# Local Properties of the Turbulent/Non-turbulent Interface

Sarp Er, Jean-Philippe Laval, John Christos Vassilicos

► To cite this version:

Sarp Er, Jean-Philippe Laval, John Christos Vassilicos. Local Properties of the Turbulent/Non-turbulent Interface. IUTAM Symposium on Turbulent/Non-Turbulent Interface in Turbulent Shear Flows (IUTAM-TNTI-2024), Oct 2024, Pékin, China. pp.48 - 65, 10.1007/978-3-031-78151-3_4. hal-05377380

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Submitted on 21 Nov 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Local Properties of the Turbulent/Non-turbulent Interface

Sarp Er, Jean-Philippe Laval, and Christos Vassilicos(✉)

Univ. Lille, CNRS, ONERA, Arts et Métiers Institute of Technology, Centrale Lille, UMR 9014 - LMFL - Laboratoire de Mécanique des Fluides de Lille - Kampé de Fériet, 59000 Lille, France

john-christos.vassilicos@cnrs.fr

https://lmfl.cnrs.fr/perso/christos-vassilicos-home/

**Abstract.** The effect of spatial resolution on local propagation velocity of the Turbulent/Non-Turbulent Interface (TNTI) is documented. A TNTI-local coordinate system and a Reynolds decomposition based on a TNTI-conditional averaging operation are introduced and used to assess the local velocity field and the mass and momentum balances at the TNTI layer. A clear jump of the TNTI-averaged Reynolds shear stress exists through the TNTI layer and the TNTI-averaged fluid velocity normal to the TNTI is determined by the correlation between the TNTI normal vector and the fluid velocity fluctuations around the TNTI-average fluid velocity. The TNTI-local fluid velocity field is compressive normal to the TNTI and stretching in the direction tangent to the TNTI, a structure that has important consequences.

**Keywords:** TNTI layer · interface · propagation velocity · fractal dimension · turbulent jets · entrainment/detrainment

## 1 Introduction

Recently, Er et al. [1] derived a relation between a globally defined average propagation velocity $v_n$ of the TNTI and a TNTI fractal dimension. Their global definition views $v_n$ as the ratio between the growth rate of the turbulent region's average volume to the average surface area of the TNTI and they estimated this surface area by assuming that it is fractal or fractal-like. They applied their approach to a temporally developing self-similar jet even though they noted that such a jet is fundamentally different from a spatially developing self-similar jet because it conserves volume rather than momentum flux and has zero rather than non-zero cross-stream mean flow velocity. The fractal dimensions that they extracted proved to be very sensitive functions of entrophy threshold in the range of entrophy thresholds within the TNTI layer, i.e. those entrophy thresholds over which the turbulence volume does not vary much. This observation echoed previous observations ([2,3]) concerning strong threshold dependence of the fractal dimension of scalar iso-surfaces.
The relation between the global $v_n$ and the fractal dimension $D_f$ turned out to capture the entrophy threshold dependence of $v_n$ via the entrophy threshold dependence of $D_f$ in the Direct Numerical Simulations (DNS) of temporally developing planar jets of [1], but only for relatively low entrophy thresholds for which $D_f$ takes well-defined values up to about $D_f \approx 2.2$. ($D_f$ is an increasing function of entrophy threshold within the TNTI layer as iso-enrophy surfaces (IES) are progressively smoother towards the outer non-turbulent edge of the layer.) At higher entrophy thresholds corresponding to the fully turbulent side of the TNTI layer the fractal dimension turned out not to be well-defined in the DNS of [1] in the sense that the box-counting algorithm does not return a power law dependence between number of boxes covering the IES and size of these boxes (as previously found by Miller & Dimotakis [4], for example, for scalar interfaces in turbulent jets). However, a power-law fit through this non-power law dependence returns a fractal dimension fit with $D_f \approx 2.36$ for the iso-enrophy surfaces at the turbulence edge of the TNTI layer. This is close to the fractal dimension 7/3 reported in many studies over the past 40 years (e.g. [5]).

A local rather than global definition of the TNTI propagation velocity $v_n^l$ has been considered in [6, 7] who showed that it leads to

$$v_n^l = -\frac{2\omega_i\omega_j s_{ij}}{|\nabla\omega^2|} + \frac{2\nu\nabla\omega_i \cdot \nabla\omega_j}{|\nabla\omega^2|} - \frac{\nu\nabla^2\omega^2}{|\nabla\omega^2|} \quad (1)$$

where $\omega_i$ is the local vorticity's component $i$ ($i = 1, 2, 3$), $s_{ij}$ is the local strain rate tensor, $\omega^2/2$ is the local entrophy and $\nu$ is the fluid's kinematic viscosity. When $v_n^l$ is negative/positive, the IES for which it is defined moves towards the non-turbulent/turbulent side of the TNTI layer signifying local entrainment/detrainment respectively.

In this paper we document the effect of spatial resolution on the local propagation velocity $v_n^l$ and then proceed with a study of the local flow field at the TNTI.

## 2 Effect of Resolution on the Local Propagation Velocity

Obtaining the local propagation velocity $v_n^l$ from DNS places strict resolution constraints. This is because first and second derivatives of entrophy/vorticity need to be computed, and the terms in Eq. 1 are normalized by the magnitude of the gradient of entrophy, which can become quite small as one moves towards the non-turbulent side of the TNTI through the viscous sublayer.

The present work is based on the direct numerical simulations PJ1, PJ2, PJ3, PJ4 and PJ5 of [1] (a detailed description of these DNS can be found in their paper). The flow fields from these simulations are interpolated in Fourier space into finer grids which consist of $N_x \times N_y \times N_z = (1536 \times 2304 \times 1536)$ elements in each direction. High resolution simulations are run for $t = 2T_{ref} \equiv 2H_J/U_J$, where $H_J$ is the initial jet width and $U_J$ is the initial streamwise velocity at the centre-plane ($x-z$ plane at $y=0$, $y$ being the coordinate along
the direction of mean flow variation of the planar jet) with global Reynolds number $Re_J \equiv U_J H_J / \nu = 3200$ and time-local Taylor length-based Reynolds number $Re_\lambda \approx 45–65$ during the time evolution of the fully turbulent jet. The PJn-HR (High Resolution) data set is produced from PJn ($n = 1, 2, 3, 4, 5$). For example, to obtain the PJ1-HR data, the velocity field of PJ1 at $t/T_{ref} = 48$ is interpolated to a fine grid and then run with exact same parameters for a time $2T_{ref}$ which becomes the highly resolved case of PJ1 at instant of $t/T_{ref} = 50$. The spatial resolution $dx = dy = dz$ increases from $dy/\eta = 0.7$ for PJn to $dy/\eta = 0.46$ for PJn-HR at instant $t/T_{ref} = 50$ ($\eta$ is the Kolmogorov scale on the centre-plane).

The PDF of the local propagation velocity can be seen in Fig. 1 for two iso- enstrophy surfaces within the TNTI layer, $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ and $\omega_{th}^2/\omega_{ref}^2 = 10^{-6}$, where $\omega_{th}^2$ is the threshold value of $\omega^2$ defining the IES and $\omega_{ref}^2$ (function of time) is $\omega^2$ integrated over the centre-plane of the jet. It can be observed from Fig. 1(a) (iso-surface $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$) that the tails of the PDF vary with spatial resolution. For the positive values of $v_n^l/u_\eta$ (detrainment) the tail of the PDF varies significantly between resolutions $dy/\eta = 1.4$ and $dy/\eta = 0.7$ but this part of the PDF converges for resolution $dy/\eta \le 0.7$. However, for the negative values of $v_n^l/u_\eta$ (entrainment), the tail of the PDF changes even from the medium resolution $dy/\eta = 0.7$ to the high-resolution $dy/\eta = 0.46$. The average value of the local propagation velocity $v_n^l$ for $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ is $\langle v_n^l \rangle_I/u_\eta = -0.268$, $\langle v_n^l \rangle_I/u_\eta = -0.544$ and $\langle v_n^l \rangle_I/u_\eta = -0.517$ for the cases with grid resolutions $dy/\eta = 1.4$, $dy/\eta = 0.7$ and $dy/\eta = 0.46$ respectively and the corresponding standard deviations are $\sigma_{v_n^l} = 1.267u_\eta$, $\sigma_{v_n^l} = 0.697u_\eta$ and $\sigma_{v_n^l} = 0.631u_\eta$ ($u_\eta$ is the Kolmogorov velocity averaged over the centre-plane).

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** PDF of $v_n^l$ at time $t/T_{ref} = 50$ for simulations with varying resolutions of $dy/\eta = 1.4$, $dy/\eta = 0.7$ (PJ4), and $dy/\eta = 0.46$ (PJ4-HR). Iso-enstrophy surface (a) $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$, (b) $\omega_{th}^2/\omega_{ref}^2 = 10^{-6}$.</figcaption></figure>
The effect of resolution is more dramatic for the IES $\omega_{th}^2/\omega_{ref}^2 = 10^{-6}$ as shown in Fig. 1(b). Now, the mean value of $v_n^l$ over the iso-surface is $\langle v_n^l \rangle_I/u_\eta = 4.484$, $\langle v_n^l \rangle_I/u_\eta = -1.014$ and $\langle v_n^l \rangle_I/u_\eta = -0.906$ and the standard deviations are $\sigma_{v_n^l} = 8.466u_\eta$, $\sigma_{v_n^l} = 1.230u_\eta$ and $\sigma_{v_n^l} = 0.665u_\eta$ for the cases with grid resolutions $dy/\eta = 1.4$, $dy/\eta = 0.7$ and $dy/\eta = 0.46$ respectively.

In conclusion, the mean and standard deviations of $v_n^l$ over the iso-surface $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ may be starting to be captured with grid resolutions higher than $\eta$. However, this may not be quite the case for $\omega_{th}^2/\omega_{ref}^2 = 10^{-6}$ at the outer edge of the TNTI layer. The PDFs of $v_n^l$ do not seem to be fully captured in either case. [1] found that iso-surfaces $\omega_{th}^2/\omega_{ref}^2 = 10^{-6}$ have a well-defined fractal dimension $D_f \approx 2.1$ thereby suggesting that this entrophy threshold, even if in the outer non-turbulent side of the TNTI layer, is a relevant part of the TNTI layer's structure. If resolution problems make it difficult to obtain fully reliable information on local entrainment/detrainment in terms of the local propagation velocity $v_n^l$, it is nevertheless possible to extract information concerning the local velocity field at the TNTI. Following [8], and various authors since then, who obtained statistics of flow field variables conditional on interface location and, up to some extent, local orientation, we now present an analysis of the local flow field at the TNTI using a local reference frame at the TNTI which takes full account of its orientation.

We begin with the definition of the local coordinate system and its positioning. Then the definition of the TNTI-averaging operation is given. Two distinct but complementary averaging operations are explained, namely the $x-z$ averaging and the TNTI-averaging. We then proceed with the presentation of profiles of flow quantities that are averaged over the TNTI. A discussion on TNTI normal profiles follows which includes the investigation of the contributions of the mean flow $\mathbf{U}$ and fluctuating velocity $\mathbf{u}'$ on the TNTI-averaged velocity components.

## 3 Definition of the Local Coordinate System at the TNTI

Similarly to many previous studies, we introduce a local coordinate system to analyze the flow field near the TNTI and perform ensemble average operation conditioned on the interface location. The detection of the iso-entrophy surfaces constituting the TNTI has been explained in [1]. The local coordinate system is defined for each point $\mathbf{X}_0$ on the IES detected by a specific $\omega_{th}^2$, taking into account the position $\mathbf{X}_0$ and the orientation of the IES at that location (the IES is computed with subgrid accuracy by linear interpolation, see Sect. 2.6 in [9]). The entrophy normal vector at each point is calculated as follows:

$$ \mathbf{n} = \nabla\omega^2/|\nabla\omega^2|. \qquad (2) $$

A local coordinate system is placed at each point on the IES, where the unit vectors of the local coordinate system are denoted by $\mathbf{e}_x$, $\mathbf{e}_y$ and $\mathbf{e}_z$. This local coordinate system is defined by a procedure starting with the positioning
of the origin of the local coordinate system on a given point $X_0$ on the IES. The condition for the local y-axis is chosen to be $\mathbf{e}_y = -\mathbf{n}$ so that the positive y-direction in the local coordinate system points towards the non-turbulent region. The unit vector $\mathbf{e}_y$ is referred to as the face normal vector since it indicates the local facing direction of the IES. The normal vector $\mathbf{n}$ is referred to as the enstrophy normal vector.

Apart from the local face normal direction, there is no evident constraint for the remaining two unit vectors which are tangential to the IES. Considering that the temporal planar jet flow has a non-zero mean velocity only in the global x-direction, we have chosen to maximize the projection of the $\mathbf{e}_x$ on the global x-axis. With the determination of the unit vectors $\mathbf{e}_y$ and $\mathbf{e}_x$, the local z-coordinate is defined by the cross-product of these two vectors. So the conditions defining the unit vectors of the local coordinate system ($X_0, \mathbf{e}_x, \mathbf{e}_y, \mathbf{e}_z$) at each point on the IES are:

$$ \begin{aligned} \mathbf{e}_x \cdot \mathbf{k} &= 0, \\ \mathbf{e}_x \cdot \mathbf{n} &= 0, \\ |\mathbf{e}_x| &= 1, \end{aligned} \tag{3} $$

$$ \mathbf{e}_y = -\mathbf{n}, \tag{4} $$

$$ \mathbf{e}_z = \mathbf{e}_x \times \mathbf{e}_y. \tag{5} $$

The vectors i, j, k represent the unit vectors of the global coordinate system and correspond to the streamwise, cross-stream, and spanwise directions.

## 4 TNTI-Averaging Operation

The positioning of the local coordinate system is a precursor step for the generation of a local grid centred on the detected IES. Once the local coordinate system is positioned and the local 3D grid is placed, the global flow field variables are interpolated at the local grid points by using the trilinear interpolation method. Given the very high spatial resolution of the simulations, a trilinear interpolation is found to be sufficient for the post-processing of the interpolated fields including the computation of the second-order derivatives. The interpolated variables include, u, v vectors and scalars such as pressure p and $\omega^2$. After the interpolation of the variables, a transformation is applied to the vector quantities by the application of a rotation operation R in order to decompose the vector fields with components in the local coordinate system ($\mathbf{e}_x, \mathbf{e}_y, \mathbf{e}_z$). The rotation operation R which relates the expression of an arbitrary vector a in the global coordinate system with its expression in the local coordinate system $a^l$ can be written as

$$ \begin{bmatrix} a^l_1 \\ a^l_2 \\ a^l_3 \end{bmatrix} = \begin{bmatrix} \mathbf{e}_x \cdot i \mathbf{e}_x \cdot j \mathbf{e}_x \cdot k \\ \mathbf{e}_y \cdot i \mathbf{e}_y \cdot j \mathbf{e}_y \cdot k \\ \mathbf{e}_z \cdot i \mathbf{e}_z \cdot j \mathbf{e}_z \cdot k \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \tag{6} $$

To ensure that the TNTI does not fold on itself and that there are no engulfed turbulent or non-turbulent regions near the interface, an additional condition
called the *no multi-cross* condition is applied. This condition checks whether the TNTI normal axis, i.e., the local y-axis, crosses the TNTI multiple times within a certain distance on each side of the IES. We apply this condition because we are interested in capturing the variation of variables across the turbulent to non-turbulent sides of the interface. Normal axis crossing the interface multiple times will make the TNTI-average profiles hard to interpret and possibly affect the parts associated with the non-turbulent part of the TNTI-averaged data and is therefore omitted also in previous studies ([10, 11]). Different methods have been used in the literature to overcome this issue, either by completely neglecting the points on the IES where the local normal axis has multiple crossings of the interface ([12]), or by defining a margin of length in order to exclude certain parts of the local profile from the conditional averaging operation ([13, 14]) or by using an envelope surface rather than the full interface with all its contortions ([15]).

In the present work, we follow the methodology applied in [12]. On the non-turbulent side of the interface, a check is made to see if there is a point where $\omega^2(y_I) > c_{mc}\omega_{th}^2$ along the interface normal axis from the IES location (which is at $y_I/\eta = 0$) towards the non-turbulent region until the maximum extent of the interpolated region. On the turbulent side of the interface, a check is made to see if there is a point having $\omega^2(y_I) < (1/c_{mc})\omega_{th}^2$ from the IES location towards the direction of the turbulent core, again until the end of the local interpolated region. In the present study, a local interpolated region with size $-50 < y_I/dy < 50$ in the interface normal direction is used, meaning that the *no multi-crossing* check is carried out in this range (which corresponds to $-23 < y_I/\eta < 23$ where $dy$ is the uniform grid size in all the directions and $\eta$ is the Kolmogorov length scale computed at the centre-plane of the jet). Here, the free parameter $c_{mc}$ is $c_{mc} = 5$ taking into account that the different thresholds analyzed are chosen with decade intervals in $\omega_{ref}^2$ values. (Thus, choosing $c_{mc} = 10$ corresponds to the next threshold considered). This parameter serves as a modifier for the sensitivity of the *no multi-cross* condition, which is useful when the local grid used for the conditional analysis is of the same order or finer than the grid used for the TNTI detection. We checked that a reasonable change in the value of $c_{mc} = 5$ does not significantly change the results presented below.

Before moving to the local analysis of the interface and the application of the TNTI-averaging operation, it is beneficial to clarify the two distinct averaging operations used in this study. Firstly, we have the averaging over the homogeneous directions where $\langle \cdot \rangle$ denotes the averaged quantity in the two homogeneous directions of the global coordinate system i.e. $x$ and $z$:

$$ \mathbf{a} = \mathbf{A} + \mathbf{a}'. \qquad (7) $$

A capital letter is used for the average in the homogeneous directions, i.e. $\mathbf{A} = \langle \mathbf{a} \rangle$.

For the second averaging operation, the local fields are averaged along the interface while keeping the local coordinates $x_I$, $y_I$, and $z_I$ constant. This operation is denoted by $\langle \cdot \rangle_I$. When applied to a vector quantity, it leads to a result in
the form

$$ \mathbf{a} = \langle \mathbf{a} \rangle_I + \mathbf{a}'' \quad (8) $$

We stress that these are two different averaging operations and that the TNTI-average values of the fluctuating component $\mathbf{a}'$ in Eq. 7 are not identically zero i.e. $\langle \mathbf{a}' \rangle_I \neq 0$. On the other hand, due to the definition of the TNTI averaging process, $\langle \mathbf{a}'' \rangle_I \equiv 0$.

For the temporally developing jet, the mean velocity profile is solely a function of y-direction at any given time, i.e. $U(y)$ (see [1]). Using Eq. 6, $U$ can be expressed in terms of the TNTI local coordinates as $U(Y_0 + \mathbf{R}^{-1}(\mathbf{X}_0) \cdot \mathbf{x}_I)$. Both averaging operations, defined by Eqs. 7 and 8, are averages over spatial surfaces (the $x-z$ surface or an IES surface in the TNTI layer). We apply these operations at an instant in time and omit the time variable from the notation. Applying the decomposition Eq. 7 to the velocity vector at a given time and any location,

$$ \mathbf{u}(x_I, y_I, z_I, \mathbf{X}_0) = U(Y_0 + \mathbf{R}^{-1}(\mathbf{X}_0) \cdot \mathbf{x}_I) + u'(x_I, y_I, z_I, \mathbf{X}_0). \quad (9) $$

Averaging the velocity vector conditionally on the TNTI location leads to

$$ \mathbf{u}(x_I, y_I, z_I, \mathbf{X}_0) = \langle \mathbf{u}(x_I, y_I, z_I, \mathbf{X}_0) \rangle_I + u''(x_I, y_I, z_I, \mathbf{X}_0). \quad (10) $$

The velocity vector components in the local coordinate system are

$$ \begin{aligned} u_l &= \mathbf{u} \cdot \mathbf{e}_x \\ v_l &= \mathbf{u} \cdot \mathbf{e}_y \\ w_l &= \mathbf{u} \cdot \mathbf{e}_z. \end{aligned} \quad (11) $$

Here, $v_l$ is the TNTI normal component, and $u_l$ and $w_l$ are components of the velocity vector, tangential to the interface. Applying now the TNTI-average to these local vector components yields

$$ \begin{aligned} u_l &= \langle \mathbf{U} \cdot \mathbf{e}_x \rangle_I + \langle \mathbf{u}' \cdot \mathbf{e}_x \rangle_I, \\ v_l &= \langle \mathbf{U} \cdot \mathbf{e}_y \rangle_I + \langle \mathbf{u}' \cdot \mathbf{e}_y \rangle_I, \\ w_l &= \langle \mathbf{U} \cdot \mathbf{e}_z \rangle_I + \langle \mathbf{u}' \cdot \mathbf{e}_z \rangle_I. \end{aligned} \quad (12) $$

The profiles of TNTI-averaged velocities and the mean local 3D fields are mostly obtained by this operation in the present paper. As the TNTI averaging operation is applied for fixed $x_I$, $y_I$, and $z_I$, it averages the $x-z$ averaged $U$ over various global y-coordinate values.

Equation 10 and Eqs. 11 imply

$$ \begin{aligned} u_l &= \langle \mathbf{u} \rangle_I \cdot \mathbf{e}_x(\mathbf{X}_0) + \mathbf{u}'' \cdot \mathbf{e}_x(\mathbf{X}_0) \\ v_l &= \langle \mathbf{u} \rangle_I \cdot \mathbf{e}_y(\mathbf{X}_0) + \mathbf{u}'' \cdot \mathbf{e}_y(\mathbf{X}_0) \\ w_l &= \langle \mathbf{u} \rangle_I \cdot \mathbf{e}_z(\mathbf{X}_0) + \mathbf{u}'' \cdot \mathbf{e}_z(\mathbf{X}_0) \end{aligned} \quad (13) $$

and applying the TNTI averaging operation we obtain

$$ \begin{aligned} u_l &= \langle \mathbf{u} \rangle_I \cdot \langle \mathbf{e}_x \rangle_I + \langle \mathbf{u}'' \cdot \langle \mathbf{e}_x \rangle_I \rangle_I + \langle \mathbf{u}'' \cdot \langle \mathbf{e}_x'' \rangle_I \rangle_I, \\ v_l &= \langle \mathbf{u} \rangle_I \cdot \langle \mathbf{e}_y \rangle_I + \langle \mathbf{u}'' \cdot \langle \mathbf{e}_y \rangle_I \rangle_I + \langle \mathbf{u}'' \cdot \langle \mathbf{e}_y'' \rangle_I \rangle_I, \\ w_l &= \langle \mathbf{u} \rangle_I \cdot \langle \mathbf{e}_z \rangle_I + \langle \mathbf{u}'' \cdot \langle \mathbf{e}_z \rangle_I \rangle_I + \langle \mathbf{u}'' \cdot \langle \mathbf{e}_z'' \rangle_I \rangle_I, \end{aligned} \quad (14) $$
where $e(X_0) = e + e''(X_0)$ with $\langle e'' \rangle_I = 0$. The second term on the right-hand side of these equations is zero due to the definition of TNTI-averaging in Eq. 8. Hence,

$$ 
\begin{aligned}
u_l &= \langle \mathbf{u} \rangle_I \cdot \langle \mathbf{e}_x \rangle_I + \langle \mathbf{u}'' \cdot \mathbf{e}_x'' \rangle_I, \\
v_l &= \langle \mathbf{u} \rangle_I \cdot \langle \mathbf{e}_y \rangle_I + \langle \mathbf{u}'' \cdot \mathbf{e}_y'' \rangle_I, \\
w_l &= \langle \mathbf{u} \rangle_I \cdot \langle \mathbf{e}_z \rangle_I + \langle \mathbf{u}'' \cdot \mathbf{e}_z'' \rangle_I,
\end{aligned} 
\quad (15) $$

By applying the decomposition in Eq. 9 to the first terms on the right-hand side of Eqs. 15 we obtain,

$$ 
\begin{aligned}
u_l &= \langle U \rangle_I \cdot \langle e_x \rangle_I + \mathbf{u}' \cdot \langle e_x \rangle_I + \langle \mathbf{u}'' \cdot e_x'' \rangle_I, \\
v_l &= \langle U \rangle_I \cdot \langle e_y \rangle_I + \mathbf{u}' \cdot \langle e_y \rangle_I + \langle \mathbf{u}'' \cdot e_y'' \rangle_I, \\
w_l &= \langle U \rangle_I \cdot \langle e_z \rangle_I + \mathbf{u}' \cdot \langle e_z \rangle_I + \langle \mathbf{u}'' \cdot e_z'' \rangle_I,
\end{aligned} 
\quad (16) $$

With respect to TNTI averaging, the fluctuations of the local velocity components are defined as follows:

$$ 
\begin{aligned}
u_l'' &= u_l - u_l \\
v_l'' &= v_l - v_l \\
w_l'' &= w_l - w_l
\end{aligned} 
\quad (17) $$

By using this definition (Eq. 17) of fluctuating velocities, normal and cross stresses can be defined in the TNTI local coordinate system e.g. $\langle v_l''^2 \rangle_I$, $\langle u_l'' v_l'' \rangle_I$ similarly to the classical turbulent stress terms. We use turbulent stresses defined in this way in our discussion of the theoretical jump conditions proposed by [16] (see Sect. 6).

## 5 TNTI-Averaged Profiles of the Flow Field Variables

It has been demonstrated in [1] that the instant $t/T_{ref} = 50$ falls within the time range where the turbulent jet exhibits self-similarity and is unaffected by boundary conditions. For the present paper, TNTI-averaged profiles have been examined at different times with the PJ1 simulation, and it has been observed that $t/T_{ref} = 50$ is representative of a wide range of times in the aforementioned time range. All results presented here are obtained from the PJn-HR simulations at $t/T_{ref} = 50$ unless stated otherwise. In order to increase statistical convergence, the results are obtained as the ensemble average over the five simulations whenever possible. In case a specific result is based on data from a single simulation rather than the ensemble average, this is explicitly mentioned in the caption.

In this section, the analysis is conducted using a local grid that spans from $-50dy$ to $50dy$ in each direction of the local coordinate system, which corresponds to a range $-23\eta$ to $23\eta$ in the local coordinates $y_I, x_I$, and $z_I$. The points $X_0$ considered on the interface are the numerical mesh points where the entrophy is closest to the entrophy threshold. It is worth mentioning that the
no multi-crossing condition results in the exclusion of 40% of the total number of points for the IES of $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$.

Now that the TNTI-averaging operation is defined, let's attempt to quantify
differences between different $\omega_{th}^2$ values. Figure 2 presents a comparison of TNTI-
average profiles of $\omega^2$ for the IES of $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ and the IES of $\omega_{th}^2/\omega_{ref}^2 =
10^{-6}$. The $\langle \omega^2 \rangle_I / \omega_{ref}^2$ profile is shown with the solid line for the IES of
$\omega_{th}^2/\omega_{ref}^2 = 10^{-6}$. The $y_I$ location where $\langle \omega^2 \rangle_I / \omega_{ref}^2 = 10^{-3}$ on this profile is
marked with a red vertical dashed line. The TNTI-average profile of $\langle \omega^2 \rangle_I / \omega_{th}^2$
for the IES of $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ is plotted on top, by offsetting its origin ($y_I/\eta = 0$
location) to coincide with the origin corresponding to the $\omega_{th}^2/\omega_{ref}^2 = 10^{-6}$ IES.
The offset between these two profiles is found to be $2.85\eta$.

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Profiles of $\langle\omega^2\rangle_I$ for IES of $\omega_{th}^2/\omega_{ref}^2 = 10^{-6}$ and $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ at $t/T_{ref} = 50$. The red line shows the $y_I$ location where $\langle\omega^2\rangle_I/\omega_{ref}^2 = 10^{-3}$ in the TNTI-average profile obtained for the IES of $\omega_{th}^2/\omega_{ref}^2 = 10^{-6}$. (a) $\langle\omega^2\rangle_I$ in log-lin plot, (b) $\langle\omega^2\rangle_I$ in lin-lin plot</figcaption></figure>

Even though three orders of magnitude difference in $\omega_{th}^2/\omega_{ref}^2$ leads to a
slight change of the spatial location of the IES i.e. $\approx 2.85\eta$, the effect of the
variation of the contortions of the IES is more dramatic on the TNTI-averaged
profiles. Figure 2 shows the TNTI-average profiles in a linear plot, in order to
see their differences more clearly. A much steeper gradient is observed for the
$\langle\omega^2\rangle_I$ profile obtained by averaging over the $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ IES. This is due
to the different geometrical structure of the IES reflected by the increasing $D_f$
with increasing $\omega_{th}^2/\omega_{ref}^2$ within the TNTI layer as shown in [1]. Due to the IES
orientation being more turbulent for $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$, the gradient obtained for
this IES is higher.

We now turn to the profiles along the interface normal direction of the TNTI-
averaged local velocity components, see Fig. 3(a), namely $\langle u_l \rangle_I$, $\langle v_l \rangle_I$,
and $\langle w_l \rangle_I$. These profiles are normalized by the centre-plane Kolmogorov
velocity $u_\eta$. A rapid change is observed in both $\langle u_l \rangle_I$ and $\langle v_l \rangle_I$ across
the TNTI. On the other hand, $\langle w_l \rangle_I$ remains zero throughout the interfacial
layer which is due to our definition of the local coordinate system which forces
$<e_z>_I$ to be aligned with the global $z$ direction (except at locations where the interface folds on itself) given that $e_x$ is chosen to align maximally with the global streamwise direction. Looking at Eq. 12 for $<w_l>_I$, both terms on its right hand side vanish. The global mean flow velocity $\mathbf{U}$ does not have a component in the global $z$-direction, the TNTI orientation $e_z$ does not have a preferential orientation around the spanwise axis and is also uncorrelated from the velocity fluctuations $\mathbf{u}'$. Consequently, both terms on the right hand side of Eq. 12 vanish and so $<w_l>_I$ remains zero along the interface normal $y_I$ axis.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** TNTI-averaged profiles of local velocity components and the local normal and shear turbulent stresses for the IES of $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ at $t/T_{ref} = 50$. ($\lambda$ is the Taylor length on the centre-plane.)</figcaption></figure>

In relation to the way that the orientation of $e_x$ is chosen, $<u_l>_I$ remains positive along the TNTI-average profile due to the jet having mean velocity in the global streamwise direction (see Eq. 12 for $<u_l>_I$). In contrast, the TNTI-averaged local normal velocity $<v_l>_I$ crosses zero at a certain point.

In the region $y_I/\eta < 0$, $<v_l>_I$ remains positive. This indicates that the interface perceives a local normal velocity pointing from the turbulent core towards the non-turbulent region. Furthermore, the negative gradient $d<\mathbf{v}_l>_I/dy_I < 0$ along the normal TNTI axis suggests that the velocity field tends to be compressive in the normal direction at the interface. In fact, the local $<\mathbf{v}_l>_I$ profile shows that the interface perceives a velocity field that moves towards it from both the turbulent and non-turbulent directions. Also, the fact that the TNTI-average velocity component $<\mathbf{v}_l>_I$ is non-zero along the $y_I$ axis is interesting in itself as the mean velocity obtained by global $x-z$ averaging does not have a component in the cross-stream direction i.e. $V(y) = 0$ given that our jet is temporally rather than spatially developing.

Figure 3(b) presents the profiles of local stresses, defined by the fluctuating velocities with respect to the TNTI-average velocity values, obtained from Eqs. 17. The normal stresses i.e., $<u''^2>_I$, $<v''^2>_I$, and $<w''^2>_I$, exhibit significantly higher magnitudes than the cross/stress term $<u''v''>_I$. These normal stresses maintain non-zero values even on the non-turbulent side of the interface,
extending as far as $y_I/\eta = 24$. However, their magnitudes decrease rapidly from the turbulent side of the IES to the non-turbulent region, particularly in the region between $y_I/\eta \approx -13.7$ and $y_I/\eta \approx -1.5$.

The profile of $\langle u''v'' \rangle_I$ displays an even steeper gradient, primarily occurring between $y_I/\eta \approx -7.2$ and $y_I/\eta \approx -2.1$. This suggests that the TNTI is characterized by a pronounced jump in $\langle u''v'' \rangle_I$ along the normal direction. It can be noted that the very high gradient of the profile of $\langle u''v'' \rangle_I$ is observed on the turbulent side of the $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ IES between $y_I/\eta \approx -2$ and $y_I/\eta \approx -10.5$.

We now analyse the decomposition presented in Eq. 16 in order to investigate the contributions of $\mathbf{U}$, $\mathbf{u}'$ and the orientation of the local coordinate system $\mathbf{e}$ (associated with the local TNTI orientation) to the components of the TNTI-averaged velocity $\mathbf{u}$. In Fig. 4, we present the profiles of $\langle u_l \rangle_I$ and $\langle v_l \rangle_I$ and of the three terms on the right-hand side of Eq. 16. The first term is the contribution from the scale product of the mean flow obtained by $x-z$ averaging and the mean orientation of the TNTI. The second term accounts for the effects of the velocity fluctuation field with respect to $x-z$ mean. And finally, the third term accounts for the velocity and orientation fluctuations with respect to the TNTI average: it is the correlation between the fluctuations of the velocity and of the interface orientation around their TNTI average.

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** TNTI-local profiles of $\langle u_l \rangle_I$ and $\langle v_l \rangle_I$ and of the terms on the right hand side of Eq. 16, for the $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ IES at $t/T_{ref} = 50$ for the PJ4-HR simulation. ($\lambda$ and $\eta$ are, respectively, the Taylor length and Kolmogorov microscale on the centre-plane.)</figcaption></figure>

Starting with the first term, the contribution of the mean velocity profile $\mathbf{U}$ varies linearly on the local $y_I/\eta$ axis for both component of the local TNTI-averaged velocity. Naturally, the highest contribution is in the component of $\langle u_l \rangle_I$, due to the positioning of the local coordinate system ensuring that the alignment of $\mathbf{e}_x$ is maximal with the global streamwise vector $\mathbf{i}$. Due to the fact that the axis goes towards the centre-plane of the jet for decreasing values of
$y_I/\eta$, the contribution of $\mathbf{U}$ increases in that direction. But there is no specific location in the local profile of $\langle \mathbf{U} \rangle_I \cdot \langle \mathbf{e}_x \rangle_I$ (and also the local profile of $\langle \mathbf{U} \rangle_I \cdot \langle \mathbf{e}_y \rangle_I$ for that matter) where it undergoes a dramatic change at the TNTI location.

On the other hand, Fig. 4(a) reveals a significant variation of $<u_l>_I$ in the $y_I$ direction when crossing the interface. It is evident that this jump is caused by TNTI averaging of the fluctuating velocity $\mathbf{u}'$ (fluctuation around the $x-z$ mean). The term $<\mathbf{u}'>_I \cdot <\mathbf{e}_x>_I$ crosses zero at $y_I/\eta = -13.8$ and for higher values of $y_I$ beyond this point, it contributes negatively to the profile of $<u_l>_I$. Although the third term also contributes to the profile of $<u_l>_I$, the primary factor driving the jump in $<u_l>_I$ across the TNTI is the second term, which includes the effects of the $\mathbf{u}'$ field.

The impact of $\mathbf{U}$ is not particularly significant on the interface normal velocity component $<v_l>_I$. The local $v_l$ profile is primarily, if not mainly, influenced by the correlation $<\mathbf{u}'' \cdot \mathbf{e}_y''>_I$ between fluctuations $\mathbf{u}''$ around the TNTI-average of the fluid velocity and orientation of the interface. In other words, the average interface normal velocity, perceived by the TNTI in the local reference frame is determined by the correlation between $\mathbf{u}''$ and $\mathbf{e}_y''$.

## 6 Jump Condition for the Tangential Velocity and the Vorticity at the Interface

The TNTI-averaged profiles presented in Sect. 5 allow us to investigate the analysis conducted by [16] on the interface. In the study of [16], the interface between the turbulent and non-turbulent regions was analyzed by placing a small control volume (CV) over a segment of the interface (interface referred to as *superlayer* in the original paper). Figure 5 shows a schematic of the CV described in [16].

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Schematic of the local control volume defined similar to [16] (*shown in dashed green line*). The subscripts *T* and *NT* are used for the quantities in the turbulent and non-turbulent sides of the control volume.</figcaption></figure>

In the analysis presented in [16], the mean velocity at the interface is expressed by two components: one aligned with the normal direction and the other in the interface's tangential direction. The positioning of the local coordinate system chosen in the present study is important in this context (Eqs. 3, 4, and 5). As shown in Fig. 3(a), the TNTI-average velocity vector $<\mathbf{u}>_I$
has components in the interface normal direction $<v_l>_I$ and in the tangential direction $<u_l>_I$, whereas the second tangential average velocity component $<w_l>_I$ is zero, compatible with the mean velocity components defined in [16] (see Fig. 3a).

The interface is considered to be a sheet-like surface with an infinitesimal thickness in [16], thus the CV covering a section of the interface is also considered to have a negligible thickness, thus neglecting the fluxes at the side surfaces of the CV and only considering the fluxes in the interface normal direction. Furthermore, the viscous stresses on either side of the interface are neglected in the analysis by W.C. Reynolds [16]. Given these assumptions, the conservation equations for mass and momentum in the TNTI normal and tangential directions are written for the CV with certain terms omitted. Starting with the mass balance for the control volume, the relation is written as;

$$[<v_l>_I]_{NT} = [<v_l>_I]_T. \qquad (18)$$

Here, $v_l$ is the mean TNTI normal velocity and the subscripts NT and T distinguish, in principle, between its value at the non-turbulent and turbulent sides of the interface (see Fig. 5). The balance is written in the TNTI local coordinate system where the mean/fluctuating velocities with respect to the TNTI-averaging operation are also expressed with components that are normal and tangential to the interface. As the CV is taken to be infinitesimally thin in the TNTI's normal direction, the faces of the CV normal to the TNTI tangential directions are neglected and we end up with a mass balance involving only fluxes in the normal direction of the interface [16]. This directly implies that the interface normal velocity should remain constant across the interface.

To assess the validity of this conclusion, Fig. 3(a) can be re-visited. It becomes apparent from this figure that the TNTI normal component $<v_l>_I$ varies across the interface, and its derivative along the $y_I$ direction exhibits high values, particularly in the proximity of the IES. Considering the CV positioned as in the work of W.C. Reynolds [16], which fully contains the jump of vorticity at the interface, there is a significant difference between the values of $<v_l>_I$ at the turbulent and non-turbulent sides of the interface. This result contradicts Eq. 18 and leads to a questioning of the assumptions made in [16] for obtaining the balance of quantities inside the CV.

The primary problem arises because the variation in the TNTI-averaged normal velocity $<v_l>_I$ directly implies the presence of a significant gradient of $<u_l>_I$ in the tangential direction of the IES. Furthermore, considering that the jump in vorticity occurs over a certain distance and is not as sharp as depicted in [16], this gradient of $<v_l>_I$ suggests significant flow of mass through the neglected side surfaces of the CV.

Secondly, the momentum balance in the normal and tangential directions are written as follows in [16],

$$[<p>_I]_{NT} = [<p>_I + \langle\rho v_l''^2\rangle_I]_T, \qquad (19)$$

$$\rho <v_l>_I ([<u_l>_I]_{NT} - [<u_l>_I]_T) - [\rho \langle v_l'' u_l'' \rangle_I]_T = 0. \qquad (20)$$
Here, $<p>_I$ denotes the TNTI-averaged pressure and $<v_l''^2>_I$ is the turbulent normal stress, which is considered only for the turbulent side in the model proposed by W.C. Reynolds [16]. In Eq. 20, $-(\rho\langle v_l'' u_l'' \rangle_I)$ is the turbulent shear stress at the boundary and similarly to Eq. 19, it has been considered only for the turbulent side of the CV.

Another important point related to this equation is that the mean velocity $<v_l>_I$ has to be defined for the first term of Eq. 20. In the analysis made by [16], the conservation of mass yields the equality given in Eq. 18 and there is no ambiguity in the choice of $<v_l>_I$ appearing in Eq. 20 because Eq. 18 states that it does not vary across the TNTI. On the other hand, a significant jump in the value of $<v_l>_I$ across the TNTI is observed in the present study (see Fig. 3a). To avoid introducing any modification to Eq. 20, $<v_l>_I$ in Eq. 20 can be defined by taking the average value $[<v_l>]_m$ of $<v_l>_I$ along the extent of the CV along the $y_I$ axis. Looking at Eq. 20, a jump in the TNTI mean tangential velocity is expected in case the turbulent shear stress is non-zero on the turbulent side of the interface. There is indeed a jump in $<u''v''>_I$ across the interface as shown in Fig. 3(b) and the values of the TNTI-averaged tangential velocity $<u_l>_I$ rise sharply on the turbulent side of the IES across the TNTI layer (see Fig. 3a). The local profiles obtained in the present study support the presence of a jump in the values of TNTI-averaged tangential velocity between the turbulent and non-turbulent sides of the interface.

Having computed the TNTI-averaged profiles of velocity, pressure, turbulent stresses and vorticity, the mass/momentum balances proposed in [16] can be checked to see if they hold near the detected IES. In order to make this check, we make use of the 1D TNTI-averaged profiles presented in Sect. 5 in the interface normal direction.

Twentieth century approaches to the TNTI regarded it as a surface with infinitesimal thickness, featuring nearly singular jumps in flow variables such as tangential velocity and span-wise vorticity. Twenty first century studies have used DNS to resolve the TNTI and perceive it as a TNTI layer with continuous variation of quantities across its thickness, even though TNTI-local gradients can indeed be very high. Due to this continuous variation, the choice of the location and the thickness of the CV to check the relations proposed in [16] becomes important.

Before presenting our results relating to the balance Eqs. 18, 19 and 20 we start by defining how the CV is placed. To do so, we plot in Fig. 6 the profiles of $\omega^2$ and of the magnitudes of the local vorticity components $\omega_x^l$, $\omega_y^l$ and $\omega_z^l$ all normalized by the mean centre-plane value of $|\omega_z|$. We also plot the gradient $-du_l/dy_I$ in the interface normal direction normalised in the same way. The centre-plane mean value of $|\omega_z|$, is used for the normalization of all the profiles as $\langle|\omega_x|\rangle_c \approx \langle|\omega_y|\rangle_c \approx \langle|\omega_z|\rangle_c$ at the centre-plane of the jet, due to local isotropy. (In relation to the term $-du_l/dy_I$, it should be noted that the derivative operation $d/dy_I$ commutes with the TNTI-averaging operation $\langle\cdot\rangle_I$.) The position of the CV's side denoted by T (turbulent side) is chosen at the maximum of the local profile $<|\omega_z^l|>_I$, i.e. at $y_I/\eta = -10.3$. The position of the CV's side denoted
by N (non-turbulent side) is chosen at $y_I/\eta = 1$ (see Fig. 6). Reasonably small changes to this choice do not change our conclusions.

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** TNTI-averaged profiles of $\langle|\omega_x^l|\rangle_I$, $\langle|\omega_y^l|\rangle_I$, $\langle|\omega_z^l|\rangle_I$, $\langle\omega^2\rangle_I$ and the gradient of local $u_l$ velocity in the TNTI normal direction $-du_l/dy_I$, normalized by the mean $|\omega_z|$ at the centre-plane of the jet. The interface location ($y_I = 0$) used for the con- ditioning of the profile is defined with the $\omega_{th}^2/\omega_{ref}^2 = 10^{-3}$ IES. Distance $y_I$ from the IES is normalized by the Kolmogorov length scale $\eta$ (bottom axis) and the Taylor length $\lambda$ (top axis) computed at the centre-plane of the jet.</figcaption></figure>

Using the local mean/fluctuating variables obtained on the basis of TNTI averaging, the balances of mass and normal/tangential momentum proposed in [16] are evaluated as follows from our DNS:

$$B_c^R \equiv \frac{[\langle v_l \rangle_I]_{NT}}{[\langle v_l \rangle_I]_T} = 0.17, \qquad (21)$$

$$B_{nm}^R \equiv \frac{[\langle p \rangle_I]_{NT}}{[\langle p \rangle_I + \langle \rho v_l''^2 \rangle_I]_T} = 1.40, \qquad (22)$$

$$B_{tm}^R \equiv \frac{[\langle v_l \rangle_I]_m ([\langle u_l \rangle_I]_{NT} - [\langle u_l \rangle_I]_T)}{[\langle u_l'' v_l'' \rangle_I]_T} = -0.83. \qquad (23)$$

The superscript R is a reminder of the relations $B_c^R = B_{nm}^R = B_{tm}^R = 1$ given by W.C. Reynolds [16]. A very large deviation from $B_c^R = 1$ is present, i.e. a large deviation from the form given in Eq. 18 of the continuity equation which states that the interface normal velocity is equal on both sides of the CV. However, as has been shown in Fig. 3(a), the value of $\langle v_l \rangle_I$ is not the same at the upper and lower sides of the CV. It varies across the thickness of the TNTI and a change in the sign of $v_l$ even occurs along the $y_I/\eta$ axis. This indicates
that there is considerable mass flux through the CV faces perpendicular to the interface which is not accounted for in the analysis of W.C. Reynolds [16].

Considering the normal momentum balance, the deviation from $B_{nm}^R = 1$ is smaller than the deviation from $B_c^R = 1$ (see Eq. 22). The deviation from the normal momentum balance given in [16] is mainly due to the neglected $<v_l''^2>_i$ term on the non-turbulent side of the interface. It can be seen in Fig. 3(b) that even though the normal stress reduces significantly across the interface, it does not go to zero in the non-turbulent region. Thus neglecting this term contributes to the 40% deviation from Eq. 19.

The tangential momentum deviation from $B_{tm}^R = 1$ is quite dramatic as the value of $B_{tm}^R$ returned by our DNS is in fact negative, let alone different from 1. This is because $<u''v''>_I$ takes positive rather than negative values.

An attempt can be made to improve the momentum balances obtained in [16]. With the introduction of the neglected terms, the equations for the normal and tangential momentum balance are:

$$ 
\begin{aligned}
[<p>_I + \rho\langle v_l''^2 \rangle_I + \rho\langle v_l>_I^2]_{NT} &= [<p>_I + \rho\langle v_l''^2 \rangle_I + \rho\langle v_l>_I^2]_T, \quad (24) \\
[\rho\langle v_l>_I\langle u_l>_I + \rho\langle u''v'' \rangle_I]_{NT} &= \rho[\rho\langle v_l>_I\langle u_l>_I + \rho\langle u''v'' \rangle_I]_T.
\end{aligned}
\quad (25) $$

The deviations from these balances (the superscript F is for “full terms”) are found to be;

$$ B_{nm}^F = \frac{[<p>_I + \rho\langle v_l''^2 \rangle_I + \rho\langle v_l>_I^2]_{NT}}{[<p>_I + \rho\langle v_l''^2 \rangle_I + \rho\langle v_l>_I^2]_T} = 1.11, \quad (26) $$

$$ B_{tm}^F = \frac{\rho[\rho\langle v_l>_I\langle u_l>_I]_{NT} - \rho[\rho\langle v_l>_I\langle u_l>_I]_T}{\rho[\rho\langle u''v'' \rangle_I]_T - \rho[\rho\langle u''v'' \rangle_I]_{NT}} = -3.08. \quad (27) $$

Equations 26 and 27 include the neglected terms in [16]. The value of $B_{nm}^F = 1.11$ (Eq. 26) is an improvement over the value $B_{nm}^R = 1.40$ in Eq. 22. However, the deviation from $B_{tm}^R = 1$ for the tangential momentum is increased when the omitted terms are included. Another way to assess the tangential momentum balance Eq. 25 is to calculate the ratio of fluxes on the top and bottom boundaries, i.e.

$$ B_{tm,2}^F = \frac{[\rho\langle v_l>_I\langle u_l>_I + \rho\langle u''v'' \rangle_I]_{NT}}{[\rho\langle v_l>_I\langle u_l>_I + \rho\langle u''v'' \rangle_I]_T} = 0.127. \quad (28) $$

The sizeable deviation from the tangential momentum balance Eq. 25 is apparent in both Eq. 27 and Eq. 28 for the CV considered here.

# 7 Conclusion

The approach in [16], placing a very thin CV on the interface, appears to have some shortcomings.
- The most important shortcoming stems from the fact that the TNTI is a TNTI layer and the jump of $\omega^2$ defining it occurs with a certain thickness. There is significant tangential velocity at the interface location and considerable average outflow of mass/momentum from the side boundaries of the CV which are neglected in the analysis of [16] where the interface is considered to be much sharper and the CV is taken with negligible thickness in the interface normal direction.

- Secondly, the $<v_l>_I$ varies significantly across the interface as shown by Fig. 3(a). This significant variation of $<v_l>_I$ across the TNTI relates to the previous point made and highlights the importance of the outflux from the side boundaries of the CV.

As the departures from the mass and momentum balances of [16] are mainly due to this normal-compression/tangent stretching feature of the TNTI local velocity field, the inclusion of the omitted terms at the CV's TNTI-tangent boundaries on both sides of the TNTI is not enough to correct the observed imbalances even though it does reduce the departure from the normal momentum balance in [16].

Finally, a jump of the TNTI-averaged Reynolds shear stress is clearly detectable through the TNTI layer and the TNTI-averaged fluid velocity normal to the TNTI is determined by the correlation between the TNTI normal vector and the fluid velocity fluctuations around the TNTI-average fluid velocity. Furthermore, spatial resolution proves to be a serious limiting factor for the accurate computational estimation of the TNTI's local propagation velocity.

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<figure><img src="image_8.png" /></figure>