3.1. Symmetries of the base equations

The basis of our analysis will be symmetries of the Euler and thermal energy equations as follows (Sadeghi & Oberlack Reference Sadeghi and Oberlack2020)

(3.1)\begin{equation} T_{t1}: \quad t^* = t+a_t, \quad \boldsymbol{x}^* = \boldsymbol{x}, \quad \boldsymbol{U}^* = \boldsymbol{U}, \quad P^* = P, \quad \varTheta^* = \varTheta \end{equation}

and

(3.2)\begin{equation} \left.\begin{gathered} T_{t2}: \quad t^* = \exp(a_{st}) t, \quad \boldsymbol{x}^* = \exp( a_{sx})\boldsymbol{x}, \quad \boldsymbol{U}^* = \exp(a_{sx}- a_{st})\boldsymbol{U}, \\ P^* = \exp[2(a_{sx}- a_{st})] P, \quad \varTheta^* = \exp(a_{\varTheta}) \varTheta. \end{gathered}\right\} \end{equation}

Further, as fully discussed in Oberlack & Rosteck (Reference Oberlack and Rosteck2010) and Sadeghi & Oberlack (Reference Sadeghi and Oberlack2020), taking the average of the transformation of the above instantaneous quantities would preserve their symmetry groups, i.e.

(3.3)\begin{equation} \bar{T}_{t1}: \quad t^* = t+a_t, \quad \boldsymbol{x}^* = \boldsymbol{x}, \quad \boldsymbol{\bar{U}}^* = \boldsymbol{\bar{U}}, \quad \bar{P}^* = \bar{P}, \quad \bar{\varTheta}^* = \bar{\varTheta} \end{equation}

and

(3.4)\begin{equation} \left.\begin{gathered} \bar{T}_{t2}: \quad t^* = \exp(a_{st}) t, \quad \boldsymbol{x}^* = \exp( a_{sx})\boldsymbol{x}, \quad \boldsymbol{\bar{U}}^* = \exp(a_{sx}- a_{st})\boldsymbol{\bar{U}}, \\ \bar{P}^* = \exp[2 (a_{sx}-a_{st})] \bar{P}, \quad \bar{\varTheta}^* = \exp(a_{\varTheta}) \bar{\varTheta}. \end{gathered}\right\} \end{equation}

New turbulent scaling laws of passive scalar transport in a temporally evolving turbulent plane jet from the TPC equation are derived based on the corresponding symmetry transformations of equations (2.3), (2.10) and (2.11), with $\nu = \alpha = 0$. The first set of essential symmetries to construct scaling laws are derived directly by transferring the induced classical symmetries of Euler and thermal energy equations, as given in (3.1)–(3.2), to their corresponding ones for the novel proposed TPC equations (2.10) and (2.11). As such, the following symmetries result for the TPC:

(3.5)\begin{equation} \left.\begin{gathered} \bar{T}_{C1}: \quad t^* = t+a_t, \quad \boldsymbol{x}^* =\boldsymbol{x}, \quad \boldsymbol{r}^* = \boldsymbol{r}, \quad H_{\varTheta\varTheta}^* = H_{\varTheta\varTheta}, \quad H_{i\varTheta}^* = H_{i\varTheta}, \\ H_{(\varTheta k)\varTheta}^*=H_{(\varTheta k)\varTheta},\quad H_{\varTheta(\varTheta k)}^*= H_{\varTheta(\varTheta k)},\quad \overline{P\varTheta}^*=\overline{P\varTheta}, \\ H_{(i k)\varTheta}^*=H_{(i k)\varTheta},\quad H_{i(\varTheta k)}^*=H_{i(\varTheta k)} \end{gathered}\right\} \end{equation}

and

(3.6)\begin{equation} \left.\begin{gathered} \bar{T}_{C2}: \quad t^* = \exp(a_{st}) t, \quad \boldsymbol{x}^* = \exp( a_{sx})\boldsymbol{x}, \quad \boldsymbol{r}^* = \exp( a_{sx})\boldsymbol{r}, \\ H_{\varTheta\varTheta}^* = \exp(2 a_{\varTheta})H_{\varTheta\varTheta},\quad H_{i\varTheta}^* = \exp( a_{\varTheta}+a_{sx}-a_{st})H_{i\varTheta}, \\ H_{(\varTheta k)\varTheta}^*= \exp( 2 a_{\varTheta}+a_{sx}-a_{st}) H_{(\varTheta k)\varTheta}, \quad H_{\varTheta(\varTheta k)}^*= \exp( 2 a_{\varTheta}+a_{sx}-a_{st}) H_{\varTheta(\varTheta k)}, \\ \overline{P\varTheta}^*= \exp( a_{\varTheta}+2 a_{sx}-2 a_{st}) \overline{P\varTheta}, \quad H_{(i k)\varTheta}^*= \exp( a_{\varTheta}+2a_{sx}-2a_{st}) H_{(i k)\varTheta}, \\ H_{i(\varTheta k)}^*= \exp( a_{\varTheta}+2a_{sx}-2a_{st}) H_{i(\varTheta k)}. \end{gathered}\right\} \end{equation}

In (3.5) and (3.6), we present symmetries implied by Euler and thermal energy equations. Now we consider statistical symmetries on the basis of the TPC equations (2.10) and (2.11), which are annotatively derived as a linear set of equations, being either the TPC equations such as (2.10) and (2.11) or one-point type of equations such as (2.3) for the present temporally enveloping plane jet. This concept was first developed in Oberlack & Rosteck (Reference Oberlack and Rosteck2010), while more details of it and its key importance for turbulence may be found in Waclawczyk et al. (Reference Waclawczyk, Staffolani, Oberlack, Rosteck, Wilczek and Friedrich2014). This statistical scaling group reads

(3.7) \begin{equation} \left.\begin{gathered} \bar{T}_{s}: \quad t^* = t,\quad \boldsymbol{x}^* = \boldsymbol{x}, \quad \boldsymbol{r}^* = \boldsymbol{r}, \quad \bar{U}_i^* = \exp( a_{ss}) \bar{U}_i,\quad \bar{\varTheta}^* = \exp( a_{ss}) \bar{\varTheta}, \\ H_{\varTheta\varTheta}^* =\exp( a_{ss}) H_{\varTheta\varTheta},\quad H_{i\varTheta}^* = \exp( a_{ss})H_{i\varTheta}, \quad H_{(\varTheta k)\varTheta}^*= \exp( a_{ss}) H_{(\varTheta k)\varTheta}, \\ H_{\varTheta(\varTheta k)}^*= \exp( a_{ss}) H_{\varTheta(\varTheta k)},\quad \overline{P\varTheta}^*= \exp( a_{ss})\overline{P\varTheta}, \\ H_{(i k)\varTheta}^*= \exp( a_{ss}) H_{(i k)\varTheta},\quad H_{i(\varTheta k)}^*= \exp( a_{ss}) H_{i(\varTheta k)}, \end{gathered}\right\} \end{equation}

while in Waclawczyk et al. (Reference Waclawczyk, Staffolani, Oberlack, Rosteck, Wilczek and Friedrich2014) it was shown that (3.7) represents a measure of intermittency, a property that has been discussed for decades as a special characteristic of turbulence.

3.2. Condition of the symmetry-invariant solutions

As indicated earlier, the main intention of this paper is to generate new turbulent scaling laws for large-scale scalar quantities, such as the mean value of the passive scalar as well as the scalar variance and turbulent scalar flux, which are the parameters most investigated theoretically, experimentally and numerically. As such, we consider the limit $r\rightarrow 0$ of the TPC in our analysis, i.e. $H_{\varTheta \varTheta }^0(\boldsymbol {x},t)=\overline {\varTheta ^2}(\boldsymbol {x},t)$ and $H_{i\varTheta }^0(\boldsymbol {x},t)=\overline {U_i \varTheta }(\boldsymbol {x},t)$. For the derivation of the turbulent scaling laws, the corresponding characteristic equations for the invariant solutions (A14) in the context of a temporally evolving turbulent plane jet, i.e. $\bar {\varTheta } (\boldsymbol {x},t)= \bar {\varTheta } (x_2,t)$, $\boldsymbol {H}^0(\boldsymbol {x},t)= \boldsymbol {H}^0(x_2,t)$,…, reduce to the following form:

(3.8)\begin{equation} \frac{\mbox{d}t}{\xi_t} = \frac{\mbox{d} x_2}{\xi_{x_2}}= \frac{\mbox{d}\bar{\varTheta}}{\eta_{\bar{\varTheta}}} =\frac{\mbox{d}H_{\varTheta\varTheta}^0}{\eta_{H_{\varTheta \varTheta}^0}} = \frac{\mbox{d}H_{i\varTheta}^0}{\eta_{H_{i \varTheta}^0}}= \frac{\mbox{d}H_{\varTheta 2\varTheta}^0}{\eta_{H_{\varTheta 2 \varTheta}^0}} = \frac{\mbox{d}H_{i 2\varTheta}^0}{\eta_{H_{i 2\varTheta}^0}}= \frac{\mbox{d}\overline{P\varTheta}^0}{\eta_{\overline{P\varTheta}^0}}, \end{equation}

the distinction being given by the form of the symmetries given as infinitesimals $\boldsymbol {\xi }$ and $\boldsymbol {\eta }$ as defined in Appendix A. Now, by adapting (A8a,b)–(A10a,b), the symmetries $\bar {T}_{t2}$, $\bar {T}_{C1}$, $\bar {T}_{C2}$ and $\bar {T}_{s}$ in (3.4)–(3.7) can be presented in terms of infinitesimals $\boldsymbol {\xi }$ and $\boldsymbol {\eta }$, in which the condition (3.8) in this case is specified as

(3.9)\begin{align} \frac{\mbox{d}t}{a_{st}t+a_t} &= \frac{\mbox{d} x_2}{a_{sx} {x_2}} = \frac{\mbox{d}\bar{\varTheta}}{[a_{\varTheta}+a_{ss}] {\bar{\varTheta}}} =\frac{\mbox{d}H_{\varTheta\varTheta}^0}{[2a_\varTheta+a_{ss}] H_{\varTheta\varTheta}^0 } = \frac{\mbox{d}H_{i\varTheta}^0}{ [a_{\varTheta}+(a_{sx}-a_{st})+a_{ss}] H_{i \varTheta}^0} \nonumber\\ &= \frac{\mbox{d}H_{\varTheta 2\varTheta}^0}{[2a_{\varTheta}+(a_{sx}-a_{st})+a_{ss}] H_{\varTheta 2\varTheta}^0} = \frac{\mbox{d}H_{i 2\varTheta}^0}{[ a_{\varTheta}+2(a_{sx}-a_{st}) +a_{ss}] H_{i2\varTheta}^0} \nonumber\\ &= \frac{\mbox{d}\overline{P\varTheta}^0}{[a_{\varTheta}+2(a_{sx}-a_{st}) +a_{ss}] \overline{P\varTheta}^0}. \end{align}

It should be noted that currently we are not looking at TPCs and therefore the $\boldsymbol {r}$ component does not occur in the latter condition.

3.3. Symmetry breaking and turbulent scaling laws

In Oberlack et al. (Reference Oberlack, Waclawczyk, Rosteck and Avsarkisov2015) and Sadeghi & Oberlack (Reference Sadeghi and Oberlack2020), the importance of a symmetry breaking and its implication for the construction of scaling laws is explained in detail. We recall that symmetry breaking is referred to as that one or more group parameters are assigned specific values because some of the symmetries are not admitted by boundary conditions or other external constraints.

A key step in constructing the scaling laws in the present flow is that the integral invariant (2.4) induces a symmetry breaking. To comprehend this we implement (3.4) and (3.7) into (2.4) to obtain

(3.10)\begin{equation} I=\int_{-\infty}^{\infty} \bar{\varTheta}^* \,\mbox{d}x_2^* \exp({-(a_\varTheta + a_{ss} +a_{sx})}). \end{equation}

For the latter being invariant under any of the above given groups, the following condition has to hold:

(3.11)\begin{equation} a_{\varTheta}+ a_{ss}+a_{sx}=0. \end{equation}

Equation (3.11) is apparently symmetry breaking and in the following we replace $a_{ss}$ using $a_{ss}=-a_\varTheta -a_{sx}$. With this consideration, (3.9) now reduces to

(3.12)\begin{align} \frac{\mbox{d}t}{a_{st}t+a_t} &= \frac{\mbox{d} x_2}{a_{sx} {x_2}} = \frac{\mbox{d}\bar{\varTheta}}{-a_{sx} {\bar{\varTheta}}} =\frac{\mbox{d}H_{\varTheta\varTheta}^0}{[a_\varTheta-a_{sx}] H_{\varTheta\varTheta}^0 } = \frac{\mbox{d}H_{i\varTheta}^0}{ -a_{st} H_{i \varTheta}^0} \nonumber\\ &= \frac{\mbox{d}H_{\varTheta 2\varTheta}^0}{[a_{\varTheta}-a_{st}] H_{\varTheta 2\varTheta}^0} = \frac{\mbox{d}H_{i 2\varTheta}^0}{[ a_{sx}-2a_{st}] H_{i2\varTheta}^0} = \frac{\mbox{d}\overline{P\varTheta}^0}{[a_{sx}-2a_{st}] \overline{P\varTheta}^0}. \end{align}

From (3.12) we now construct the scaling laws for a temporally evolving plane turbulent jet. Integration of (3.12) leads to a set of invariants which are taken as the new independent and dependent variables as follows:

(3.13)$$\begin{gather} \tilde{x}_2=\frac{x_2}{(t+t_0)^{n}}, \end{gather}$$

(3.14)$$\begin{gather}\tilde{\varTheta}(\tilde{x}_2)=\frac{\bar{\varTheta}({x}_2,t)}{(t+t_0)^{{-}n}}, \end{gather}$$

(3.15)$$\begin{gather}\tilde H_{{\varTheta\varTheta}}^0(\tilde{x}_2)=\frac{H_{\varTheta\varTheta}^0({x}_2,t)}{(t+t_0)^{{-}m-n}}, \end{gather}$$

(3.16)$$\begin{gather}\tilde H_{{i \varTheta}}^0(\tilde{x}_2)=\frac{H_{i\varTheta}^0({x}_2,t)}{(t+t_0)^{{-}1}}, \end{gather}$$

(3.17)$$\begin{gather}\tilde H_{\varTheta 2\varTheta}^0(\tilde{x}_2)=\frac{H_{\varTheta 2\varTheta}^0({x}_2,t)}{(t+t_0)^{{-}m-1}}, \end{gather}$$

(3.18)$$\begin{gather}\tilde H_{i 2\varTheta}^0(\tilde{x}_2)=\frac{H_{i 2 \varTheta}^0({x}_2,t)}{(t+t_0)^{n-2}}, \end{gather}$$

(3.19)$$\begin{gather}\widetilde{\overline{P\varTheta}}^0 (\tilde{x}_2)=\frac{\overline{P\varTheta}^0({x}_2,t)}{(t+t_0)^{n-2}}, \end{gather}$$

where

(3.20a–c)\begin{equation} n=\frac{a_{sx}}{a_{st}}, \quad m={-}\frac{a_{\varTheta}}{a_{st}}, \quad t_0=\frac{a_t}{a_{st}}. \end{equation}

Here, the variables marked with a tilde are the invariants of the system composed of (2.3), (2.4), (2.10) and (2.11) and $t_0$ is a virtual (temporal) origin. By transferring to the classical notation, (3.15)–(3.19) are presented as follows:

(3.21)$$\begin{gather} \tilde{H}_{\varTheta \varTheta}^0(\tilde{x}_2) = \frac{R_{\theta\theta}^0({x}_2,t)+ \bar{\varTheta}({x}_2,t)^2}{(t+t_0)^{{-}m-n}}, \end{gather}$$

(3.22)$$\begin{gather}\tilde{H}_{i\varTheta}^0(\tilde{x}_2) = \frac{R_{i\theta}^0({x}_2,t)+\bar{U}_i({x}_2,t)\bar{\varTheta}({x}_2,t)}{(t+t_0)^{{-}1}}, \end{gather}$$

(3.23)$$\begin{gather}\tilde{H}_{\varTheta2\varTheta}^0(\tilde{x}_2) = \frac{R_{\theta2\theta}^0({x}_2,t)+2R_{2\theta}^0({x}_2,t)\bar{\varTheta}({x}_2,t)}{(t+t_0)^{{-}m-1}}, \end{gather}$$

(3.24)$$\begin{gather}\tilde{H}_{i2\varTheta}^0(\tilde{x}_2) = \frac{R_{i2\theta}^0({x}_2,t)+R_{i2}^0({x}_2,t)\bar{\varTheta}({x}_2,t)+R_{2\theta}^0({x}_2,t)\bar{U}_i({x}_2,t)}{(t+t_0)^{n-2}} \end{gather}$$

and

(3.25)\begin{equation} \widetilde{\overline{P\varTheta}}^0(\tilde{x}_2) = \frac{\overline{p\theta}^0({x}_2,t)+\bar P({x}_2,t) \bar{\varTheta}({x}_2,t)}{(t+t_0)^{n-2}}, \end{equation}

where we have used the relation between the $H$-notation and $R$-notation

(3.26)$$\begin{gather} H_{\varTheta \varTheta}^0(\boldsymbol{x},t)= R_{\theta\theta}^0(\boldsymbol{x},t)+ \bar{\varTheta}(\boldsymbol{x},t)^2, \end{gather}$$

(3.27)$$\begin{gather}H_{i\varTheta}^0(\boldsymbol{x},t)=R_{i\theta}^0(\boldsymbol{x},t)+ \bar{U}_i(\boldsymbol{x},t)\bar{\varTheta}(\boldsymbol{x},t), \end{gather}$$

(3.28)$$\begin{gather}H_{\varTheta2\varTheta}^0(\boldsymbol{x},t)= R_{\theta2\theta}^0(\boldsymbol{x},t)+2R_{2\theta}^0(\boldsymbol{x},t)\bar{\varTheta}(\boldsymbol{x},t), \end{gather}$$

(3.29)$$\begin{gather}H_{i2\varTheta}^0(\boldsymbol{x},t)= R_{i2\theta}^0(\boldsymbol{x},t)+R_{i2}^0(\boldsymbol{x},t)\bar{\varTheta}(\boldsymbol{x},t)+ R_{2\theta}^0(\boldsymbol{x},t)\bar{U}_i(\boldsymbol{x},t) \end{gather}$$

and

(3.30)\begin{equation} \overline{P\varTheta}^0(\boldsymbol{x},t)= \overline{p\theta}^0(\boldsymbol{x},t)+ \bar P(\boldsymbol{x},t)\bar{\varTheta}(\boldsymbol{x},t). \end{equation}

For a temporally evolving turbulent jet, we have $\bar {U}_2=\bar {U}_3=0$ and with this (3.22) reduces to

(3.31)$$\begin{gather} \tilde{H}_{1\varTheta}^0(\tilde{x}_2) = \frac{R_{1\theta}^0({x}_2,t)+ \bar{U}_1({x}_2,t)\bar{\varTheta}({x}_2,t)}{(t+t_0)^{{-}1}}, \end{gather}$$

(3.32)$$\begin{gather}\tilde{H}_{2\varTheta}^0(\tilde{x}_2) = \frac{R_{2\theta}^0({x}_2,t)}{(t+t_0)^{{-}1}}, \end{gather}$$

(3.33)$$\begin{gather}\tilde{H}_{3\varTheta}^0(\tilde{x}_2) = \frac{R_{3\theta}^0({x}_2,t)}{(t+t_0)^{{-}1}}, \end{gather}$$

and (3.24) reduces to

(3.34)$$\begin{gather} \tilde{H}_{12\varTheta}^0(\tilde{x}_2) = \frac{R_{12\theta}^0({x}_2,t)+R_{12}^0({x}_2,t) \bar{\varTheta}({x}_2,t)+R_{2\theta}^0({x}_2,t)\bar{U}_1({x}_2,t)}{(t+t_0)^{n-2}}, \end{gather}$$

(3.35)$$\begin{gather}\tilde{H}_{22\varTheta}^0(\tilde{x}_2) = \frac{R_{22\theta}^0({x}_2,t)+R_{22}^0({x}_2,t)\bar{\varTheta}({x}_2,t)}{(t+t_0)^{n-2}} \end{gather}$$

and

(3.36)\begin{equation} \tilde{H}_{32\varTheta}^0(\tilde{x}_2) = \frac{R_{32\theta}^0({x}_2,t)+R_{32}^0({x}_2,t) \bar{\varTheta}({x}_2,t)}{(t+t_0)^{n-2}}. \end{equation}

Additionally, $U_3$ decorrelates from $U_2$ and $\varTheta$, and hence the terms $R_{3\theta }^0$, $R_{32}^0$ and $R_{32\theta }^0$ vanish identically and therefore do not need to be considered.

The scaling law (3.32) corresponds to the classical scaling law (1.2) and we illustrate this in the following section. However, other scaling laws like (3.21) or (3.31) have a significantly different structure compared with the corresponding classical laws such as (1.3). If, for example, we rearrange the scaling law (3.21) in the classical notation on the basis of the correlations from the fluctuations, then we have to use the scaling law (3.14). So we see that after a rearrangement to $R_{\theta \theta }^0$

(3.37)\begin{equation} R_{\theta\theta}^0({x}_2,t) = \tilde{H}_{\varTheta \varTheta}^0(\tilde{x}_2) (t+t_0)^{{-}m-n} - \tilde{\varTheta}(\tilde{x}_2)^2 (t+t_0)^{{-}2 n}, \end{equation}

it is composed of two different power laws, which have different decay exponents $-m-n$ and $-2n$.

Hence, in order to compare with plain canonical power laws, the comparison of the above theoretical results against the new DNS data obtained in a temporally evolving plane jet in the next section is exclusively based on the $H$-correlations.

4.1. Numerical scheme and DNS details

The subsequent analysis is based on highly resolved DNS of a temporally evolving turbulent plane jet flow. It should be pointed out that several previous works have commented on turbulent scaling laws and considered whether the shape of the self-similar profiles and the scaling exponents might depend on the DNS initial conditions (Oberlack & Zieleniewicz Reference Oberlack and Zieleniewicz2013; Sadeghi et al. Reference Sadeghi, Oberlack and Gauding2018; Sadeghi & Oberlack Reference Sadeghi and Oberlack2020; Sadeghi et al. Reference Sadeghi, Oberlack and Gauding2020). To address the possible effects of different DNS initial conditions, we validate the new scaling laws using two different sets of DNS with slightly different initial conditions and Reynolds numbers.

A detailed description of the set-up of the simulation and a validation of the results is provided by Hunger, Gauding & Hasse (Reference Hunger, Gauding and Hasse2016), Hunger et al. (Reference Hunger, Dietzsch, Gauding and Hasse2018) and Sadeghi et al. (Reference Sadeghi, Oberlack and Gauding2018). In the following, the main features of the DNS concerning the passive scalar that was not highlighted in Sadeghi et al. (Reference Sadeghi, Oberlack and Gauding2018) are summarized. The DNS solves the incompressible Navier–Stokes equations and an advection–diffusion equation (2.1) for a passive scalar $\varTheta (\boldsymbol {x},t)$. The computational domain has periodic boundary conditions in streamwise $x_1$ and spanwise $x_3$ directions, while free-slip boundary conditions are used in the cross-wise $x_2$ direction. The flow is statistically homogeneous in $x_1$–$x_3$ planes. Statistics are averaged over these planes and depend only on time $t$ and the cross-wise coordinate $x_2$. Simulation DNS1 is performed on a numerical grid with $3072 \times 1024 \times 1024$ points, where the size of the computational domain equals $L_{x_1} = 60$, $L_{x_2} = 24.4$ and $L_{x_3} = 20$. A uniform equidistant mesh is used for the inner part of the domain, while the outer part is slightly coarsened towards the cross-wise boundaries. The simulation DNS2 is performed on an equidistant mesh with $3072 \times 3072 \times 1536$ points, where the size of the computational domain is $L_{x_1} = 40$, $L_{x_2} = 40$ and $L_{x_3} = 30$. In both cases, the size of the computational domain is large enough to minimize confinement effects at later integration times. The DNS is well resolved, since the grid width is smaller than or equal to the Kolmogorov length scale.

For non-dimensionalization, the initial centreline velocity $\bar {U}_1(0)$ and the initial jet thickness $\delta _{0.5}(0)$ are used. The passive scalar $\varTheta (\boldsymbol {x},t)$ is non-dimensionalized by its centreline value at initialization. The initial Reynolds number is defined as $\mathit {Re}_0 = \bar {U}_1(0)\delta _{0.5}(0)/\nu$ and equals 2200 for case DNS1 and 2500 for case DNS2. The initial Péclet number is defined as $\mathit {Pe}_0=\mathit {Re}_0 \mathit {Sc}$, with the Schmidt number $Sc$ being the ratio of kinematic viscosity $\nu$ and molecular diffusivity $\alpha$. The Schmidt number is set to unity for both cases. A careful initialization of the velocity field is important for temporally evolving turbulent flows. The initial velocity profile is composed of two mirrored hyperbolic-tangent mean profiles. Its initial velocity profile is perturbed around the position of the highest shear rate with broadband random Gaussian fluctuations to facilitate a quick transition to fully developed turbulence. The fluctuating field satisfies a prescribed energy spectrum, defined as

(4.1)\begin{equation} E(\kappa,t=0) \propto \kappa^4 \exp ({-}2 (\kappa/\kappa_0)^2), \end{equation}

where $\kappa$ denotes the wavenumber and $\kappa _0$ is the wavenumber at which the maximum of the initial energy spectrum occurs. The value of $\kappa _0$ is set to 6.0 for DNS1 and 6.4 for DNS2. The initial fluctuation intensity is 2 % for both cases. The initial scalar profile is identical to the mean velocity profile, but non-perturbed. As a consequence, scalar fluctuations are created solely through the convection by the turbulent velocity field.

4.2. Validation of the new scaling laws

In this subsection, the scaling laws (3.13)–(3.19) derived in § 3 are validated against the DNS data (DNS1 and DNS2) of the temporally evolving jet. According to Lie symmetry analysis, the variable $x_2$ is self-similar when it is scaled by a power-law function of time with arbitrary exponent $n$ (3.13). We may estimate $n$ from a variety of the scaled variables for $x_2$. We select the mean scalar half-width $\delta _{\theta }$ defined as the distance between the location along $x_2$ at which the mean scalar is equal to half of the maximum mean scalar and the centreline as the relevant scale, which is also widely used in the context of turbulent jet flows in the literature. In figure 2, the temporal evolution of $\delta _{\theta }$ is plotted for DNS1 and DNS2. According to the similarity analysis, the length scale should evolve in a power-law fashion, as required by (3.13). It is found that $\delta _{\theta }$ perfectly follows a power law (for $t \gtrsim 21$) as

(4.2)\begin{equation} \delta_{\theta} \propto (t+t_0)^{n}, \end{equation}

with $t_0=-1.70$ and $n = 0.55$ for DNS1 and $t_0=-5$ and $n = 0.56$ for DNS2. It can be observed that the two DNS datasets exhibit very similar exponents $n$, while the virtual origin $t_0$ is slightly further departed. We will later observe that even slight changes in $n$ and $t_0$ may affect the shape of the self-similar profiles (e.g. height). The scaling behaviour for the mean temperature contains in the classical notation in (1.1) the temporal variation of $\varTheta _s(t)$, which essentially corresponds to the variation of the centreline at $x_2=0$. Comparing this with the symmetry-induced law in (3.14), we find

(4.3)\begin{equation} \varTheta_{s} \propto (t+t_0)^{{-}n}, \end{equation}

where $n$ is expected to exhibit the same value as above, i.e. $n = 0.55$ for DNS1 and $n = 0.56$ for DNS2. This is also reflected using the symmetry-induced scalings (3.13) and (3.14) in the integral invariant (2.4), i.e. the temperature scale should scale inversely with $\delta _\theta$.

Figure 2. The temporal variation of $\delta _{\theta }$. The dashed lines are the DNS data (note that DNS1 data are shifted up 1.5 times). The solid lines are power-law fits to the DNS data $\delta _{\theta } \propto (t+t_0)^{n}$ in (4.2). In DNS1, $t_0=-1.70$ and $n = 0.55$; in DNS2, $t_0=-5$ and $n = 0.56$. The range of DNS data up to $t=15$ is strongly transient and is not shown here.

To verify the evolution of this scale, we use the maximum magnitude of the mean scalar $\bar {\varTheta }$ on the centreline ($\bar {\varTheta }_c$) as the relevant scale, again as it is widely used in the literature, and plot the temporal evolution of $\bar {\varTheta }_c$ in figure 3. As can be seen, $\bar {\varTheta }_c$ is nicely fitted with (4.3) with the previously obtained $t_0$ and $n$ for DNS1 and DNS2. This again highlights that the exponent $n$ can be solely identified by the mean scalar half-width $\delta _{\theta }$ measurements, and no additional information is needed for the rest of the present analysis.

Figure 3. The temporal variation of $\bar {\varTheta }_c$. The dashed lines are the DNS data (note that DNS1 data are shifted up 2 times). The solid lines are power-law fits to the DNS data $\bar {\varTheta }_c \propto (t+t_0)^{-n}$ in (4.3). The respective $n$ and $t_0$ may be taken from the caption of figure 2.

The self-similar solutions of the mean scalar profiles $\bar {\varTheta }(x_2,t)$ or rather $\tilde {\varTheta }(\tilde {x}_2)$, based the on symmetry result (3.14), are plotted in figure 4, with $t_0=-1.70$ and $n = 0.55$ for DNS1 (for $t\geq 22$) and with $t_0=-5$ and $n = 0.56$ for DNS2 (for $t\geq 25$). As can be seen, there is an excellent agreement between $\tilde {\varTheta }(\tilde {x}_2)$ at different $t$ for both DNS1 and DNS2. It is also apparent that the self-similar solution for the mean scalar $\bar {\varTheta }(x_2,t)$ based on a symmetry analysis recovers the classical similarity solutions for this quantity where a scalar scale such as $\bar {\varTheta }_c$ is proportional to the denominator of (3.14) and a length scale such as $\delta _{\theta }$ is proportional to the denominator of (3.13). Therefore, interchangeably they can be used for the self-similarity of $\bar {\varTheta }(x_2,t)$ and also the higher moments (i.e. $\varTheta _{s} = \bar {\varTheta }_c$ and $L_{\theta } = \delta _{\theta }$ in (1.1)), as shown in figure 1(a). Additionally, it is apparent that the self-similarity shapes for each set of DNS data are dependent on their own virtual origin and power-law exponent, and can be expected to be different from one condition to another as previously noted. We will expect similar differences in other profiles as well. We now contrast the traditional self-similarity, i.e. (1.2) and (1.3), with the new symmetry-based self-similar solutions (3.21), (3.31) and (3.32), comparing it also to the DNS data. As shown in § 3, the second-order moment velocity–scalar vector $H_{i\varTheta }^0({x}_2,t)$ is to be scaled with $(t+t_0)^{-1}$ according to (3.16). To verify this solution, both DNS1 and DNS2 are fitted to (3.31) and (3.32) ($i=1,2$ in (3.16)) and presented in figures 5 and 6, respectively. As can be observed, the DNS data nicely collapse based on the new scaling laws. For this, no parameter had to be adjusted and, to be consistent, even the virtual origin $t_0=-1.7$ or $t_0=-5$ were taken from the previous fitting.

Figure 4. The self-similar solution (3.14), compared with DNS1 (a) and DNS2 (b).

Figure 5. The self-similar solution (3.31), compared with DNS1 (a) and DNS2 (b).

Figure 6. The self-similar solution (3.32), compared with DNS1 (a) and DNS2 (b).

Then, we consider the self-similarity of $H_{\varTheta \varTheta }^0({x}_2,t)$ and this quantity was found to scale with $(t+t_0)^{-m-n}$ (3.15). The exponent $n$ was previously given from a length-scale analysis, i.e. $n = 0.55$ for DNS1; $n = 0.56$ for DNS2. Unlike the scalar–velocity second-order moments, the power-law constant for $H_{\varTheta \varTheta }^0$ cannot be obtained using Lie group analysis alone as $m$ is a free exponent which is likely to be dependent on external conditions (such as initial conditions or virtual origin). This exponent can, however, be estimated by employing the numerical results if a self-similar solution such as derived in (3.15) exists. As such, the DNS data are fitted to (3.21) (which is an extended version of (3.15)) and presented in figure 7 for both DNS1 and DNS2. It is observed that the DNS data nicely collapse based on the new scaling laws, with the parameters from fitting the scaling laws to the data of $m=0.48$ and $m+n=1.03$ for DNS1 and $m=0.50$ and $m+n=1.06$ for DNS2. Additionally, the present solutions show that other than the exponent $n$, the exponent $m$ is also slightly affected by different DNS initial conditions, while this leads to affect the shape of self-similar profiles, even though very small $m$ are not markedly departed in the present cases. With the latter it becomes apparent that the statistical symmetry (3.7) plays a weak role for the present scaling laws, as may be taken from relation (3.11). Using this relation, isolating for $a_{ss}$ and dividing by $a_{st}$ we obtain ${a_{ss}}/{a_{st}} = - ({a_\varTheta }/{a_{st}}) - {a_{sx}}/{a_{st}} = m-n =-0.07$ for DNS1, or $m-n =-0.06$ for DNS2. Additionally, since $m$ is slightly smaller than $n$, based on (3.21) and (3.26), this may suggest that (even though not explicitly) the scalar variance $\overline {\theta ^2}$ decays more slowly than $\bar {\varTheta }_c^2$, and subsequently, the relative intensity of the scalar fluctuations, defined as

(4.4)\begin{equation} I_{\theta \theta}= {\overline{\theta^2}}^{1/2}/\varTheta_c , \end{equation}

slightly increases with time (see figure 8). Such an attitude on the relative intensity of the scalar (or velocity fluctuations) has been observed in a large number previous works concerning turbulent jet flows (e.g. Lockwood & Moneib Reference Lockwood and Moneib1980; Dahm & Dimotakis Reference Dahm and Dimotakis1990; Dowling & Dimotakis Reference Dowling and Dimotakis1990; Lubbers et al. Reference Lubbers, Brethouwer and Boersma2001; Mi et al. Reference Mi, Nobes and Nathan2001; Quinn Reference Quinn2006; Sadeghi & Pollard Reference Sadeghi and Pollard2012; Sadeghi, Lavoie & Pollard Reference Sadeghi, Lavoie and Pollard2015). When the relative intensity of the scalar fluctuations $I_{\theta \theta }$ deviates from constancy, one should expect that (1.3) does not fully hold over time or downstream of the jet, which is other evidence that classical scaling may not lead to a complete self-similarity of the scalar fluctuations. The present approach, however, proposed a novel similarity-based form of the second and higher moments, which has a different structure compared with the classical scaling. Therefore, if $m$ and $n$ are not equal, which may lead to unsatisfactory results based on (1.3) as shown in figure 1, the new self-similar solutions for higher moments are still nicely validated.

Figure 7. The self-similar solution (3.21), compared with DNS1 with $t_0=-1.70$ and $m=0.48$ (a) and compared with DNS2 with $t_0=-5$ and $m=0.50$ (b).

Figure 8. The temporal variation of $I_{\theta \theta }$ as defined in (4.4). The dashed lines are the DNS data (note that DNS1 data are shifted up 1.5 times). The solid lines are power-law fits to the DNS data. In DNS1, $I_{\theta \theta } \propto (t+t_0)^{0.08}$; in DNS2, $I_{\theta \theta } \propto (t+t_0)^{0.09}$.

Finally, we verify the new derived turbulent scaling laws for the higher moments of the velocity–scalar and the pressure–scalar terms (3.17)–(3.19) or rather their extended versions (3.23), (3.24) and (3.34), (3.35). The higher moments are indeed known as rather complex parameters in turbulence where their self-similarity behaviour is of particular interest. From (3.17)–(3.19) it is quite apparent that their self-similar behaviour should be naturally obtained based on the exponents and parameters that have been already identified from the second moments and length-scale evolution. For the sake of brevity, therefore, only the results for DNS1 are presented for higher moments. We begin with the self-similarity of ${H}_{\varTheta 2\varTheta }^0({x}_2,t)$, where in (3.23) it was found to scale as $(t+t_0)^{-m-1}$. The exponent $m$ was previously given from the self-similarity of $H_{\varTheta \varTheta }^0({x}_2,t)$, i.e. $m = 0.48$ for DNS1. Using this $m$, the DNS data are fitted to (3.23) and presented in figure 9. It can be seen that the DNS data nicely collapse based on the new scaling laws, while using the previously obtained parameters from fitting the scaling laws to the data. According to (3.18), the third-order moment velocity–scalar vector ${H}_{i2\varTheta }^0({x}_2,t)$ should scale as $(t+t_0)^{n-2}$, while again $n$ is already known. This solution is verified by fitting DNS1 to (3.34) and (3.35) ($i=1,2$ in (3.18)) and presented in figure 10. It is observed that the DNS data and the new scaling laws for the third-order moment velocity–scalar vector show a very good agreement. Finally, we consider the self-similarity of ${\overline {P\varTheta }}^0({x}_2,t)$, where this quantity was also found to scale with $(t+t_0)^{n-2}$. As such, we fit the DNS data to (3.24) and present the self-similar solutions in figure 11. It is seen that the new scaling law for the pressure–scalar term is also adequately verified using the DNS data. While for the higher moments, we did not have to fit any data to get new exponents as the self-similarity only depends on previously derived parameters from the lower-order moments. The very good agreement between DNS and the derived scaling laws even for higher moments confirms the accuracy of the present theoretical approach and the significance of a symmetry method to connect scaling laws of different moments.

Figure 9. The self-similar solution (3.23), compared with DNS1 with $t_0=-1.70$ and $m=0.48$.

Figure 10. The self-similar solutions (3.34) (a) and (3.35) (b), compared with DNS1 with $t_0=-1.70$ and $n=0.55$.

Figure 11. The self-similar solution (3.25), compared with DNS1.

As a final note, we should highlight that while in this study we derived and validated scaling laws for temporally evolving plane jets, we also performed additional analysis to derive the equivalent scaling laws for spatially developing plane jets, presented in detail in Appendix C. This will allow the new scaling laws to be well compared also with the experimental or DNS data of spatially developing jet flows.