLNS except for two substitutions,
µ̃ → µ̃ + µt ,
κ̃ → κ̃ + κt =
cp µ̃ cp µt
,
+
Pr
Prt
(3.10a,b)
where κt is the eddy diffusivity. Equation (3.10a,b) is used likewise in the algebraic
models, and is one of the simplest eLNS models widely used in previous works (Alizard
et al. 2015; Pickering et al. 2021; Chen et al. 2023), though the derivation and justification
were not elaborated before. Detailed expressions of LLNS and LeLNS can be found from
973 A36-11
X. Chen, C. Cheng, J. Gan and L. Fu
Chen et al. (2023). From the mathematical point of view, using (3.10a,b) means intensified
damping effects due to turbulence, especially in the outer region where µt ≫ µ̃. This point
is crucial in understanding the properties of the linear operators in § 4.
Next, we address the calculation of µt and Prt . From the Boussinesq assumption, µt is
′′ v ′′ /(∂ ũ/∂y). Also, Pr is obtained from its definition using u
′′ v ′′
computed as µt = −ρ̄ u
t
′′ T ′′ . If the fluctuation statistics are not available, µ can be evaluated from the mean
and v
t
streamwise momentum (3.1b) as
dp̄
1 − y+ /Reτ
d
dũ
µt
µ̃
0=− +
=
.
(3.11)
−
(µ̃ + µt )
→
+
+
dx dy
dy
µ̃w
dũ /dy
µ̃w
For more general flows where (3.11) is inapplicable, µt can be estimated using, e.g. the
algebraic models. Also, Prt can be simply assumed a constant 0.9. In this work, µt and
Prt are determined from the DNS statistics for the benchmark case results. Equation (3.11)
and the constant Prt are adopted in § 6 for the parameter study.
3.3. Response to stochastic forcing
Ensemble-averaged variable correlation Φ (see (2.7)) is required to compute HL , γ 2
and A2pm . Based on § 3.2, Φ can be obtained by solving a stochastically forced linear
system using only mean flow input (Madhusudanan et al. 2019). As in (2.3), Fourier
decomposition is applied on f ′′ for the component fˆ ′′ . After substituting (2.3) into (3.5)
or (3.9), the equation for a single mode with kx =
/ 0 or kz =
/ 0 is
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
∂ q̂′′
= L̂q̂′′ + fˆ ′′ ,
∂t
(3.12)
where L̂(q̃, kx , kz ) can be either L̂LNS or L̂eLNS , fˆ ′′ = [ f̂ρ′′ , f̂u′′ , f̂v′′ , f̂w′′ , f̂T′′ ]T is the forcing
term and q̂′′ is the response. The energy norm of q̂′′ is defined as (Chu 1965)
2h
2h
ρ̄c
R
T̃
v
T̂ ′′† T̂ ′′ dy =
ρ̂ ′† ρ̂ ′ +
q̂′′H M q̂′′ dy,
q̂′′ 2 = (q̂′′ , q̂′′ )E =
ρ̄ û′′H û′′ +
ρ̄
T̃
0
0
(3.13)
where M is the energy weight matrix.
When (3.12) is driven by a stochastic forcing, the response is also stochastic if the system
is linearly stable. Following those for incompressible flows (Gupta et al. 2021), the forcing
is fˆ = Bfˆ 0 where B( y) is a modelling matrix and fˆ 0 is a white noise signal. Specifically,
fˆ 0 is assumed to be a δ-correlated Gaussian white noise with zero mean,
fˆ ′′0 = 0,
′ ′
′
′
fˆ ′′0 ( y, t)fˆ ′′H
0 ( y , t ) = Iδd ( y − y )δd (t − t ),
(3.14)
where I is the identity matrix and δd is the Dirac function. The introduction of B( y) allows
for varying forcing amplitude for different variables at different y. The specific expression
of B will be provided later in (3.17). Mathematically, Φ is obtained by solving the algebraic
Lyapunov equation (see Farrell & Ioannou (1993) for more details) as
L̂Φ + Φ L̂† +BBH = 0.
(3.15)
Note that L̂ contains wall-normal derivatives, so L̂† is defined in an adjoint manner.
Also, the vector product is with respect to (3.13). Equation (3.15) is discretized using the
Chebyshev collocation point method. By default, Ny = 301 points are used, abundant to
973 A36-12
Estimating coherent velocity and temperature structures
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
ensure grid independence (Bae et al. 2020). The Lyapunov equation is solved using the
M ATLAB software. For boundary conditions, a no-slip, isothermal wall is assumed on
both sides. Therefore, the wall fluctuations satisfy û′′w = v̂w′′ = ŵ′′w = T̂w′′ = 0, and ρ̂w′ is
solved through the continuity equation. The solver verification can be found from Chen
et al. (2023).
In the incompressible model of Madhusudanan et al. (2019), B is set to I for both
LNS and eLNS, meaning that the stochastic forcing has uniform amplitude along the
wall-normal direction. This is also adopted in previous compressible works (Alizard
et al. 2022; Chen et al. 2023). To improve the estimation below the logarithmic region,
Gupta et al. (2021) design two new models for incompressible flows, termed ‘W-model’
and ‘λ-model’, where B is y-dependent or even scale-dependent. The simpler W-model
is considered here, where the forcing amplitude for the momentum equation varies in
proportion to νt = µt /ρ̄. For compressible flows, f̃ρ and f̃T are present and need to be
modelled. Without comprehensive knowledge of the forcing statistics, we consider the
simplest form here, which is instructive and free from ad hoc implementations. First, B is
set to be diagonal, i.e. B = diag([Bρ νt νt νt BT ]T ), where the amplitude functions Bρ and
BT are to be determined. In this way, the off-diagonal terms, i.e. the cospectra of the forcing
components and also the anisotropy (Foysi, Sarkar & Friedrich 2004), are not modelled.
Second, B is assumed to be only related to the mean flow, independent of length scales λx
and λz . The extended SRA (3.7) and the DNS data of Coleman et al. (1995) suggest that
the following relations are nearly satisfied:
∂ T̃
∂ T̃
ρ̄
1
1
∂
ρ̄
′′
′
′′
u′′ ≈
Trms
=
(3.16a,b)
=
u′′rms , ρrms
urms ,
rms
Prt ∂ ũ
Prt ∂ ũ
Prt T̃ ∂ ũ
where the subscript rms is for root mean square, and the approximation is due to the
slightly varying mean pressure. The mean flow gradients appear in their absolute values
because the fluctuation r.m.s. is positive. For channel flows, ∂ T̃/∂ ũ > 0 and ∂ ρ̄/∂ ũ < 0
hold in the whole field. From (3.16a,b), the forcing amplitude is modelled as
⎛
T ⎞
1
∂
ρ̄
∂
T̃
1
νt , νt , νt , νt ,
νt ⎠ .
(3.17)
B = diag ⎝
Prt ∂ ũ
Prt ∂ ũ
Since the SLSE quantities in § 2.3 contain only the ratios of cospectra, multiplying B by
a non-zero constant does not affect the SLSE results. The examination for (3.16a,b) is
′
′′ from
and Trms
shown in figure 2 using the present DNS data (benchmark case). Both ρrms
′
(3.16a,b) are in line with DNS, though ρrms is slightly underestimated. Notably, the fact
′
′′ are primarily determined by u′′ and their mean flow gradients hints
and Trms
that ρrms
rms
that they are advection dominated. The larger deviation in the outer region indicates more
pronounced thermodynamic processes where the mean flow gradients are small and the
local Mach number is high.
In summary, two linear models used in this work are (1) the LNS model (3.5) and
(2) the eLNS model ((3.9) and (3.10a,b)). In both cases, (3.17) is used for variable control.
There is also a notation ‘eLNS-CD’ below, to be introduced in § 5.4. In § 3.1, we use the
Favre averages to derive the linear model, so we should also use q̃ to calculate L̂ and then
Φ. As Mab here is relatively low, the difference between q̃ and q̄ is very small (Huang
et al. 1995; Cheng & Fu 2022b). We have compared Φ using the two mean flows, and
the relative difference is less than 1 %. Thereby, for ease of later comparison with the
ODE-based mean flow (§ 6), we will use q̄ to compute L̃ and Φ throughout.
973 A36-13
X. Chen, C. Cheng, J. Gan and L. Fu
(a) 0.06
(b) 0.06
DNS
(3.16a,b)
ρ′rms/ρb
′′ /T
Trms
w
0.04
0.02
0
0.04
0.02
0
0.2
0.4
0.6
0.8
1.0
0
0
0.2
y/h
0.4
0.6
0.8
1.0
y/h
Figure 2. Wall-normal distributions of the (a) density and (b) temperature fluctuations (r.m.s.) from the DNS
data and (3.16a,b). Note that u′′rms is also from DNS.
4. Properties of the linear operator
As introduced in § 3.2, the linear operators L̂LNS and L̂eLNS were considered in
previous works, but their mathematical features were not elaborated. The framework
in § 3.3 requires a globally stable system, and the response amplification depends on
the non-normality of the linear operator (Trefethen 1997). Thereby, we scrutinize the
eigenspectra and pseudospectra of the linear operators in this section. The categorization
of different modes is also discussed. Significant differences will be noted from the
incompressible case, and the discussions have important implications for later SLSE
results.
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
4.1. Spectra of the linear operator
From (3.12), the eigenvalues of L̂ reflect the temporal modal instability of q̂′′ , defined
as L̂q̌′′ = −iωq̌′′ . Here ω = ωr + iωi is the eigenvalue with ωr the frequency and ωi the
growth rate, and q̌′′ is the right-eigenfunction. The eigenspectra of L̂LNS and L̂eLNS are
displayed in figures 3(a) and 3(b) at λx = 8h and λz = 2h. As will be shown in § 5, the
LCS at this (λx , λz ) is approximately the highest within the length scales considered. All
the eigenmodes in figure 3 are stable (ωi < 0), meeting the requirement of a globally
stable system. The distributions of ωr are similar between the LNS and eLNS spectra.
A prominent difference is that the damping rates |ωi | of the eLNS modes are nearly
two orders of magnitude higher. This is because after the substitutions in (3.10a,b),
the damping second-order derivative terms are highly increased. The proof from the
growth-rate decomposition is provided in Appendix C to support the explanation.
Next, the categorization of eigenmodes is discussed, which is crucial for interpreting
later linear-model results. The classic four mode branches are identified in both LNS
and eLNS spectra, namely the vortical, entropy, and fast and slow acoustic branches, as
denoted in different colours (Balakumar & Malik 1992). Notably, the term vortical branch
hereinafter is not restricted to vortices but a general term for all kinematically dominated
modes, analogous to the incompressible case. The reference frequencies of four branches,
as labelled in figure 3, are obtained by solving the fluctuations on an inviscid uniform
973 A36-14
Estimating coherent velocity and temperature structures
(b)
0
–0.1
0
–5
Vort./Entrp.
Fast acout.
Slow acout.
–40
–10
–20
0
40
20
–40
–20
ωr h/Ub
–4
0.4
–0.4
–1
–2
–1
0
1
–0.4
–0.4
3
4
.4
–3
–1
–1
–0.8
–2
–1
0
–0
–1
–1
4
–0.
–0.4 Slow acout.
–3
–0
Vort./Entrp.
–1
–1
0
2
4
–0.
Fast acout.
–0.4
0.4
–1
ωih/Ub
(d )
0
4
–0.
Fast acout.
–0.4
–0.8
–3
–2
–1
–1
4
–0.4
Slow acout.
–0.4
40
.4
–0.
–1
–1
–1
0
20
Vort./Entrp.
–1
ωih/Ub
(c)
0
ωr h/Ub
–1
–0.2
–1
ωih/Ub
(a)
1
2
3
4
ωr h/Ub
(e)
104
( f ) 104
Vort./
Entrp.
LNS
G
Gnormal
GUb/h
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
103
Slow
acout.
Slow
acout.
102
101
101
–2
–1
0
1
2
3
4
ωf h/Ub
G
Gnormal
103
Fast
acout.
102
100
–3
Vort./
Entrp.
eLNS
100
–3
–2
Fast
acout.
–1
0
1
2
3
4
ωf h/Ub
Figure 3. (a,b) Spectra, (c,d) pseudospectra (log10 ε), and (e, f ) the maximum response gain of the linear
operators for (a,c,e) LNS and (b,d, f ) eLNS in the benchmark case (λx = 8h, λz = 2h). The vertical dashed
lines in (a,b,e, f ) are from (4.1), and the black dots in (c,d) denote the eigenvalues.
mean flow, as
ωr,vort = ωr,ent = ũc kx
1 2
2
k + kz
ωr,acout = ũc kx ±
Mac x
⎫
for vortical, entropy modes, ⎬
for fast/slow acoustic modes,⎭
(4.1)
973 A36-15
X. Chen, C. Cheng, J. Gan and L. Fu
(a)
(b)
(c)
1.5
1.5
1.0
1.0
0.5
0.5
ρ̌ ′/ρb
ǔ ′′/U
1.5
b
Ť ′′/Tw
p̌ ′ /p–
1.0
|q̌′′|
0.5
0
0
0
0.5
y/h
1.0
0
0
0.5
1.0
0
y/h
0.5
1.0
y/h
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Figure 4. Eigenfunctions of the linear operator in the eLNS model (λx = 8h, λz = 2h): (a) acoustic mode
(ωh/Ub = 3.312 − 0.038i); (b) vortical mode (ωh/Ub = 0.870 − 0.059i) and (c) entropy mode (ωh/Ub =
0.866 − 0.075i).
where the subscript c denotes the channel centre (y = h), analogous to the free stream in
boundary layers. The derivation for (4.1) and corresponding eigenfunctions are detailed
in Appendix B. In the free stream, the acoustic modes are isentropic and dilatational,
the entropy modes are isobaric and static, and the vortical modes are solenoidal with
zero thermodynamic components, so they can be easily distinguished. In the channel
flow, however, these modes are more complicated due to the non-uniform mean flow and
viscosity. For illustration, figure 4 plots the eigenfunctions of the least stable vortical,
acoustic and entropy modes in the eLNS spectra, where the linearized pressure fluctuation
p̌′ /p̄ = ρ̌ ′ /ρ̄ + Ť ′′ /T̃. The modes of lower ωi exhibit common features. The acoustic
modes (figure 4a) can be recognized from others based on ωr . Also, they have large
pressure fluctuations, while the vortical and entropy modes are nearly isobaric. There
are a series of acoustic modes in figure 3 with different wall-normal wavenumbers (ky ),
and ωr,acout in (4.1) represents the lower (upper) frequency limit with zero ky for the fast
(slow) branches. In comparison, the vortical and entropy modes have close ωr , so they
cannot be distinguished directly in figure 3. The vortical modes (figure 4b) have large
velocity components, and the non-zero ρ̌ ′ and Ť ′′ are mainly from the passive advection
due to mean-flow gradients. Their entropy fluctuations are also non-zero since the mean
flow is not isentropic. In contrast, the entropy modes (figure 4c) are dominated by the
thermodynamic components, while their velocity components are not zero due to the
viscous and diffusive terms. The vortical modes are not isentropic and the entropy modes
are kinematic, so they can be coupled with each other and the modes with smaller ωi
in the vortical/entropy branch may not be as clearly distinguished as in figure 4. The
eigenfunctions of different types of modes in the LNS model are similar in shapes to
figure 4. The differences are contributed by the eddy terms related to µt and κt (see
Appendix B). The effects of different modes on the response behaviour will be further
discussed in § 4.2.
4.2. Pseudospectra of the linear operator
All the eigenmodes of L̂ are asymptotically stable, so its transient behaviour is determined
by external forcing and the non-normality of the linear operator (Trefethen 1997).
973 A36-16
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Estimating coherent velocity and temperature structures
The ε-pseudospectra is a quantitative measure of non-normality. By definition, a smaller ε
means that the eigenvalues are more sensitive to perturbations on L̂, so the non-normality
is stronger. Also, in response to a harmonic external forcing of frequency ωf , ε measures
the largest gain of the response G(ωf ) = max (−iωf I − L̂)−1 , which equals 1/ε0 with
ε0 the ε at the imaginary axis (see Chomaz (2005), for more details). The pseudospectra
and G(ωf ) of the two linear operators are depicted in figures 3(c)–3( f ) to provide insights
into the response characteristics. To distinguish from the resonant effects, the gain if L̂ is
normal, Gnormal (ωf ), is also plotted. For the LNS case, strong non-normality appears for
the vortical modes since G can be over one order of magnitude higher than Gnormal , which
is known to be an essential feature of the Orr–Sommerfeld operator. The non-normality
of the entropy modes cannot be directly measured because their ω are close to the vortical
modes. In comparison, the acoustic modes are of little non-normality as G results mostly
from the resonant effects. Therefore, we can expect that the behaviour of q̂′′ is largely
determined by the vortical modes, similar to the incompressible case. Hanifi, Schmid &
Henningson (1996) show that including the acoustic modes or not negligibly affects the
transient growth results for a laminar flow (LNS case). A similar examination will be
performed later for the turbulent mean flow in the stochastically driven system. When the
linear operator is eddy viscosity enhanced (figures 3d and 3f ), the non-normality of the
acoustic modes is slightly affected, but that of the vortical (and possibly entropy) modes is
severely weakened, reflected from the larger ε and much smaller G/Gnormal . Consequently,
the acoustic (and also entropy) modes, and thus the compressibility effects, are expected
to play more vital roles in the fluctuation evolution.
It is worth mentioning that Symon et al. (2021) and Kuhn et al. (2022) also report in their
incompressible cases that eddy viscosity decreases the non-normality of the linear operator
for structures of high λx /λz . Symon et al. (2021) conclude that this effect of counteracting
non-normality in the eLNS model mitigates the trade-off between turbulent production and
nonlinear energy transfer, leading to better prediction for streamwise-elongated structures
than the LNS model. In the compressible case studied here, however, additional acoustic
and entropy modes are present, so the weakened non-normality of the vortical modes can
lead to more complicated consequences.
To support the above observations and further quantify the effects of acoustic and
entropy modes, the energy amplification factor of the response V is computed. Here, V
is defined as the variance of q̂′′ , obtained as the trace of Φ with respect to the energy
norm (Hwang & Cossu 2010). Two V are needed, with acoustic (or entropy) modes
included and excluded, respectively, to distinguish their contributions. Equation (3.15)
contains the contributions of all modes, so a framework needs to be developed first that can
give the cospectra considering only partial modes. Following that in the transient growth
analysis (Hanifi et al. 1996), the response is restricted to the space spanned by partial
eigenfunctions,
q̂′′ ≈
Nm
m=1
cm (t)q̌′′m = Q̌′′ C(t),
(4.2)
where m and Nm are the indexes and total number of the modes selected, and cm is a
time-dependent coefficient. In matrix form, Q̌′′ = [q̌′′1 , . . . , q̌′′Nm ] and C = [c1 , . . . , cNm ]T .
The resulting cospectrum is Φ = q̂′′ q̂′′H = Q̌′′ CC H Q̌′′H . Similar to (3.15), the
governing equation for CC H is derived as
ΛCC H + CC H ΛH + [Q̌′′H (BBH )−1 Q̌′′ ]−1 = 0,
(4.3)
973 A36-17
X. Chen, C. Cheng, J. Gan and L. Fu
(a)
95
rVna (%)
103
40
0.3
20
90
85
(d )
80
50
60
1.0
40
55
100
λz /h
0
101
(e)
λz/h
20
rVna (%)
40
100
1
2
4
8
0
rV (%)
ne
100
4.0
70
UbV/h
60
20
60
0.3
80
0
rVna (%)
100
100
101
8
4.0
0
101
λz/h
4
2
45
100
20
40
95
55
V (all modes)
V (no acout.)
rV
na
102
10–1
60
1
102
101
10–1
1.0
60
60
80
90
λz/h
80
80
UbV/h
4.0
100
104
(b)
rVna (%)
100
(c)
105
80
60
1.0
40
20
80
0.3
1
2
80
4
8
0
λx/h
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Figure 5. (a,b) Energy amplification factors for all modes (V) and when acoustic modes are excluded (Vnacout )
(left axis) and their ratio rVna = Vnacout /V (right axis) at different spanwise wavelengths (λx = 8h), and the
contours of (c,d) rVna and (e) rVne = Vnentrp /V at different length scales. Panels (a,c) are for the LNS model
and (b,d,e) are for eLNS.
where Λ = diag([ω1 , . . . , ωNm ]). Therefore, the response when only partial eigenmodes
are included can be obtained by solving (4.3). Note that BBH can be non-invertible as the
element νt in (3.17) tends to zero at the wall. A small positive constant is added to the
diagonal of B for solving (4.3) only to avoid the singularity. Also, one drawback of (4.3) is
its inapplicability to the highly non-normal case where Q̌′′H Q̌′′ has a very large condition
number, hence nearly non-invertible.
Figures 5(a) and 5(b) plot the distributions of V for the LNS and eLNS models at
different λz (λx = 8h). The V in the LNS case is over one order of magnitude higher
than that in the eLNS case, consistent with the stronger non-normality in figure 3(c).
Note that the higher V in the LNS case does not directly influence the SLSE quantities
since they contain only the ratios of cospectra. The effects of acoustic modes are studied
first. In figure 5(a), the energy growth with acoustic modes excluded (Vnacout ) is quite
close to V, supporting the observation in figures 3(c) and 3(e) that the acoustic modes are
in secondary roles for the linear operator in terms of non-normality. As a quantitative
measure, the ratio rVna = Vnacout /V is plotted in percentage. In most of the λz range
973 A36-18
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Estimating coherent velocity and temperature structures
displayed, the non-acoustic components contribute over 90 % of the total energy growth
of the response, similar to the laminar case (Hanifi et al. 1996). For the eLNS model in
figure 5(b), however, the non-acoustic components contribute only approximately 50 % of
the total energy growth, and the contributions of acoustic modes are prominent, consistent
with the inference from figures 3(d) and 3( f ).
To gain further knowledge on the scale dependence of the acoustic-mode effects, the
contours of rVna are depicted in figures 5(c) and 5(d) for the two models. The λx and
λz ranges displayed are of interest for SLSE, as will be specified in § 5. The rVna in the
LNS case is over 80 % in most scale ranges, while that in the eLNS case is within 45 %
to 60 %. More importantly, rVna tends to increase at a larger scale in the LNS model,
and is over 95 % when λx > 2h and λz > 0.5h. In contrast, rVna diminishes as λx and λz
rise in the eLNS model, indicating that the large-scale structures are especially affected
by the acoustic modes. In the same way, the effects of entropy modes are quantified
in figure 5(e) for the eLNS case, using rVne = Vnentrp /V with Vnentrp the energy growth
excluding entropy modes. The rVne in the LNS case is not displayed as the matrix excluding
the entropy modes is highly non-normal (see figure 3c). In the eLNS case, rVne is within
60 % to 80 % in most scale ranges, so the entropy modes are also non-negligible for
the response growth, though their influences are weaker than the acoustic modes since
rVne > rVna . Note that the contours of rVne are not as smooth as rVna in figure 5(d), due
to the coupling effects between the vortical and entropy modes, as discussed in § 4.1.
Nevertheless, the qualitative trend in figure 5(e) is not affected. The entropy-mode effect
in the LNS case is not demonstrated, so its difference from the eLNS case is not reflected.
The results in § 5 suggest that the acoustic components play a larger role than the entropy
ones for SLSE in the present case, so the difference of rVna (figures 5(c) and 5(d)) is
important, and the acoustic components will be more focused on below.
We provide more interpretations on the trends in figure 5. For an inviscid uniform
mean flow (free stream, µ̃ = µt = 0), the eigenfunctions of different modes in (4.1) are
perpendicular to each other under the energy norm in (3.13), so L̂ is normal (George &
Sujith 2011). It becomes non-normal in the presence of mean flow gradients and viscosity.
The latter two effects are the strongest in the near-wall region, while the acoustic and
entropy modes are prevailing in the outer region, so the non-normality between the vortical
and other three branches is weak (figure 3c). Meanwhile, as λx and λz increase from
small scales, the amplified structure moves away from the wall into the outer region. The
decrease of mean flow gradients leads to less coupled mode branches and thus increasing
rVna (closer to the free stream condition), as observed in figure 5(c). When the model
is µt -enhanced (eLNS), however, the trends are reversed simply because µt is zero at
the wall and largest in the outer region with µt ≫ µ̃ (∼70 times in the present case).
This means, from the view of equations, that the damping effects due to the modelled
eddy terms are highly intensified (µ̃ → µ̃ + µt ). Consequently, the larger-scale structure
is more affected by µt , and the dissipative second-order derivative terms for µt result in
strong coupling between the acoustic, entropy modes and others. This is exactly what we
observe in figures 5(d) and 5(e).
In summary, there are four branches of eigenmodes in both LNS and eLNS cases,
and the vortical branch has strong non-normality to support the transient behaviour of
the fluctuations (response) subject to stochastic forcing. For the LNS model, the acoustic
modes have little non-normality and increasingly small contributions to the energy growth
as the length scales (both λx and λz ) increase. In contrast, the non-normality of the vortical
modes diminishes in the eLNS model due to the damping effects of µt on the eigenmodes.
As a result, the acoustic and entropy modes can account for 20 % to 55 % of the energy
973 A36-19
X. Chen, C. Cheng, J. Gan and L. Fu
growth; their contribution goes higher for larger-scale structures. The above is a significant
difference between the LNS and eLNS models, not present in incompressible flows.
For the eLNS model, the fact that the acoustic and entropy modes, and thus
compressibility effects, can have such a high contribution to the fluctuation growth
deserves serious concern, considering the great similarities reported before to
incompressible flows (see § 1). We examine below whether this prominent role of
acoustic and entropy components in the eLNS model is an intrinsic defect introduced
by turbulence modelling, through comparison with DNS. In fact, we show in § 5 that
the prominent acoustic and entropy components can be problematic, especially for the
density/temperature fields, and will seek remedies.
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
5. General flow structures and SLSE results
5.1. General flow structures
We first present an overview of the general structures of velocity and temperature fields in
the benchmark case. Comprehensive analyses have been reported in previous works based
on experiments and DNS for both channel and boundary-layer flows (Smith & Smits 1995;
Ringuette et al. 2008; Pirozzoli & Bernardini 2011; Cogo et al. 2022; Cheng & Fu 2023a),
so here we only provide a brief description to set the grounds for later discussions.
After the velocity transformation of Griffin et al. (2021), the mean streamwise velocity
+
∗
+
ũ+
GFM ( y ) matches the universal incompressible profile ūincp ( y ) within and below the
logarithmic region, as shown in figure 6(d). Therefore, we can follow the criterion in
incompressible flows to name different regions in terms of y∗ and y/h. Three wall-normal
heights are selected, namely y∗ = 11 in the buffer layer (near-wall region), y∗ = 107 (y =
0.14h) in the logarithmic layer and y = 0.5h (y∗ = 385) in the outer region, to display the
instantaneous fields in different regions using DNS. Hence, the overall flow organization
can be recognized. Notably, Abe & Antonia (2017) show in their incompressible case
that y ≈ 0.5h is the region where large-scale structures of the streamwise velocity and
temperature (as a passive scalar) fluctuations dominate. In the near-wall region (figures 6c
and 6g), both u′′+ and T ′′+ exhibit streamwise-elongated streaks, connected with the
‘sweep’ and ‘ejection’ events in the near-wall self-sustaining cycles (Hamilton, Kim &
Waleffe 1995; Schoppa & Hussain 2002). Meanwhile, u′′+ and T ′′+ strongly resemble
each other morphologically. As a quantitative measure, Cheng & Fu (2023a) show that
′′ ), is approximately
the correlation coefficient of u′′+ and T ′′+ , Ru′′ T ′′ = u′′ T ′′ /(u′′rms Trms
∗
one at y 12. Further away from the wall into the logarithmic region (figures 6b and
6f ), dominant velocity streaks are in larger length scales, while the streamwise-elongated
feature is less obvious for T ′′ . In other words, T ′′ is more isotropic than its near-wall
counterpart, filled with jellyfish-shaped structures. The observation position in figures 6(a)
and 6(e) is out of the logarithmic region. Large scale u′′ streaks are still identifiable, but
streamwise T ′′ streaks are hardly seen with obvious spanwise extensions. Cheng & Fu
(2023a) demonstrate that the spanwise length scales of u′′ and T ′′ corresponding to their
pre-multiplied spectra peaks are actually comparable to each other throughout the channel;
T ′′ has a shorter peak streamwise length scale within and above the logarithmic region,
thus tending to be more isotropic.
In the following, we focus on predicting the coherent velocity and temperature
fluctuations using SLSE based on the measurement at outer locations. Here y∗m is fixed
at the centre of the logarithmic region (Marusic et al. 2010), and the estimations are
conducted at y∗p < y∗m . In analogy to the incompressible flow (Townsend 1976), the centre
∗1/2
of the logarithmic region is evaluated to be 3.9Reτ
973 A36-20
, which gives y∗m = 107.
Estimating coherent velocity and temperature structures
(d )
2
3
(e)
103
2
y = 0.5h
0
4
–3
5
102
z/h
0
4
–5
101
–2
–4
( f )2
y = 0.14h
0 y∗
–2
–4
(c) 2
0
z/h
z/h
0
–2
–4
(b) 2
0.3
ũ + = y ∗
y ∗ = 11
–2
–4
(g) 2
7
–0.3
0
4
1
0
z/h
(a)
–1
0
4
3.2
(1/0.4) log ( y ∗) + 5.5
–2
–4
0
x/h
4
–7 100
0
0
z/h
z/h
0
10
ũ +GFM
20
–2
–4
0
4
–3.2
x/h
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Figure 6. Overview of the turbulent field for the benchmark case: (a–c) instantaneous streamwise velocity
fluctuation (u′′+ ), (d) transformed mean velocity profile and (e–g) instantaneous temperature fluctuation (T ′′+ );
(a,e) are at y = 0.5h (y∗ = 385), (b, f ) are at y = 0.14h (y∗ = 107) and (c,g) are at y∗ = 11 (top view).
5.2. Estimation for streamwise velocity
2 and A
The two components of HL,uu , i.e. γuu
pm,uu , are calculated first. They describe the
spectral coherence and amplitude ratio of û′′ at different heights. Afterwards, HL,uu is used
to estimate the instantaneous field. In general, the SLSE results for û′′ from DNS and linear
models closely resemble those from the incompressible case (Madhusudanan et al. 2019;
Gupta et al. 2021), so the discussion is presented briefly, highlighting crucial points.
Panels (a−c) in figures 7 and 8 are the results from the DNS data. The three estimation
locations are y∗p = 70, 40 and 10, ranging from the logarithmic layer down to the buffer
layer. Limited by the computational domain, the largest length scales displayed are λx ≈ 8h
and λz ≈ 4h. As in incompressible flows (Baars et al. 2016; Madhusudanan et al. 2019),
large-scale structures have stronger coherence than the small scale, especially in the
streamwise direction. This is in line with figure 6 where the streamwise-elongated streaky
motions dominate the instantaneous velocity field. Meanwhile, the coherence is weakened
with the decrease of y∗p (figures 7a–7c), simply because y∗m and y∗p are farther apart. At
2 > 0.5 only appears at λ > 4h, or λ∗ > 3600, which is much
y∗ = 10, the region with γuu
x
x
larger than the dominant length scale of the u′′ streaks (λ∗x ≈ 1000, Yao & Hussain 2020).
However, though close to the wall, the motion of the largest scale is still coherent with
that at y∗m = 107, reminiscent of the AEM. From figures 8(b) and 8(c), the structures
with λz < 0.5h have relatively large Apm,uu , which means narrow (in terms of λz ) streaky
motions are continuously enhanced approaching the wall. This is consistent with figure 6
where the fluctuation amplitude rises as y∗ decreases.
2 and A
We next focus on the linear model results. In general, γuu
pm,uu from the eLNS
model are closer to DNS than the LNS model. By considering the wall-normal variation
2 and A
of the forcing amplitude (3.17), γuu
pm,uu near the wall can also be well predicted
by eLNS, as shown in figures 7(i) and 8(i), though Apm,uu at small λz ( 0.5h) tends to
be under-predicted. In comparison, the LNS results have increasingly larger deviations
973 A36-21
X. Chen, C. Cheng, J. Gan and L. Fu
λz /h
(a)
(b)
(c)
4.0
4.0
4.0
1.0
1.0
1.0
0.3
0.3
1
2
4
8
λz /h
2
4
8
(e)
(f)
4.0
4.0
4.0
1.0
1.0
1.0
(d )
0.3
0.3
1
2
4
8
(g)
λz /h
0.3
1
2
4
8
(i)
4.0
4.0
4.0
1.0
1.0
1.0
0.3
1
2
4
8
2
4
8
1
2
4
8
2
4
8
0.3
1
(h)
0.3
1
0.3
1
λx /h
2
4
8
λx /h
1
λx /h
γ 2uu
0
0.5
1.0
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Figure 7. Contours of 2-D LCS for u′′ (y∗m = 107) at (a,d,g) y∗p = 70, (b,e,h) y∗p = 40 and (c, f,i) y∗p = 10.
Panels (a−c) are from the DNS data, (d−f ) are from the LNS model and (g−i) are from the eLNS model. The
dotted lines denote λx = λz .
2 and A
towards the wall, and γuu
pm,uu are both underestimated. In particular, Apm,uu is
smaller than unity, i.e. |û′′p |2 < |û′′m |2 in most regions in figures 8(e) and 8( f ), which
is opposite to the DNS trend. This indicates that the structures modelled by LNS are
localized in the wall-normal direction. The introduction of µt enhances the structural
coherence between different heights, and can reasonably reflect the wall-normal coherence
2 in
and amplitude variation of the velocity fluctuations. Notably, in the region λx < λz , γuu
the eLNS model has some deviations from the DNS data, not present in incompressible
cases. This difference is related to compressibility effects and will be further discussed in
§ 6.
After obtaining HL , the large-scale coherent portion of the instantaneous field u′′p at y∗p
can be estimated using (2.4a) and the DNS snapshots u′′ at y∗m , reflecting the superposition
effects in IOIM. As pointed out by Baars et al. (2016), the scales at very small γ 2 can
be erroneously estimated due to the possibly large HL , then one may obtain intense
small-scale motions in some regions of little coherence with the fields measured. To
avoid this, a threshold γth2 = 0.05 is introduced, and HL with γ 2 < γth2 is forced to zero
(Madhusudanan et al. 2019). In this way, the small-scale motions of little coherence are
∗
artificially ruled out. The resulting snapshots of u′′+
p at yp = 70 and 10, estimated from
973 A36-22
Estimating coherent velocity and temperature structures
λz /h
(a)
(b)
(c)
4.0
4.0
4.0
1.0
1.0
1.0
0.3
2
4
8
λz /h
1
2
4
8
(e)
(f)
4.0
4.0
4.0
1.0
1.0
1.0
(d )
0.3
2
4
8
(g)
1
2
4
8
(h)
(i)
4.0
4.0
4.0
1.0
1.0
1.0
0.3
2
4
8
2
4
8
1
2
4
8
2
4
8
0.3
0.3
1
1
0.3
0.3
1
λz /h
0.3
0.3
1
1
λx /h
2
4
8
λx /h
1
λx /h
|û ′′(yp)|2
0
1
2
|û ′′(ym)|2
Figure 8. Contours of the amplitude ratio for u′′ (Apm,uu , y∗m = 107) at (a,d,g) y∗p = 70, (b,e,h) y∗p = 40 and
(c, f,i) y∗p = 10. Panels (a−c) are from the DNS data, (d−f ) are from the LNS model and (g−i) are from the
eLNS model. The dotted lines denote λx = λz .
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
different models, are depicted in figure 9. The corresponding 2-D velocity spectra of u′′+
p ,
′′+†
2 ′′+
′′+†
( ym )DNS ,
Φu+p up ( yp ) = û′′+
p ( yp )ûp ( yp ) = |HL,uu ( yp , ym )| û ( ym )û
(5.1)
are also shown for more quantitative evaluation. Compared with figures 6(b) and 6(c),
small-scale structures (λx h, λz 0.3h) are nearly missing in figures 9(a) and 9(d)
due to the weak coherence at those scales, hence not considered by the superposition
effects. Also, the dominant structures in figures 9(a) and 9(d) have similar λx and λz .
These footprints posed by the motions at y∗m on y∗p are crucial in the IOIM (Baars et al.
2016). For the linear models, the eLNS results in figures 9(c) and 9( f ) bear a strong
resemblance to the DNS counterparts, which is expected from the relative agreement of
2 and A
γuu
pm,uu in figures 7 and 8. Nevertheless, the structures of λz 0.5h are somewhat
underestimated, which may be improved by further modelling the scale dependence of B
(Gupta et al. 2021). In comparison, the LNS model estimates weak u′′+ , especially down
to the buffer layer. The velocity field and 2-D spectrum in figure 9(e) are nearly uniform
under the contour levels, indicative of missing footprints from the wall-attached motions
in the logarithmic region.
In summary, the SLSE results for streamwise velocity are quite similar to the
incompressible case, suggesting weak compressibility effects on the linearized momentum
equation under current flow conditions. By introducing µt and κt , the spectral linear
973 A36-23
X. Chen, C. Cheng, J. Gan and L. Fu
(ii)
4
1
4 8
(ii)
2
0
4
(c) (i)
z/h
1
4 8
z/h
1.0
0.3
–2
–4
4
0
–4.5 0
4.5
up′′+
1
4 8
λx /h
x/h
kxkzΦ+u u
p p
0 0.2 0.4
1
4 8
1.0
0.3
0
4
(ii)
4.0
0
1.0
0.3
–2
–4
0
4
1
x/h
–2.5 0
2.5
kxkzΦ+u u
up′′+
4 8
λx /h
p p
0 0.15 0.30
Figure 9. Estimated instantaneous
and the pre-multiplied 2-D spectra at (a–c) = 70 and (d–f ) y∗p = 10
using the snapshot at y∗m = 107 and the kernel function from the (a,d) DNS data and models of (b,e) LNS and
(c, f ) eLNS.
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
u′′+
p
0
2
λz /h
0
4 8
4.0
–2
–4
4.0
1
(ii)
( f )(i)
(ii)
2
4
2
λz /h
0.3
–2
–4
0
(e) (i)
1.0
1.0
0.3
–2
–4
4.0
0
0
λz /h
0.3
0
4.0
λz /h
1.0
(b) (i)
z/h
z/h
0
(ii)
2
4.0
–2
–4
z/h
(d) (i)
λz /h
z/h
2
λz /h
(a) (i)
y∗p
coherence and amplitude ratio of û′′ between different heights can be well captured by the
linear models (eLNS), though the structures of small λz can be somewhat underestimated.
This agreement gives us confidence to perform a parameter study for the velocity
fluctuations in § 6. Instead of collecting large amounts of time-resolved measurements or
simulations, the kernel function of velocity can be obtained using the eLNS model if one
only has a mean flow. Then SLSE can be performed based on one snapshot of the velocity
field.
5.3. Estimation for temperature
The SLSE results for temperature fluctuations are investigated in a similar style to § 5.2.
2 and H
In contrast to the velocity counterparts, the features of γTT
L,TT were rarely reported
in previous studies, so the implications from DNS are discussed first.
2 and A
∗
The γTT
pm,TT from DNS are plotted in panels (a−c) in figures 10 and 11 at yp =
2 is very similar to γ 2 . In particular, figure 10(c) is
70, 40 and 10. The distribution of γTT
uu
nearly indistinguishable from figure 7(c) at y∗p = 10. An important implication is that in the
buffer layer, in addition to the nearly perfect correlation between u′′ and T ′′ (see Ru′′ T ′′ and
2 and γ 2 ) are basically identical
figure 6), the spectral coherence with the measurement (γuu
TT
2 is no
for û′′ and T̂ ′′ , which benefits the compressible turbulence modelling. Moreover, γTT
2
∗
lower than γuu at the three yp displayed, and is even higher in some regions with λx < λz .
Therefore, the wall-normal spectral coherence of the temperature fluctuation is at least not
weaker than the streamwise velocity. Intense inner–outer interaction is anticipated, and
973 A36-24
Estimating coherent velocity and temperature structures
λz /h
(a)
(b)
(c)
4.0
4.0
4.0
1.0
1.0
1.0
1
2
4
8
λz /h
1
2
4
8
(e)
(f)
4.0
4.0
4.0
1.0
1.0
1.0
(d )
1
2
4
8
(g)
1
2
4
8
(h)
(i)
4.0
4.0
4.0
1.0
1.0
1.0
1
2
4
8
2
4
8
1
2
4
8
2
4
8
0.3
0.3
0.3
1
0.3
0.3
0.3
λz /h
0.3
0.3
0.3
1
λx /h
2
4
8
λx /h
0
0.5
1
λx /h
1.0
γ 2TT
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Figure 10. Contours of 2-D LCS for T ′′ (y∗m = 107) at (a,d,g) y∗p = 70, (b,e,h) y∗p = 40 and (c, f,i) y∗p = 10.
Panels (a−c) are from the DNS data, (d−f ) are from the LNS model and (g−i) are from the eLNS model.
the view is supported that T ′′ is also a wall-attached quantity (Cheng & Fu 2023a). The
2 and γ 2 are all located at λ ≈ h at different y∗ , different from the impression
peaks of γuu
z
p
TT
from figure 6 that T ′′ tends to be more isotropic as y∗ rises. High values of Apm,TT appear
at λz < 0.5h, the same as that for Apm,uu . The maximum Apm,TT exceeds six at y∗p = 10
(figure 11c), which means that |T̂p′′ |2 with small λz increases rapidly towards the wall
with respect to |T̂m′′ |2 , due to local large temperature gradients.
The linear model results for T ′′ are discussed below. In general, the LNS model
2 and A
∗
′′
underestimates γTT
pm,TT at all yp selected, the same as that for u . The Apm,TT
in figures 11(e) and 11( f ) are less than unity in most regions, so the T̂ ′′ predicted is
more localized with insufficient wall-normal connection. Nevertheless, the feature of λx
2 is captured in the LNS model, and γ 2 peaks at λ ≈ h. The
preference over λz for γTT
z
TT
2 in
behaviour of the eLNS model, however, is quite different from those for u′′ . The γTT
figures 10(g)–10(i) differs considerably from the DNS data, nearly isotropic in terms of
λx and λz . Also, Apm,TT at all y∗p is underestimated. Thereby, the eLNS model fails to
give agreeable results with DNS for estimating temperature fluctuations. Revisiting the
assumptions in § 3.2, (3.8a) treats T ′′ in an isotropic way, which may be responsible for
2 in the eLNS model. Nevertheless, our numerical tests show
the isotropic behaviour of γTT
that using (3.8b) instead does not improve obviously the results, though some degree
of anisotropy is introduced. The discussion in § 4 suggests that acoustic and entropy
components are significant in the eLNS model, but no supporting signs are observed
973 A36-25
X. Chen, C. Cheng, J. Gan and L. Fu
λz /h
(a)
(b)
(c)
4.0
4.0
4.0
1.0
1.0
1.0
0.3
0.3
1
2
4
8
λz /h
2
4
8
(e)
(f)
4.0
4.0
4.0
1.0
1.0
1.0
(d )
0.3
0.3
1
2
4
8
(g)
λz /h
0.3
1
2
4
8
(i)
4.0
4.0
4.0
1.0
1.0
1.0
0.3
1
2
4
8
2
4
8
1
2
4
8
2
4
8
0.3
1
(h)
0.3
1
0.3
1
λx /h
2
4
8
1
λx /h
λx /h
|T̂ ′′(yp)|2
0
1
2
0
1.5
3.0
|T̂ ′′(ym)|2
0
3
6
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Figure 11. Contours of the amplitude ratio for T ′′ (Apm,TT , y∗m = 107) at (a,d,g) y∗p = 70, (b,e,h) y∗p = 40 and
(c, f,i) y∗p = 10. Panels (a−c) are from the DNS data, (d−f ) are from the LNS model and (g−i) are from the
eLNS model.
from the displayed DNS data. Therefore, we check the eLNS results below for possible
explanations of the differences.
5.4. Cospectrum decomposition
To figure out the difference between the eLNS and DNS results, Φ is scrutinized. As
discussed in § 4, multiple branches of modes are present. One way to identify different
mechanisms is to conduct a mode decomposition for Φ, known as the Karhunen–Loéve
or proper orthogonal decomposition (POD). The eigenvalue problem is constructed as
Φ q̌′′j = θj q̌′′j , where θj is the eigenvalue, and the matrix–vector multiplication is defined in
terms of the energy norm. Note that the symbol q̌′′j is reused to represent the eigenfunctions
for clarity, though the physical meaning is not identical to that in § 4. Since Φ is Hermite,
θj is real. Also, q̌′′j satisfies the orthogonal relation (q̌′′i , q̌′′j )E = Eb δij (Eb is the energy
dimension). Different from (4.1), θj measures the energy of the jth POD mode (q̌′′j ) and
′′
′′
j θj = V. If the pairs (θj , q̌j ) are sorted in the descending order of θj , then q̌1 represents
the leading energy-containing structure of the response.
The response with λx = 8h, λz = 2h is considered first using the eLNS model, close to
2 and γ 2 . Note that Φ discussed below is the solution of (3.15), not (5.1).
the maximum γuu
TT
973 A36-26
Estimating coherent velocity and temperature structures
(a)
0.9
θ1 = 9.24
ρ̌ ′/ρb
ǔ ′′/Ub
p̌′/p̄
(3.16a,b) for ρ̌ ′
(b)
1.5
0.6
1.0
0.3
0.5
θ5 = 0.76
(5.2a,b) for ρ̌ ′
(5.2a,b) for ǔ ′′
ρ̌ ′/ρb
ǔ ′′/Ub
p̌′/p̄
|q̌ ′′|
0
0
0.2
0.4
0.6
0.8
1.0
0
0
0.2
y/h
0.4
0.6
0.8
1.0
y/h
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Figure 12. Profiles of different components of the cospectrum POD modes in the eLNS model (λx = 8h,
λz = 2h): (a) the most energetic mode (θ1 ) and (b) the fifth most energetic mode (θ5 ). Equations (3.16a,b) and
(5.2a,b) are used to approximate other components based on ǔ′′ (for (a)) and p̌′ (for (b)).
After POD, at least two types of modes are identified, resulting from the different branches
in § 4. The most energetic mode q̌′′1 is shown in figure 12(a), where ǔ′′ is the largest
component. Meanwhile, ρ̌ ′ and Ť ′′ (not shown) can be well approximated by (3.16a,b) and
ǔ′′ , analogous to figure 2. This demonstrates that ρ̌ ′ and Ť ′′ of this mode are dominated
by advection, and q̌′′1 is in line with the trends in DNS. Therefore, this mode is termed an
advection mode, which can be considered as a subset of the vortical modes. The dilatation
of the advection mode is tiny, and p̌′ /p̄ is also small, thus analogous to the incompressible
counterpart. Meanwhile, the advection mode takes the form of streamwise-elongated
velocity (ǔ′′ ), density and temperature streaks forced by streamwise vortices (v̌ ′′ , w̌′′ ), as
observed in figure 6 and reported in previous experimental, DNS and linear-model results
(Coleman et al. 1995; Williams et al. 2018; Chen et al. 2023). A 3-D view of this mode
will be shown later. Modes q̌′′2 , q̌′′3 and q̌′′4 have similar features to q̌′′1 , but q̌′′5 is different. As
shown in figure 12(b) for q̌′′5 , p̌′ and ρ̌ ′ are much larger than ǔ′′ , and ρ̌ ′ is in high amplitude
at the wall, so this mode experiences strong dilatation. This mode is of an acoustic nature,
as discussed in § 4.1. To prove this, the analytical eigenfunction of the acoustic mode in
the free stream is used (see Appendix B), leading to the following relations:
p̌′
ρ̌ ′
=
,
ρ̄
γ0 p̄
ǔ′′
kx
p̌′
=
,
a
kx2 + ky2 + kz2 γ0 p̄
(5.2a,b)
where a is the speed of sound. Equation (5.2a,b) is examined in figure 12(b) with ky = 0.
Even though (5.2a,b) is derived based on a uniform mean flow, remarkable agreement
is observed for both ρ̌ ′ and ǔ′′ , confirming that q̌′′5 is of an acoustic nature except in the
near-wall region. The large ρ̌ ′ at the wall is due to the prescribed boundary condition
Ťw′′ = 0. Furthermore, a series of acoustic modes exist with different ky , as shown in
figure 13. These modes with ky =
/ 0 strongly oscillate in the interior region with (5.2a,b)
also satisfied, and the maximum amplitudes of ρ̌ ′ and p̌′ remain nearly constants. The
θj of these modes are labelled in figures 12 and 13, non-dimensionalized by h/Ub . The
energy ratio of the most energetic acoustic and advection modes is 8.2 % (θ5 /θ1 ), so these
acoustic modes have non-negligible contributions to Φ. In comparison, the same ratio
in the LNS case is only 0.06 %. Therefore, the conclusion in § 4 is supported since the
973 A36-27
X. Chen, C. Cheng, J. Gan and L. Fu
θ7 = 0.57
(a) 2
θ11 = 0.39
(b) 2
θ15 = 0.27
(c) 2
ρ̌ ′/ρb
ǔ ′′/Ub
p̌′/p̄
(5.2a,b) for ρ̌ ′
|q̌ ′′|
1
0
1
0
0.5
1.0
0
1
0
0.5
y/h
1.0
0
0
0.5
y/h
1.0
y/h
Figure 13. Profiles of different components of the cospectrum POD modes in the eLNS model (λx = 8h, λz =
2h): the (a) seventh, (b) eleventh and (c) fifteenth energetic modes. Equation (5.2a,b) is used to approximate
other components based on p̌′ .
acoustic components are negligible for the response in the LNS case, but non-negligible
in the eLNS case. Moreover, no counterparts to the entropy modes are found in at least
the top thirty most energetic modes in both the LNS and eLNS cases, suggesting their
relatively small contribution than the acoustic components at this length scale.
Considering § 4, and observing the great similarities between u′′ and T ′′ in §§ 5.2 and
5.3, it is conjectured that the significant role of acoustic and entropy components to the
fluctuation growth in the eLNS model is an intrinsic defect introduced by the turbulence
modelling. To verify this, Φ is decomposed to exclude the contributions of acoustic and
entropy modes, denoted as Φ ae , and the remaining part is Φ nae . The eigenvalue problem
above does not allow a strict decomposition as Φ = Φ ae + Φ nae , so the singular value
decomposition is deployed instead, taking the form of
σj ψ̌ j φ̌ H
(5.4)
Φ=
σj ψ̌ j φ̌ H
σj ψ̌ j φ̌ H
j ,
j +
j =
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
j
j∈ae
Φ ae
j∈nae
Φ nae
where σj is the singular value, and ψ̌ j and φ̌ j are the decomposition modes. As Φ is
Hermite, ψ̌ j = φ̌ j ; σj and ψ̌ j have analogous behaviours to θj and q̌′′j , hence are not
elaborated for conciseness. The same criteria as in § 4 are deployed to identify the acoustic
and entropy modes, based on their eigenfunctions (see figure 4). It is worth mentioning
the difference of the decompositions in figure 5 and (5.4). The former is to exclude the
acoustic (or entropy) eigenmodes of the linear operator before solving the response, while
the latter is a posterior decomposition of the response with all eigenmodes included. The
two strategies are not equivalent, and the latter is adopted here because the acoustic and
entropy modes are not fully decoupled with the vortical modes (see § 4), especially in the
near-wall region.
The cospectrum decomposition results using (5.4) are displayed in figure 14, still with
λx = 8h, λz = 2h. They are normalized by the maximum of û′′ û′′† . Interestingly, ρ̂ ′ ρ̂ ′† ,
û′′ û′′† and û′′ T̂ ′′† have clear attributions, suggesting a rather ‘perfect’ decomposition.
Here, ‘perfect’ means that we can clearly identify which physical process is responsible
for the generation of different components. Specifically, nearly all ρ̂ ′ ρ̂ ′† are contributed
973 A36-28
Estimating coherent velocity and temperature structures
0
y′/h
(a)
0.2 0.4
y′/h
0 0.06 0.12
(c)
1.0
1.0
0.5
0.5
0.5
0.5
0.5
0
1.0
0.5
1.0
(f)
0.5
0
1.0
(g)
0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.5
0
1.0
0.5
0
1.0
( j)
0.5
0
1.0
(k)
0
1.0
1.0
1.0
0.5
0.5
0.5
0.5
0.5
0
1.0
0.5
0
1.0
(n)
0.4
Φ
Φnae
DNS
0.2
0
0
0.5
y/h
1.0
0.5
0
1.0
0
1.2
0.2
0.30
0.6
0.1
0.15
0
0.5
y/h
1.0
0.5
1.0
0.5
1.0
0.5
y/h
1.0
( p)
(o)
0
1.0
(l)
1.0
(m)
0.5
(h)
1.0
(i)
0 0.12 0.24
(d)
1.0
0
y′/h
0.5 1.0
1.0
(e)
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
0
(b)
0
0
0.5
y/h
1.0
0
0
Figure 14. Cospectra of different components in the eLNS model (λx = 8h, λz = 2h): (a,e,i,m) ρ̂ ′ ρ̂ ′† ,
(b, f,j,n) û′′ û′′† , (c,g,k,o) T̂ ′′ T̂ ′′† and (d,h,l,p) |û′′ T̂ ′′† |. They are non-dimensionalized by ρb , Ub and Tw
accordingly. Panels (a−d) are the original results containing all modes, (e−h) are from Φ nae and (i−l) are
from Φ ae . The coordinates y and y′ are for the two variables of cospectra, respectively (see (2.7)). Panels
(m−p) are the comparisons with Φ DNS at y′ = y.
by Φ ae , while the contribution of Φ nae is tiny (figures 14e and 14i). From figure 12(a),
the latter part can be approximated by u′′ and the mean flow gradients, representing the
advection effect. In contrast, û′′ û′′† and û′′ T̂ ′′† are mostly contributed by Φ nae , nearly
unaffected by the acoustic and entropy components. Thereby, the eLNS results for u′′ in
§ 5.2 basically agree with DNS, as in the incompressible case. Also, it is suggested that the
u′′ –u′′ and u′′ –T ′′ couplings are dominated by advection and other vortical motions, while
the acoustic- and entropy-induced motions have little coherence for the couplings. For
T̂ ′′ T̂ ′′† , Φ nae and Φ ae have nearly equal contributions. The former is more concentrated
towards the wall, represented by (3.16a,b), while the latter extends to the channel centre
populating neutral acoustic and entropy waves. To further justify the decomposition, the
normalized Φ from DNS is also computed. Since we are mainly concerned with the
relative amplitudes of Φ nae and Φ ae , only the Φ with y = y′ are extracted for comparison,
973 A36-29
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
X. Chen, C. Cheng, J. Gan and L. Fu
as shown in figures 14(m)–14(p). For û′′ û′′† and û′′ T̂ ′′† , Φ and Φ nae are close to each
other and they basically follow Φ DNS , though differences in shapes are observable. The Φ
in the eLNS model varies more mildly than the DNS data, possibly due to using a smooth
µt . For ρ̂ ′ ρ̂ ′† and T̂ ′′ T̂ ′′† , however, Φ is several times higher than Φ DNS , exhibiting
large values around the centreline. After the decomposition, Φ nae is much closer to Φ DNS ,
also tending to diminish towards the channel centre. Thereby, we can conclude that the
acoustic and entropy components Φ ae in the eLNS model barely exist in the DNS data.
In short, in the eLNS model, û′′ û′′† and û′′ T̂ ′′† remain little influenced by the acoustic
and entropy components, while ρ̂ ′ ρ̂ ′† and T̂ ′′ T̂ ′′† are largely affected, which can be
responsible for the deviation of HL,TT in § 5.3. For improvement, we use Φ nae in the
following to estimate temperature and also the velocity–temperature coupling. The results
are denoted as eLNS-CD (‘CD’ for cospectrum decomposition).
Before presenting the eLNS-CD results, we need to examine whether the dominant
advection mode in figure 12 is a universal feature at different length scales. The scales
(λx , λz ) = (8h, 0.3h), (0.3h, 2h) and (0.3h, 0.3h) are considered in addition to (8h, 2h)
in figure 12, covering the large and small streamwise/spanwise scales studied in § 5.3.
The velocity and temperature components of the most energetic modes at these scales
are plotted in figure 15. Their 3-D structures are depicted in figure 16, where u′′(kx ,kz ) =
ǔ′′(kx ,kz ) ( y) exp[i(kx x + kz z)] (from (2.3)). The energetic large-scale mode (8h, 2h) wanders
across different layers, crucial for the inner–outer interaction as discussed above. This
type of mode is also characterized by geometrically self-similar hierarchies (McKeon
2019). The mode (8h, 0.3h) is also an advection mode as in figure 12(a), exhibiting
streamwise-elongated streaky motions very close to the wall. The peaks of ǔ′′ and Ť ′′
are at y∗ = 14 and 11, respectively, within the buffer layer. Also, ρ̌ ′ and Ť ′′ result primarily
from the advection process, well approximated by (3.16a,b). In comparison, the mode
(0.3h, 2h) is spanwise elongated (since λz > λx ) with large amplitudes of v̌ ′′ and w̌′′ . This
mode is not dominated by streamwise advection, but also results from vortical modes since
the thermodynamic components are small. For the mode (0.3h, 0.3h) in figures 15(c) and
16(d), v̌ ′′ is the largest component and ǔ′′ ∼ w̌′′ as λx ∼ λz . This mode is mostly amplified
around y ≈ 0.5h, and its oblique structure can be associated with the spanwise meandering
of large-scale motions (Hutchins & Marusic 2007; Abe et al. 2018). Meanwhile, its Ť ′′
component is in low amplitude, much lower than the approximation by (3.16a,b). In short,
with the decrease of λx , the most energetic structure changes gradually from the motions
dominated by streamwise advection to spanwise elongation. Consequently, the cospectrum
decomposition at small λx may not be as ‘perfect’ as in figure 14. From §§ 5.2 and 5.3, the
structures with smaller λx tend to have weaker coherence for the velocity and temperature
fluctuations between different heights, so they can be of secondary importance for
SLSE.
2 and A
The γTT
pm,TT from the eLNS-CD model are displayed in figures 17 and 18 at
the same y∗m and y∗p as in § 5.3. The DNS results in figures 10 and 11 are re-plotted for
easier comparison. The cospectrum decomposition requires categorization of the POD
modes, so the eLNS-CD results are not as smooth as those by eLNS, as discussed for
figures 5(e) and 15. As can be seen, after removing the acoustic and entropy components,
2 and A
2
γTT
pm,TT are in good agreement with DNS. To be specific, γTT from eLNS-CD
is not as isotropic as that in figure 10 (eLNS results), and the scale selection in terms
2 does not monotonically increase with λ . Meanwhile, the
of λz is captured, so γTT
z
high Apm,TT at λz < 0.5h is correctly reflected, showing a prominent improvement over
figures 11(g)–11(i).
973 A36-30
Estimating coherent velocity and temperature structures
(a)
(b)
θ1 = 0.08
3
ǔ ′′/U
b
v̌ ′′/Ub
w̌ ′′/Ub
Ť ′′/Tw
2
(c)
θ1 = 0.04
1.2
λx = 8h
λz = 0.3h
λx = 0.3h
λz = 2h
θ1 = 0.02
1.2
λx = 0.3h
λz = 0.3h
0.8
0.8
0.4
0.4
|q̌′′|
1
0
0
0.5
1.0
0
0
0.5
y/ℎ
1.0
0
0
0.5
y/ℎ
1.0
y/ℎ
Figure 15. Different components of the most energetic POD modes (θ1 ) for the eLNS cospectrum at different
length scales (λx , λz ): (a) (8h, 0.3h), (b) (0.3h, 2h) and (c) (0.3h, 0.3h).
(b)
1
y/ℎ
–2
–1
0
1
1
0 x/ℎ
0
z/ℎ
–1
0
1
–1
–2
y/ℎ
(a)
1
0 x/ℎ
0
z/ℎ
2
1
–1
2
0.05 0.30 0.55 0.80 1.05
(d)
–2
y/ℎ
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
1
–1
0
1
0 x/ℎ
0
z/ℎ
1
–1
1
–2
y/ℎ
(c)
0.05 0.30 0.55 0.80 1.05
–1
0
1
0 x/ℎ
0
z/ℎ
2
0.05 0.30 0.55 0.80 1.05
1
–1
2
0.05 0.30 0.55 0.80 1.05
u′′(kx ,kz ) /max|u′′(kx ,kz ) |
= 0.6 of the most energetic mode of the eLNS cospectrum for (λx ,
Figure 16. Isosurface
λz ) of (a) (8h, 2h), (b) (8h, 0.3h), (c) (0.3h, 2h) and (d) (0.3h, 0.3h), flooded by the mean velocity ũ/Ub .
Finally, the instantaneous T ′′ at y∗m is utilized to estimate the coherent temperature
structures (Tp′′ ) at different y∗p . The estimated 2-D spectra ΦT+p Tp ( yp ) as in (5.1) are also
computed. First, the estimations using HL,TT from DNS are depicted in figures 19(a) and
19(d). Compared with figure 9(a), relatively more energy resides at λx ∼ h, contributing
to the more isotropic pattern of temperature than the streamwise velocity counterpart.
The estimated Tp′′ at y∗p = 10 is dominated by streamwise streaks and resembles a lot the
velocity counterpart in figure 9(d) except for smaller amplitudes. Similar to figure 9, the
LNS model severely underestimates the amplitude of Tp′′ at nearly all scales, especially
close to the wall where the two plots in figure 19(e) are nearly uniform. In comparison,
973 A36-31
X. Chen, C. Cheng, J. Gan and L. Fu
λz/h
(a)
(b)
(c)
4.0
4.0
4.0
1.0
1.0
1.0
0.3
2
4
8
1
2
4
8
1
(e)
(f)
4.0
4.0
4.0
1.0
1.0
1.0
(d )
λz/h
0.3
0.3
1
0.3
2
4
8
4
8
2
4
8
0.3
0.3
1
2
2
1
λx/h
4
8
1
λx/h
0
0.5
λx/h
1.0
γ 2TT
Figure 17. Contours of 2-D LCS for T ′′ (y∗m = 107) at (a,d) y∗p = 70, (b,e) y∗p = 40 and (c, f ) y∗p = 10. Panels
(a−c) are from the DNS data and (d−f ) are from the eLNS model after the cospectrum decomposition
(eLNS-CD). The dotted lines denote λx = λz .
λz/h
(a)
(b)
(c)
4.0
4.0
4.0
1.0
1.0
1.0
1
2
4
8
λz/h
1
2
4
8
1
(e)
(f)
4.0
4.0
4.0
1.0
1.0
1.0
(d )
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
0.3
0.3
0.3
1
2
4
8
1
2
λx/h
0
1
4
8
2
4
8
0.3
0.3
0.3
2
4
λx/h
2
0
T ′′
1.5
3.0
(Apm,TT , y∗m
8
1
λx/h
|T̂ ′′( yp)|2
|T̂ ′′( ym)|2
0
y∗p
3
6
y∗p
Figure 18. Contours of the amplitude ratio for
= 107) at (a,d)
= 70, (b,e)
= 40 and
(c, f ) y∗p = 10. Panels (a−c) are from the DNS data and (d−f ) are from the eLNS-CD model. The dotted lines
denote λx = λz .
the eLNS-CD results exhibit significant improvements over the LNS and eLNS models,
as expected from figures 17 and 18. The estimated large-scale structures (λx 3h) agree
well with the DNS data, but the smaller-scale motions in figure 19(c) are obviously
2 in figure 17(d) at small λ .
under-predicted due to the underestimated γTT
x
973 A36-32
Estimating coherent velocity and temperature structures
4
(c) (i)
1
λz/ℎ
0
0.3
0
4
x/ℎ
–0.9 0
0.9
Tp′′+
0
1
4 8
0.3
0
4
λx/ℎ
kxkzΦ+TpTp
0 0.01 0.02
4 8
(ii)
4.0
0
–2
–4
1
1.0
2
1.0
4 8
4.0
( f ) (i)
4.0
1
(ii)
–2
–4
4 8
(ii)
2
λz/ℎ
z/ℎ
z/ℎ
λz/ℎ
0
4
2
1.0
0.3
–2
–4
0
(e) (i)
4.0
0
0.3
–2
–4
4 8
(ii)
2
z/ℎ
1
1.0
λz/ℎ
4
4.0
0
z/ℎ
z/ℎ
λz/ℎ
0
(b) (i)
z/ℎ
1.0
0.3
–2
–4
(ii)
2
4.0
0
–2
–4
(d ) (i)
(ii)
2
λz/ℎ
(a) (i)
1.0
0.3
0
4
x/ℎ
–1.1 0
1.1
Tp′′+
1
4 8
λx/ℎ
kxkzΦ+TpTp
0 0.025 0.050
Figure 19. Estimated instantaneous Tp′′+ and the pre-multiplied 2-D spectra at (a,b,c) y∗p = 70 and (d,e, f ) y∗p =
10 using the snapshot at y∗m = 107 and the kernel function from the (a,d) DNS data and models of (b,e) LNS
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
and (c, f ) eLNS-CD.
Therefore, the eLNS model can reflect reasonably the wall-normal coherence and
amplitude variation of the temperature fluctuation, but it can be deteriorated by
the overdamped vortical modes and thus relatively intensified acoustic and entropy
components due to the changed mathematical properties of the linear operator. Through
a procedure of cospectrum decomposition (eLNS-CD), the interference from the acoustic
and entropy components can be removed to obtain SLSE results that basically agree with
DNS data. Meanwhile, an important implication is that the coherence of temperature
is dominated by the vortical components, including advection and other kinematically
dominated motions, strongly analogous to the incompressible case. The compressibility
effects, exerting influence through the acoustic and entropy components, are of secondary
importance.
5.5. Estimation for velocity–temperature coupling
Cheng & Fu (2023a) show that SLSE can be designed to estimate the velocity–temperature
coupling (§ 2.3). Here we provide the results from the linear models to examine
2 and H
2
whether γuT
L,uT can be reasonably predicted. First of all, γuT and Apm,uT =
(|T̂p′′ |2 /|û′′m |2 )1/2 from DNS are displayed in figures 20(a)–20(c). Two types of y∗m and
y∗p combinations are considered, namely y∗m = y∗p and y∗m =
/ y∗p . In the former case, the
measured u′′ is used for predicting the local T ′′ at the same height, while in the latter
case, T ′′ at another height is predicted. In figure 20(a), y∗m = y∗p = 14 are both in the
buffer layer, and u′′ and T ′′ have close morphological connections (see § 5.1). Therefore,
2 is larger than 0.6 in most length scales displayed, and the streamwise-elongated
γuT
973 A36-33
X. Chen, C. Cheng, J. Gan and L. Fu
λz/h
(a)
(b)
(c)
4.0
4.0
4.0
1.0
1.0
1.0
0.3
1
(d )
4
4.0
0.3
8 (e)
4.0
1
4
0.3
8 (f)
4.0
1
λz/h
(g)
4
8
4.0
8
0.9
4
8
4
8
0.3
0.3
1
4
1.0
1.0
0.3
8
0.8
0.9
1.0
4
1
(h)
4
8
(i)
1
4.0
4.0
λz/h
0.8
1.0
1.0
1
0.3
0.3
0.3
1
λz/h
( j)
4
8
(k)
4
8
(l)
4.0
4.0
4.0
1.0
1.0
1.0
0.3
4
λx/h
γ2uT
0
0.5
8
1
γ2uT
1.0
1
0.3
0.3
1
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
1
0.8
0
λx/h
0.5
4
1.0
1
8
|T̂ ′′ (yp)|2
|û ′′ (ym)|2
λx/h
0.1
0.4
0.7
Figure 20. Contours of 2-D LCS and amplitude ratio for the u′′ –T ′′ coupling from the (a−c) DNS data, and
the models of (d−f ) LNS, (g−i) eLNS and (j−l) eLNS-CD. Panels (a,d,g,j) are at y∗m = y∗p = 14, and other
panels are at y∗m = 107, y∗p = 14. Panels (c, f,i,l) are for the amplitude ratio (|T̂ ′′ ( yp )|2 /|û′′ ( ym )|2 )1/2 , and
2 . The dotted lines denote λ = λ . The dash-dot lines in ( f,i) are the contours in the
other panels are for γuT
x
z
saturating region.
streaky motions (larger λx and smaller λz ) are more coherent in terms of the coupling.
Meanwhile, Apm,uT ≈ 0.35 (non-dimensionalized by Ub , Tw ) is nearly uniform at different
′′ in (3.16a,b).
scales (not shown), close to the mean flow coefficient between u′′rms and Trms
∗
∗
∗
Figures 20(b) and 20(c) are for the case ym =
/ yp , where ym = 107 is at the centre of
2 and A
∗
the logarithmic region and yp is in the buffer layer. Here, γuT
pm,uT have similar
′′
′′
′′
′′
distributions to those for u –u and T –T , which is reasonable considering the strong
2 and γ 2 , and A
analogies between γuu
pm,uu and Apm,TT .
TT
2
From figure 20, γuT and Apm,uT from LNS and eLNS considerably deviate from the
2 at different y∗ and y∗ are underestimated in both models. In addition,
DNS data. The γuT
m
p
Apm,uT in the LNS case is high for spanwise-elongated motions (λz > λx ) but low for the
streamwise-elongated (λx > λz ), opposite to the DNS trend. Also, Apm,uT in the eLNS
973 A36-34
Estimating coherent velocity and temperature structures
(a)
(b)
9.0
9.0
0.9
λz/h
3.0
3.0
0.5
0.9
1.0
0.1
0.3
0.3
0.5
1.0
0.1
0.3
1.0
3.0
λx/h
9.0
24.0
0.3
DNS
ODE
1.0
3.0
9.0
24.0
λx/h
2 and (b) γ 2 using the mean flows from DNS and the ODE-based solver,
Figure 21. Contours of (a) γuu
TT
∗
respectively. Here, ym is fixed at 107, and y∗p = 40. The levels displayed are 0.1, 0.3, 0.5, 0.7 and 0.9, as
labelled.
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
case exhibits an overall higher value at nearly all λx and λz . From figure 14, û′′ T̂ ′′† by
2 and A
eLNS is little affected by the acoustic and entropy components, but γuT
pm,uT are
′′
′′†
influenced since they are related to T̂ T̂ . After the cospectrum decomposition, the
eLNS-CD results in figures 20(j)–20(l) are much closer to DNS. The scale dependence of
2 and A
∗
∗
∗ / y∗ cases. Nevertheless,
γuT
pm,uT are correctly captured, for both ym = yp and ym =
p
2
an obvious deviation is that γuT for the spanwise-elongated motions is underestimated in
figure 20(j). As discussed in figure 15(b), wall-normal and spanwise motions dominate at
these scales, affecting the performance of the cospectrum decomposition. In summary,
the eLNS model can be used to estimate the velocity–temperature coupling after the
cospectrum decomposition (eLNS-CD). Analogous to the coherence of temperature,
the coupling of the coherent velocity and temperature fluctuations is also dominated
by advection and other vortical components, while the compressibility effects are of
secondary importance. Thereby, the present results support and provide more proof for
the conclusions of Cheng & Fu (2023a).
6. Parameter study
Observing the consistency between the DNS data and the linear models, a parameter study
is performed to investigate the variations of γ 2 and Apm with different Mab and Reτ . As
the compressible DNS database is very limited, the ODE solver in § 2.2 is utilized to
obtain the mean flows of varying Mab and Reτ . First, we need to confirm that the SLSE
results predicted by the ODE-based mean flow are as reliable as those using the DNS
2 and γ 2 are
mean flow. The benchmark case is calculated, and the comparisons of γuu
TT
∗
∗
shown in figure 21 using the two mean flows. The ym and yp are the same as those in § 5,
with y∗p = 40 in the logarithmic layer. Note that the result comparisons for other y∗p are
similar and thus not shown for conciseness. For an overall comparison, the ranges of λx
and λz are much larger than those in § 5, without the limitation of the DNS computational
domain. Only the eLNS model is selected, and the LNS model is not considered due to
2 and γ 2 using the DNS and ODE mean flows
the relatively large deviation. Basically, γuu
TT
agree with each other within the length scales displayed, demonstrating the reliability of
the ODE solver in predicting SLSE results.
973 A36-35
X. Chen, C. Cheng, J. Gan and L. Fu
Mab
Re∗τ,c
Reτ
Reb
T̃c /Tw
∗
y+
m (ym = 107)
∗
y+
p (yp = 70)
∗
y+
p (yp = 10)
0.3
1.5
3.0
4.0
780
780
780
780
782
1165
2360
3620
15000
19400
29700
37900
1.00
1.37
2.53
3.72
109
156
309
471
71.2
101
198
301
10.1
12.9
21.9
31.4
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Table 1. Parameters for channel flow cases with different Mab but the same Re∗τ,c .
Semi-local units have been extensively proven to be capable of collapsing various
turbulent quantities to their incompressible counterparts, such as the mean velocity profile
(Griffin et al. 2021), Reynolds stress (Huang et al. 1995), the spanwise spacing of
near-wall streaks (Morinishi et al. 2004), kinetic energy budget (Duan et al. 2010) and
streamwise inclination angles (Bai, Cheng & Fu 2023). One interesting point is to examine
whether semi-local units collapse the 2-D LCS and amplitude ratio. This is conducive
to the modelling of inner–outer interaction for compressible flows. In addition to the
benchmark case, three more cases are considered, with varying Mab but the same Re∗τ,c .
Their computational parameters are listed in table 1. The Mab = 0.3 case is close to the
incompressible condition with nearly uniform ρ̄ and T̃, and the fourth case has a high Mab
of 4.
With Mab increased, Reτ and Reb grow rapidly due to the stronger variation of the
thermodynamic properties. These two Reynolds numbers are positively correlated with
the required grid points for DNS, so the computational cost dramatically increases with
the rise of Mab , which is largely responsible for the limited DNS database for compressible
channel flows.
+
For different cases, y∗m and y∗p are kept the same, though the corresponding y+
m and yp
increase by over three times with Mab lifted. Meanwhile, the streamwise and spanwise
wavelengths of fluctuations are all expressed in the outer scale, i.e. λx /h and λz /h.
2 and A
∗
2
The contours of γuu
pm,uu are depicted in figure 22 at yp = 70 and 10. The γuu
∗
at yp = 70 varies slightly with Mab increased from 0.3 to 4.0, which means that the
compressibility effects on the spectral coherence of streamwise velocity in the logarithmic
region can be scaled using semi-local units, supporting Morkovin’s hypothesis. When y∗p
2 among cases are more obvious,
is down to 10 in the buffer layer, the differences of γuu
especially at large λz . Nevertheless, there is still good collapse for streamwise-elongated
streaky motions with λx 5h and λz 0.5h. In figures 22(b) and 22(d), the region
Apm,uu > 1, i.e. |û′′p |2 > |û′′m |2 , is mainly concentrated at λx > λz , little influenced by
2 is low and the
Mab . In contrast, Apm,uu at λx < λz exhibits Mab dependence, where γuu
dominant POD mode for the cospectrum is not an advection mode (see § 5.4). In summary,
2 and A
γuu
pm,uu of the streamwise-elongated (λx > λz ) structures with high coherence
exhibit good Mab independence within and below the logarithmic region when using
semi-local units. These structures have high Apm,uu , so they are pronounced and tend to
be dominant for the estimated u′′ ( y∗p ), beneficial for constructing the compressible IOIM.
In contrast, the spanwise-elongated structures with λx < λz are more influenced by Mab .
2
They tend to be of secondary importance for SLSE because of the relatively low γuu
and Apm,uu .
The parameter study regarding temperature is not conducted for two reasons. First, heat
transfer needs to be included in the incompressible case for reference (Antonia et al. 2009),
but such cases are not currently available in our dataset. Second, the conclusions presented
973 A36-36
Estimating coherent velocity and temperature structures
(a)
(b)
9.0
9.0
Mab = 0.3
Mab = 1.5
Mab = 3.0
Mab = 4.0
λz/h
3.0
1.0
1.0
0.9
1.0
3.0
1.0
0.7
1.2
0.4
0.3
0.1
0.3
λz/h
(c)
0.3
1.0
3.0
9.0
24.0
0.3
0.9
1.0
3.0
9.0
24.0
9.0
24.0
(d)
9.0
9.0
3.0
3.0
1.0
1.0
0.9
0.9
0.5
0.1
0.3
1.5
0.3
1.7
0.3
1.0
3.0
λx/h
9.0
24.0
0.3
1.0
3.0
λx/h
2 and (b,d) A
∗
∗
Figure 22. Contours of (a,c) γuu
pm,uu in the cases of varying Mab but the same Reτ,c . Here, ym is
fixed at 107, and (a,b) y∗p = 70 and (c,d) y∗p = 10. The levels are labelled in the figures. The green dotted lines
denote λx = λz .
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
above concerning the effects of acoustic and entropy modes on temperature are limited to
the case of Mab = 1.5, awaiting future examination at higher Mab using the DNS data.
7. Discussions
7.1. Other linear models with turbulence modelling
As introduced in § 3.2, there are a number of combinations of turbulence modelling
relations to linearize the fluctuation equations, leading to different linear models. Since
(3.6) is used, they can all be considered as eddy viscosity enhanced ones, so for ease of
writing, the eLNS model presented above using (3.10a,b) is termed the benchmark eLNS
model. We have examined other eLNS models, including
(i) using (3.8b) instead of (3.8a) to linearize the turbulent heat flux;
(ii) adding k′′ as an independent variable (so six equations with the basic variable set
q′′ = [ρ ′ , u′′ , v ′′ , w′′ , T ′′ , k′′ ]T ) to linearize the dissipation rate and the k′′ equation;
(iii) linearizing ρ ′ (in Nρ′′ , Nu′′i and NT′′ ) using the SRA in (3.16a,b) (similar to (3.8)).
They are termed the adjusted eLNS models collectively. Same as the benchmark eLNS
model, non-negligible contributions of acoustic and entropy components to Φ are all
present in these adjusted models, because the eigenmodes are damped by µt and the
non-normality of the linear operator (in particular, the vortical mode branch) decreases
as in § 4. Consequently, the SLSE results from these adjusted models are all largely
973 A36-37
X. Chen, C. Cheng, J. Gan and L. Fu
different from DNS regarding the temperature fluctuations, similar to the benchmark
eLNS model (see § 5.3). Thereby, detailed comparisons among these eLNS models are
not provided considering the collective failure, and the main focus has been paid to the
mathematical features of the linear model for possible explanations of the deviation, as
has been elaborated in §§ 4 and 5.4. The procedure of cospectrum decomposition is thus
introduced to obtain basically agreeable results with the DNS data.
7.2. Implications for calculating boundary layer flows
As shown in figure 13, there are a series of acoustic modes in the POD of Φ with different
ky , oscillating in the outer region. Bounded by the two walls at y = 0 and 2h in the channel
flow, ky is discretized so only those with λy = 2h/n are allowed (n is an integer). In
analogy, the acoustic modes exist in boundary layer flows, and their ky are determined
by the height of the computational domain. Different from the channel flow, the location
of the upper boundary is in the free stream and is artificially selected in the boundary
layers, so the results of the acoustic modes and thus Φ can be dependent on the domain
height, which seems not physically appropriate. When the response mode is relatively
supersonic (phase speed ω/kx larger than the free stream acoustic wave, either downstream
or upstream), the acoustic components in the free stream are highly pronounced, so
the responses from input–output or resolvent analyses can be tightly dependent on the
computational domain, as noted by Bae et al. (2020) and Madhusudanan & McKeon
(2022). The present work suggests that such a prominent role of acoustic modes in the
linear models is not in line with DNS, within at least the Ma considered. As one solution
candidate, the cospectrum decomposition procedure can be used to identify and remove the
acoustic components. Detailed comparisons with experimental or DNS data for boundary
layer flows are anticipated to assess further the behaviour of the linear models in different
configurations.
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
7.3. Implications for the turbulence modelling simulations
The discussions in § 4 concerning L̂eLNS and µt in compressible flows may be of greater
significance in some RANS-related applications. The linear operator can be encountered,
e.g. in the sensitivity analysis of RANS modelling to field variables, and in some spatial
numerical schemes in the computational fluid dynamics where linearization of the inviscid
and viscous fluxes towards basic variables is required (see e.g. MacCormack 1985). Also,
the linear operator can, at least partially, govern the transient dynamics when the turbulent
flow is perturbed around its statistically equilibrium state, simulated using, e.g. unsteady
RANS simulations, detached eddy simulations or wall-modelled large eddy simulations.
The diminishing non-normality of the linear operator due to µt can overdamp the vortical
modes, hence leading to possibly overemphasized roles of the acoustic and entropy modes.
Nevertheless, this effect can be difficult to isolate due to the combined complexities of
flow physics (shocks, separation, etc.), more sophisticated turbulence models, numerical
schemes, grid resolutions and so on. More future explorations are anticipated in this
direction.
8. Summary
In this work, the linear models, based on stochastically forced linearized NS equations,
are deployed for SLSE of the velocity and temperature fluctuations in compressible
turbulent channel flows. The benchmark case has Mab = 1.5 and Reb = 20 020.
The spectral coherence and amplitude ratio (γ 2 and Apm ) of the fluctuation between
973 A36-38
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
Estimating coherent velocity and temperature structures
different wall-normal heights are obtained using the linear models with only mean flow
input. Afterwards, the coherent portion of the instantaneous field at other heights can be
estimated through HL based on one snapshot at the measurement height. Three aspects are
mainly focused on, namely the derivation and mathematical features of the linear models,
quantitative comparison of the SLSE quantities with the DNS data, and the structural
coherence of velocity and temperature fluctuations inferred by the SLSE results.
First, the derivation of the linear model is discussed carefully. The full expressions of
the fluctuation equations are provided, and the assumptions for linearization are detailed.
Two types of models are considered. The first is the LNS model with all the nonlinear
terms collected into the forcing term, while the second is the eddy viscosity enhanced
one (eLNS model), using turbulence modelling relations for possible linearization of the
nonlinear terms. The mathematical properties of the linear operator are scrutinized by
studying their eigenspectra and pseudospectra. There are four branches of eigenmodes
for L̂LNS and L̂eLNS , and the vortical branch has strong non-normality to support the
transient behaviour of the fluctuations. For the LNS model, the acoustic modes have
little non-normality and increasingly small contributions to the energy growth as length
scales (both λx and λz ) increase. In contrast, the non-normality of the vortical modes
diminishes in the eLNS model, closely related to the damping effects of µt (and κt ) on the
eigenmodes. Consequently, the acoustic and entropy modes can account for 20 % to 55 %
of the response energy growth, which largely affects the SLSE results for temperature.
The SLSE results (γ 2 , Apm and HL ) for streamwise velocity are quite similar to the
incompressible case. The eLNS model outperforms the LNS model, giving basically
agreeable results with the DNS data, though the structures of small λz are still somewhat
underestimated. However, both models have considerable deviations from DNS regarding
temperature and velocity–temperature coupling. A decomposition of the eLNS cospectrum
(Φ nae and Φ ae ) shows that the acoustic and entropy components hardly affect the
streamwise-velocity cospectrum, but have non-negligible contributions (especially the
acoustic ones) to those of density and temperature, which is not supported by the DNS
data. Thereby, the prominent role of acoustic and entropy components for the response
growth is an intrinsic defect of the eLNS model introduced by the turbulence modelling
related to µt . A procedure of cospectrum decomposition is thus designed to remove
the acoustic and entropy components (eLNS-CD). The resulting SLSE quantities for
temperature and velocity–temperature coupling show a noticeable improvement over
the eLNS model, and are basically agreeable with DNS, except in the region of
small λx . Thereby, the coherent velocity and temperature fluctuations are dominated by
advection and other vortical motions, while the compressibility effects are of secondary
importance.
Finally, a parameter study is conducted for four cases with varying Mab (0.3–4) but
2 and A
the same Re∗τ,c . Semi-local units well collapse γuu
pm,uu to the incompressible case
for the streamwise-elongated structures of high coherence, which benefits the modelling
of inner–outer interaction for compressible flows. In contrast, the spanwise-elongated
structures (λx < λz ) are more influenced by Mab . Moreover, the behaviours of other linear
models with different turbulence modelling relations are discussed. Some implications
are presented for calculating the boundary layer flows and dealing with RANS-related
computations.
Future works will be focused on assessing the behaviours of the linear models for
compressible cases at higher Mab and Re and incompressible cases with heat transfer,
using the DNS or experimental data. Improvements on the linear models are anticipated
973 A36-39
X. Chen, C. Cheng, J. Gan and L. Fu
through, e.g. modified linearization of turbulent eddy terms and more realistic modelling
of the forcing.
Funding. This research was supported by the Center for Ocean Research in Hong Kong and Macau, a joint
research center between QNLM and HKUST. L.F. acknowledges the fund from the Research Grants Council
(RGC) of the Government of Hong Kong Special Administrative Region (HKSAR) with RGC/ECS Project
(no. 26200222) and the fund from the Project of Hetao Shenzhen–Hong Kong Science and Technology
Innovation Cooperation Zone (no. HZQB-KCZYB-2020083).
Declaration of interests. The authors report no conflict of interest.
Data availability statement. The data that support the findings of this study are available from the
corresponding author upon reasonable request.
Author ORCIDs.
Xianliang Chen https://orcid.org/0000-0002-7540-3395;
Cheng Cheng https://orcid.org/0000-0002-7961-793X;
Jianping Gan https://orcid.org/0000-0001-9827-7929;
Lin Fu https://orcid.org/0000-0001-8979-8415.
Appendix A. Fluctuation equation for turbulent kinetic energy
Based on (3.3), the equation for k′′ is derived as
′′
∂k
∂k′′
∂k
∂k
ρ̄
+ ρ ′ + (ρ ′ ũj + ρ̄u′′j )
+ ũj
∂t
∂xj
∂t
∂xj
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
′′ u′′ )
= −ρ̄(u′′i u′′j − u
i j
′′
∂ ũi
′′ u′′ ∂ui
− ρ̄ u
i j
∂xj
∂xj
′ ′′
′ u′′
∂p
∂p
u
∂
j
j
′′ k′′ − u′′ u
′′ ′′
[ρ̄(u′′j k′′ − u
−
−
i i uj )] −
j
∂xj
∂xj
∂xj
′ u′′
′′
′ u′′
′′
′′
′′
∂u
∂u
∂τ
∂τ
∂u
∂u
ij
i
j
ij
i
j
+ p′
− p′
−
+
− τij′ i − τij′ i
∂xj
∂xj
∂xj
∂xj
∂xj
∂xj
′′ ′′
′
∂ρ ′ k′′ ∂ρ (ũj + uj )k
′ ′′ ∂k
′ ′′ ∂ ũi
′′ ∂ ũi
.
+ ρ uj
+ ρ ui
+
+ (ũj + uj )
−
∂t
∂xj
∂xj
∂t
∂xj
(A1)
As in (3.4c), there are fluctuations of turbulent heat flux, pressure dilatation and dissipation
rate, so these terms are responsible for the energy transfer between cv T ′′ and k′′ . The first
two terms on the right-hand side are the fluctuating turbulent production term, which can
be linearized using (3.6). The resulting term takes the same form as the linearized viscous
dissipation term (the last term on the left-hand side in (3.4c)) except for replacing µ̃ with
µt . Equation (A1) can be included in (3.4) to solve the problem in § 3.3 if k′′ is treated as
the sixth independent variable, as discussed in § 7.1.
Appendix B. Eigenmodes in inviscid uniform mean flow
An inviscid uniform mean flow (using the channel-centre value, µ̃ = µt = 0) is
considered. Since there are no wall-normal gradients of the mean flow, the fluctuation
is assumed to be periodic in the wall-normal direction with a wavenumber ky (see (2.3)).
973 A36-40
Estimating coherent velocity and temperature structures
The linear operator L̂c then takes the form of
⎡
ikx ũc
ikx ρ̄
iky ρ̄
⎢
T̃
⎢ ikx R c
ikx ũc
0
⎢
ρ̄c
⎢
⎢
T̃c
L̂c = ⎢
0
ikx ũc
⎢ iky R ρ̄
c
⎢
⎢
⎢ ik R T̃c
0
0
⎣ z ρ̄
c
0
ikx (γ0 − 1)T̃c iky (γ0 − 1)T̃c
ikz ρ̄
0
0
ikx ũc
ikz (γ0 − 1)T̃c
0
⎤
⎥
ikx R ⎥
⎥
⎥
⎥
iky R ⎥
⎥.
⎥
⎥
ikz R ⎥
⎦
(B1)
ikx ũc
After applying the energy norm, the five eigenvalues of L̂c are obtained, as shown in
(4.1). The orthogonal eigenfunctions (along with the linearized pressure and entropy
fluctuations) q̌ = [ρ̌ ′ , ǔ′′ , v̌ ′′ , w̌′′ , Ť ′′ , p̌′ , š′′ ]T are specified below.
Fast and slow acoustic modes:
T
ky k z
1
kx
q̌ = √
±ρ̄, a, a, a, ±(γ0 − 1)T̃, ±γ0 p̄, 0 ,
(B2a)
2γ0 p̄
ǩ
ǩ
ǩ
Entropy mode:
√
γ0 − 1
q̌ = √
[ρ̄, 0, 0, 0, −T̃, 0, −γ0 s̄]T ,
γ0 p̄
(B2b)
Vortical mode:
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
q̌ = √
1
[0, tix a, tiy a, tiz a, 0, 0, 0]T ,
γ0 p̄
i = 1, 2.
(B2c)
Here, ǩ2 equals kx2 + ky2 + kz2 (notation ǩ distinguished from the turbulent kinetic energy
k), and [tix , tiy , tiz ]T (i = 1, 2) are two unit vectors perpendicular to [kx , ky , kz ]T . The mean
entropy is s̄ = p̄/ρ̄ γ0 . Different modes in (B2) are perpendicular to each other in terms of
(3.13) (George & Sujith 2011), so L̂c is normal. To classify different modes in the channel
flows (see §§ 4.2 and 5.4), the eigenfunction is normalized first, to be applicable to flows of
different Mab , as q̌nrm = [ρ̌ ′ /ρ̄, ǔ′′ /a, v̌ ′′ /a, w̌′′ /a, Ť ′′ /T̃, p̌′ /p̄]T . Afterwards, (B2) is used
to judge whether the pressure, kinematic or thermodynamic component has the largest
amplitude (see figure 4).
Appendix C. Growth-rate decomposition for eigenmodes
The growth-rate decomposition can quantify the contributions of each term in the
governing equation to the eigenmode growth rates (see e.g. Chen, Wang & Fu (2022a)
and Chen et al. (2022b) for more detail). Hence the growth-rate difference between
the LNS and eLNS eigenmodes can be explained. The decomposition is realized after
left-multiplying q̂′′H M to (3.12) and adding the complex conjugate. Note that fˆ is not
considered for the eigenmodes. The classified terms are defined as follows. The production
is sum of the terms related to v̌ ′′ and the mean flow gradients (∂ ũ/∂y, ∂ ρ̄/∂y and ∂ T̃/∂y).
The pressure dilatation is p̌∇ · ǔ and the pressure work is ǔ · ∇ p̌. The viscous and
diffusive terms are those related to µ̃ and κ̃, and the eddy viscous and diffusive terms
are those related to µt and κt . The decomposition results for the least stable three types
of eigenmodes are listed in table 2 (see their shapes in figure 4). The difference in ωi
973 A36-41
X. Chen, C. Cheng, J. Gan and L. Fu
Modes
Prod.
Acoustic, LNS
Acoustic, eLNS
0.0024
0.0024
Vortical, LNS
Vortical, eLNS
0.0000
−0.0001
Entropy, LNS
Entropy, eLNS
0.0000
0.0000
PreD.
PreW.
VsDf.
EVsDf.
sum (ωi )
−0.0003
−0.0072
0.0003
0.0072
−0.0085
−0.0083
0.0000
−0.0324
−0.0061
−0.0383
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
−0.0039
−0.0011
−0.0048
−0.0016
0.0000
−0.0581
0.0000
−0.0734
−0.0039
−0.0593
−0.0048
−0.0750
Table 2. Growth-rate decomposition for the least stable eigenmodes of the LNS and eLNS linear operators
(shapes in figure 4, λx = 8h, λz = 2h). The results are non-dimensionalized by Ub /h. Term abbreviations:
Prod. for production; PreD. for pressure dilatation; PreW. for pressure work; VsDf. for viscosity and diffusivity;
EVsDf. for eddy viscosity and diffusivity.
between the LNS and eLNS eigenmodes is mainly caused by the eddy terms, the same for
the acoustic, vortical and entropy branches. Thereby, the conclusion in § 4.1 is supported
that the eddy terms related to µt and κt severely stabilize the eigenmodes.
https://doi.org/10.1017/jfm.2023.768 Published online by Cambridge University Press
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