3.1 Lie groups of the compressible Euler equation

Euler equations can be obtained from the set of equations (1.1) upon neglecting the effects of viscosity and heat conductivity, that is, by setting $\mu = \kappa = 0$ :

(3.1a) \begin{eqnarray}\frac{\partial \rho}{\partial t} + u_j \frac{\partial \rho}{\partial x_j} &=& - \rho \frac{\partial u_j}{\partial x_j},\qquad\ \ \end{eqnarray}

(3.1b) \begin{eqnarray} \frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} &=& - \frac{1}{\gamma M^2} \frac{1}{\rho} \frac{\partial p}{\partial x_i}, \end{eqnarray}

(3.1c) \begin{eqnarray} \ \ \ \ \ \frac{\partial T}{\partial t} + u_j \frac{\partial T}{\partial x_j} &=& -(\gamma - 1) T \frac{\partial u_k}{\partial x_k}, \end{eqnarray}

(3.1d) \begin{eqnarray} \ \ p &=& \rho T.\end{eqnarray}

For an arbitrary value of the specific heat ratio ( $\gamma$ ), the following are the symmetry transformations of Euler equations [Reference Ibragimov6, Reference Ovsiannikov14]:

1. (1) Time translation symmetry $\left(T_{a_t}\right)$ : The equations (3.1) are invariant, when time is translated by an arbitrary constant, $a_t \in \mathbb{R}$ . The global form of symmetry transformation, $T_{a_t}$ , is

   (3.2a) \begin{equation} T_{a_t}\,:\, \quad t^* = t + a_t, \quad {\boldsymbol{x}}^* = {\boldsymbol{x}}, \quad \rho^* = \rho, \quad {\boldsymbol{u}}^* = {\boldsymbol{u}}, \quad T^* = T, \quad p^* = p \end{equation}
   with the symmetry operator:
   (3.2b) \begin{equation} X = \frac{\partial}{\partial t}. \end{equation}

2. (2) Spatial translation symmetry $\left(T_{a_{x,i}}\right)$ : The three symmetry groups represented by $T_{a_{x,i}},$ $i = 1, 2, 3$ indicate the invariance of the system (3.1) with respect to arbitrary spatial translation by $a_{x,i} \in \mathbb{R}, i = 1,2, 3$ . The global transformation is

   (3.2c) \begin{equation} T_{a_{x,i}}\,: \quad t^* = t, \quad x_i^* = x_i + a_{x,i}, \quad \rho^* = \rho, \quad {\boldsymbol{u}}^*={\boldsymbol{u}}, \quad T^* = T, \quad p^* = p , \end{equation}
   and the symmetry operator is
   (3.2d) \begin{equation} X = \frac{\partial}{\partial x_1}, \quad X = \frac{\partial}{\partial x_2}, \quad X = \frac{\partial}{\partial x_3}, \end{equation}
   for $a_{x,1}, a_{x,2}$ and $a_{x,3}$ , respectively.

3. (3) Galilean symmetry $\left(T_{a_{g,i}}\right)$ : The Euler equations are invariant in a coordinate frame that is moving at constant velocity, $a_{g,i} \in \mathbb{R}, i = 1, 2, 3$ :

   (3.2e) \begin{equation} T_{a_{g,i}}\,: \quad t^* = t, \> x_i^* = x_i + a_{g,i}t, \> \rho^* = \rho, \> u_i^* = u_i + a_{g,i}, \> T^* = T, \> p^* = p , \end{equation}
   with the symmetry operators given by:
   (3.2f) \begin{equation} X = t \frac{\partial}{\partial x_1} + \frac{\partial}{\partial u_1}, \quad X = t \frac{\partial}{\partial x_2} + \frac{\partial}{\partial u_2}, \quad X = X = t \frac{\partial}{\partial x_3} + \frac{\partial}{\partial u_3},\end{equation}
   for $a_{g,1}, a_{g,2}$ and $a_{g,3}$ , respectively.

4. (4) Rotational symmetry $\left(T_{a_{r,i}}\right)$ : This symmetry signifies the invariance of Euler equations (3.1) under rotation, represented by three symmetry groups, $a_{r,i} \in \mathbb{R}, i = 1, 2, 3$ . For example, rotation about $x_1$ axis, written in terms of the group parameter, $a_{r,1}$ , is given by:

   (3.2g) \begin{equation} \begin{aligned} T_{a_{r,1}}\,:\quad &t^* = t, x_2^* = \left(\cos{a_{r,1}}\right) x_2 - \left(\sin{a_{r,1}}\right) x_3, x_3^* = \left(\sin{a_{r,1}}\right) x_2 + \left(\cos{a_{r,1}}\right) x_3, \\ &\rho^* = \rho, u_2^* = \left(\cos{a_{r,1}}\right) u_2 - \left(\sin{a_{r,1}}\right) u_3, u_3^* = \left(\sin{a_{r,1}}\right) u_2 + \left(\cos{a_{r,1}}\right) u_3, \\ &x_1^* = x_1,\quad u_1^* = u_1,\quad T^* = T,\quad p^* = p, \end{aligned}\end{equation}
   with
   (3.2h) \begin{equation} X = x_2 \frac{\partial}{\partial x_3} - x_3 \frac{\partial}{\partial x_2} + u_2 \frac{\partial}{\partial u_3} - u_3 \frac{\partial}{\partial u_2}.\end{equation}
   Similarly, the rotation about $x_2$ axis, represented by $a_{r,2}$ , is given by:
   (3.2i) \begin{equation} \begin{aligned} T_{a_{r,2}}\,:\quad &t^* = t, x_1^* = \left(\cos{a_{r,2}}\right) x_1 - \left(\sin{a_{r,2}}\right) x_3, x_3^* = \left(\sin{a_{r,2}}\right) x_1 + \left(\cos{a_{r,2}}\right) x_3, \\ &\rho^* = \rho, u_1^* = \left(\cos{a_{r,2}}\right) u_1 - \left(\sin{a_{r,2}}\right) u_3, u_3^* = \left(\sin{a_{r,2}}\right) u_1 + \left(\cos{a_{r,2}}\right) u_3, \\ &x_2^* = x_2,\quad u_2^* = u_2,\quad T^* = T,\quad p^* = p, \end{aligned}\end{equation}
   with
   (3.2j) \begin{equation} X = - x_3 \frac{\partial}{\partial x_1} + x_1 \frac{\partial}{\partial x_3} - u_3 \frac{\partial}{\partial u_1} + u_1 \frac{\partial}{\partial u_3}.\end{equation}
   Finally, rotation about $x_3$ axis can be written, using the group parameter $a_{r,3}$ , as:
   (3.2k) \begin{equation} \begin{aligned} T_{a_{r,3}}\,: \quad &t^* = t, x_1^* = \left(\cos{a_{r,3}}\right) x_1 - \left(\sin{a_{r,3}}\right) x_2, x_2^* = \left(\sin{a_{r,3}}\right) x_1 + \left(\cos{a_{r,3}}\right) x_2, \\ &\rho^* = \rho, u_1^* = \left(\cos{a_{r,3}}\right) u_1 - \left(\sin{a_{r,3}}\right) u_2, u_2^* = \left(\sin{a_{r,3}}\right) u_1 + \left(\cos{a_{r,3}}\right) u_2, \\ &x_3^* = x_3,\quad u_3^* = u_3,\quad T^* = T,\quad p^* = p, \end{aligned} \end{equation}
   and the symmetry operator is
   (3.2l) \begin{equation} X = - x_2 \frac{\partial}{\partial x_1} + x_1 \frac{\partial}{\partial x_2} - u_2 \frac{\partial}{\partial u_1} + u_1 \frac{\partial}{\partial u_2}. \end{equation}
   Consequently, the Euler equations are invariant under any linear combination of the above three rotation groups, leaving the equations invariant under arbitrary rotation.

5. (5) Scaling of time ( $T_{a_{s,t}}$ ): When time, velocity, temperature and pressure are scaled as shown below, the Euler equations are invariant:

   (3.2m) \begin{equation} T_{a_{s,t}}\,:\, t^* = e^{a_{s,t}} t, x_i^* = x_i, \rho^* = \rho, u_i^* = e^{-a_{s,t}} u_i, T^* = e^{-2a_{s,t}} T, p^* = e^{-2a_{s,t}} p, \end{equation}
   where, the group parameter, $a_{s,t} \in \mathbb{R}$ , and the corresponding symmetry operator is
   (3.2n) \begin{equation} X = t \frac{\partial}{\partial t} - u_i \frac{\partial}{\partial u_i} - 2T \frac{\partial}{\partial T} - 2p \frac{\partial}{\partial p}. \end{equation}

6. (6) Scaling of space ( $T_{a_{s,x}}$ ): Similarly, the following scaling transformation of space and dependent variables leaves the Euler equations invariant:

   (3.2o) \begin{equation} T_{a_{s,x}}\,:\, t^* = t, x_i^* = e^{a_{s,x}} x_i, \rho^* = \rho, u_i^* = e^{a_{s,x}} u_i, T^* = e^{2a_{s,x}} T, p^* = e^{2a_{s,x}} p, \end{equation}
   with $a_{s,x} \in \mathbb{R}$ . The symmetry operator is given by:
   (3.2p) \begin{equation} X = x_i \frac{\partial}{\partial x_i} + u_i \frac{\partial}{\partial u_i} + 2T \frac{\partial}{\partial T} + 2p \frac{\partial}{\partial p}. \end{equation}

7. (7) Scaling of density $\left(T_{a_{s,\rho}}\right)$ : The scaling symmetry that is unique to compressible flows is the scaling of density and pressure, represented by the group parameter, $a_{s,\rho} \in \mathbb{R}$ :

   (3.2q) \begin{equation} T_{a_{s,\rho}}\,:\quad t^* = t, x_i^* = x_i, \rho^* = e^{a_{s,\rho}} \rho, u_i^* = u_i, T^* = T, p^* = e^{a_{s,\rho}} p \end{equation}
   whose symmetry operator is
   (3.2r) \begin{equation} X = \rho \frac{\partial}{\partial \rho} + p \frac{\partial}{\partial p}. \end{equation}

In addition to the above symmetry groups, monatomic gases, that is, gases with specific heat ratio $\gamma = 5/3$ , satisfy an additional ‘projective’ symmetry transformation $\left(T_{a_{pr}}\right)$ :

(3.3a) \begin{equation}\begin{aligned} T_{a_{pr}}\,: \quad &t^* = \frac{t}{1-a_{pr}t}, x_i^* = \frac{x_i}{1-a_{pr}t}, \rho^* = \rho \!\left(1 - a_{pr}t\right)^3, T^* = T\!\left(1-a_{pr}t\right)^2 \\ &p^* = p \!\left(1-a_{pr}t\right)^5, u_i^* = a_{pr} x_i + u_i \!\left(1 - a_{pr}t\right),\end{aligned}\end{equation}

where, $a_{pr} \in \mathbb{R}$ , and the resulting symmetry operator is

(3.3b) \begin{equation} X = t^2 \frac{\partial}{\partial t} + x_i t \frac{\partial}{\partial x_i} - 3 \rho t \frac{\partial}{\partial \rho} - 2 T t \frac{\partial}{\partial T} - 5 p t \frac{\partial}{\partial p} + \left(x_i - tu_i\right) \frac{\partial}{\partial u_i}.\end{equation}

The projective group is demonstrative of the power of Lie group analysis in identifying the relevant physics exhibited by the differential equations. For more discussion on this group, the reader is referred to Chapter 12 of [Reference Cantwell3].

When viscosity and heat conductivity are considered, the scaling of space and time groups merge into a single scaling group. As a result, in compressible flows with constant viscosity and heat conductivity, there are two scaling groups, as opposed to Euler equations with three scaling groups. In compressible flows, $\mu$ and $\kappa$ are functions of temperature, leading to more symmetry breaking. Therefore, in this paper, we perform symmetry analysis of CF hierarchy upon neglecting the viscosity and heat conduction terms. In turbulent flows, in the limit of zero viscosity ( $\mu \to 0$ ), the assumption is not too restrictive. When $\mu \to 0$ , the action of viscosity is limited to the smallest scales, thus justifying inviscid consideration for the rest of the scales. Furthermore, viscosity and heat conductivity also break projection symmetry.

3.2 Lie group analysis of the CF hierarchy

Let us begin our investigation of the CF hierarchy by looking at the single-point CF equation. The equation for single-point CF, obtained from equation (2.5) by substituting $n=1$ , is given below:

(3.4) \begin{equation} \begin{aligned} \left[\mathrm{i} \frac{\partial}{\partial t} + \frac{\partial^2}{\partial x_j^{(1)} \partial \> {}^{(1)}v_j } \right] \frac{\partial}{\partial {}^{(1)}\varrho} \varphi_1 = & - {}^{(1)}\varrho \left[\frac{\partial }{\partial x_j^{(2)}}\left. \frac{\partial^{3} \varphi_{2}}{\partial \>{}^{(2)}v_j \> \partial \>{}^{(1)}\varrho^2 }\right|_{{}^{(2)}\varrho, {}^{(2)}{\boldsymbol{v}}, {}^{(2)}\theta = 0} \right]_{{\boldsymbol{x}}^{(2)}={\boldsymbol{x}}^{(1)}}\\[4pt] & -\mathrm{i}\frac{{}^{(1)}v_i}{\gamma M^2} \left[ \frac{\partial }{\partial x_i^{(2)}}\left. \frac{\partial^{2} \varphi_{2}}{\partial \>{}^{(2)}\varrho \> \partial \>{}^{(2)}\theta}\right|_{{}^{(2)}\varrho, {}^{(2)}{\boldsymbol{v}}, {}^{(2)}\theta = 0} \right]_{{\boldsymbol{x}}^{(2)}={\boldsymbol{x}}^{(1)}}\\[4pt] & -(\gamma-1) {}^{(k)}\theta \left[ \frac{\partial}{\partial x_j^{(2)}} \left. \frac{\partial^{3} \varphi_{2}}{\partial {}^{(1)}\varrho \> \partial \>{}^{(1)}\theta \> \partial \>{}^{(2)}v_j} \right|_{{}^{(2)}\varrho, {}^{(2)}{\boldsymbol{v}}, {}^{(2)}\theta = 0} \right]_{{\boldsymbol{x}}^{(2)}={\boldsymbol{x}}^{(1)}}. \end{aligned}\end{equation}

The advantage of using the CF formulation, over the conventional PDF formulation [Reference Praturi, Plümacher and Oberlack16], can easily be seen in the above equation. The equation (3.4) is local, where as the PDF hierarchy is non-local as it includes integral terms. In equation (3.4), $\varphi_1$ and $\varphi_2$ are the dependent variables and $t, {\boldsymbol{x}}^{(1)}, {\boldsymbol{x}}^{(2)}, {}^{(1)}\varrho, {}^{(2)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(2)}{\boldsymbol{v}}, {}^{(1)}\theta$ and ${}^{(2)}\theta$ are the independent variables.

To compute the symmetries, we make use of one of the infinite classes of the ‘generalised form of the symmetry operators’ that are equivalent to Lie point groups [Reference Olver13]. We specifically consider the ‘evolutionary’ form of the generalised symmetry operators in which the infinitesimals for the independent variables are zero. This precludes us from considering the transformation of independent variables explicitly. Specifically, the infinitesimal transformation of the CFs, $\varphi_1^*$ and $\varphi_2^*$ , are represented as:

(3.5a) \begin{equation} \varphi_1^* = \varphi_1 + a \tilde{\eta}^{\varphi_1} + \mathcal{O}\!\left(a^2\right), \quad \varphi_2^* = \varphi_2 + a \tilde{\eta}^{\varphi_2} + \mathcal{O}\!\left(a^2\right),\end{equation}

where, a is the group parameter and,

(3.5b) \begin{equation}\begin{aligned} \tilde{\eta}^{\varphi_1} =\ &\eta^{\varphi_1} - \xi^t \frac{\partial \varphi_1}{\partial t} - \xi^{x^{(1)}_1} \frac{\partial \varphi_1}{\partial x^{(1)}_1} - \xi^{x^{(1)}_2} \frac{\partial \varphi_1}{\partial x^{(1)}_2} - \xi^{x^{(1)}_3} \frac{\partial \varphi_1}{\partial x^{(1)}_3} - \xi^{{}^{(1)}\varrho} \frac{\partial \varphi_1}{\partial \> {}^{(1)}\varrho} \\ &- \xi^{{}^{(1)}v_1} \frac{\partial \varphi_1}{\partial \> {}^{(1)}v_1} - \xi^{{}^{(1)}v_2} \frac{\partial \varphi_1}{\partial \> {}^{(1)}v_2} - \xi^{{}^{(1)}v_3} \frac{\partial \varphi_1}{\partial \> {}^{(1)}v_3} - \xi^{{}^{(1)}\theta} \frac{\partial \varphi_1}{\partial \> {}^{(1)}\theta},\end{aligned}\end{equation}

and

(3.5c) \begin{equation}\begin{aligned} \tilde{\eta}^{\varphi_2} =\ & \eta^{\varphi_2} - \xi^t \frac{\partial \varphi_2}{\partial t} - \xi^{x^{(1)}_1} \frac{\partial \varphi_2}{\partial x^{(1)}_1} - \xi^{x^{(1)}_2} \frac{\partial \varphi_2}{\partial x^{(1)}_2} - \xi^{x^{(1)}_3} \frac{\partial \varphi_2}{\partial x^{(1)}_3} - \xi^{{}^{(1)}\varrho} \frac{\partial \varphi_2}{\partial \> {}^{(1)}\varrho} \\ &- \xi^{{}^{(1)}v_1} \frac{\partial \varphi_2}{\partial \> {}^{(1)}v_1} - \xi^{{}^{(1)}v_2} \frac{\partial \varphi_2}{\partial \> {}^{(1)}v_2} - \xi^{{}^{(1)}v_3} \frac{\partial \varphi_2}{\partial \> {}^{(1)}v_3} - \xi^{{}^{(1)}\theta} \frac{\partial \varphi_2}{\partial \> {}^{(1)}\theta} \\ &- \xi^{x^{(2)}_1} \frac{\partial \varphi_2}{\partial x^{(2)}_1} - \xi^{x^{(2)}_2} \frac{\partial \varphi_2}{\partial x^{(2)}_2} - \xi^{x^{(2)}_3} \frac{\partial \varphi_2}{\partial x^{(2)}_3} - \xi^{{}^{(2)}\varrho} \frac{\partial \varphi_2}{\partial \> {}^{(2)}\varrho} \\ &- \xi^{{}^{(2)}v_1} \frac{\partial \varphi_2}{\partial \> {}^{(2)}v_1} - \xi^{{}^{(2)}v_2} \frac{\partial \varphi_2}{\partial \> {}^{(2)}v_2} - \xi^{{}^{(2)}v_3} \frac{\partial \varphi_2}{\partial \> {}^{(2)}v_3} - \xi^{{}^{(2)}\theta} \frac{\partial \varphi_2}{\partial \> {}^{(2)}\theta}\end{aligned}\end{equation}

are the characteristics of the symmetry transformation [Reference Olver13]. Here, $\eta^{\varphi_1}$ and $\eta^{\varphi_2}$ are the infinitesimals for $\varphi_1, \varphi_2$ ; and $\xi^t$ , $\xi^{x^{(k)}_i}, \xi^{{}^{(k)}\varrho}, \xi^{{}^{(k)}v_i}$ and $\xi^{{}^{(k)}\theta}$ are the infinitesimals for time, space and spectral parameters for density, velocity and temperature, respectively; and $i=1-3$ , $k = 1, 2$ . It should be noted that all of the infinitesimals, $\eta^{\varphi_1}, \eta^{\varphi_2}, \xi^t, \xi^{x^{(k)}_i}, \xi^{{}^{(k)}\varrho}, \xi^{{}^{(k)}v_i}$ and $\xi^{{}^{(k)}\theta}$ are explicit functions of $t, x_i$ and the dependent variables. This implies that we do not consider the infinitesimals to depend on the derivatives of the dependent variables, $\varphi_n$ , for all n. In the evolutionary form of generalised symmetries, the derivatives of $\varphi_1$ and $\varphi_2$ transform as:

(3.6a) \begin{equation} \left.\frac{\partial \varphi_1}{\partial t} \right|^* = \frac{\partial \varphi_1}{\partial t} + a \frac{D}{D t} \tilde{\eta}^{\varphi_1} + \mathcal{O}\!\left(a^2\right),\end{equation}

and

(3.6b) \begin{equation} \left.\frac{\partial^2 \varphi_1}{\partial {}^{(1)}\varrho \partial t} \right|^* = \frac{\partial^2 \varphi_1}{\partial \> {}^{(1)}\varrho \>\partial t} + a \frac{D}{D t} \frac{D}{D {}^{(1)}\varrho} \tilde{\eta}^{\varphi_1} + \mathcal{O}\!\left(a^2\right),\end{equation}

etc. This implies that the infinitesimals for the derivatives of the dependent variables (say time derivative of $\varphi_1$ ) in the evolutionary form of the generalised symmetry formulation can be obtained by taking the total derivatives (here, with respect to time) of the infinitesimal of the dependent variable ( $\tilde{\eta}^{\varphi_1}$ ). As the CF equations contain high-order derivatives (the order of the highest derivative increases as the order of CF is increased), the handling of derivatives is simplified in the evolutionary form of symmetry operators. This operator has also been used to derive the symmetries of integro-differential equations [Reference Wacławczyk, Grebenev and Oberlack20], as the limits on the integral do not undergo any transformation.

To compute the symmetries of CF equation (3.4), we apply the evolutionary form of the symmetry operator with the infinitesimals as defined above. Thus, the obtained determining equation is of the form:

(3.7) \begin{equation} \begin{aligned} \mathrm{i} \frac{D^2 \tilde{\eta}^{\varphi_1}}{D t D \> {}^{(1)}\varrho} &+ \frac{D^3 \tilde{\eta}^{\varphi_1}}{D \>{}^{(1)}v_j D x^{(1)}_j D \> {}^{(1)}\varrho} = - {}^{(1)}\varrho \left[ \frac{D}{D x^{(2)}_j} \!\left( \frac{D^3 \tilde{\eta}^{\varphi_2}}{D \> {}^{(2)}v_j D \> {}^{(1)}\varrho^2} \right)_{{}^{(2)}\varrho, {}^{(2)}{\boldsymbol{v}}, {}^{(2)}\theta = 0} \right]_{{\boldsymbol{x}}^{(2)}={\boldsymbol{x}}^{(1)}} \\[4pt] &- \mathrm{i}\frac{{}^{(1)}v_i}{\gamma M^2} \left[ \frac{D}{D x^{(2)}_i} \!\left( \frac{D^2 \tilde{\eta}^{\varphi_2}}{D \> {}^{(2)}\varrho D \> {}^{(2)}\theta} \right)_{{}^{(2)}\varrho, {}^{(2)}{\boldsymbol{v}}, {}^{(2)}\theta = 0} \right]_{{\boldsymbol{x}}^{(2)}={\boldsymbol{x}}^{(1)}} \\[4pt] &- {}^{(1)}\theta (\gamma - 1) \left[ \frac{D}{D x^{(2)}_j} \!\left( \frac{D^3 \tilde{\eta}^{\varphi_2}}{D \> {}^{(2)}v_j D \> {}^{(1)}\varrho D \> {}^{(1)}\theta} \right)_{{}^{(2)}\varrho, {}^{(2)}{\boldsymbol{v}}, {}^{(2)}\theta = 0} \right]_{{\boldsymbol{x}}^{(2)}={\boldsymbol{x}}^{(1)}}. \end{aligned}\end{equation}

The point evaluations such as ${}^{(2)}\varrho, {}^{(2)}{\boldsymbol{v}}, {}^{(2)}\theta = 0$ and ${\boldsymbol{x}}^{(2)}={\boldsymbol{x}}^{(1)}$ do not need to be transformed as we utilise the evolutionary form of the symmetry operator. Similarly, the determining equation for the governing equation of two-point CF can also be derived.

Finally, upon utilising the computer algebra system, Maple [Reference Maple10], for part of the symmetry calculations, the following are the infinitesimals of the symmetries admitted by the single-point CF equation (3.4):

(3.8a) \begin{eqnarray} \xi^t &=& \frac{a_{pr}}{2} t^2 + a_{s,t} t + a_t,\qquad\qquad\qquad\qquad\qquad\qquad\ \ \end{eqnarray}

(3.8b) \begin{eqnarray} \xi^{x^{(1)}_1} &=& \frac{a_{pr}}{2} x^{(1)}_1 t + a_{s,x} x^{(1)}_1 + a_{g,1} t - a_{r,3} x^{(1)}_2 - a_{r,2} x^{(1)}_3 + a_{x,1}, \end{eqnarray}

(3.8c) \begin{eqnarray} \xi^{x^{(1)}_2} &=& \frac{a_{pr}}{2} x^{(1)}_2 t + a_{s,x} x^{(1)}_2 + a_{g,2} t - a_{r,1} x^{(1)}_3 + a_{r,3} x^{(1)}_1 + a_{x,2}, \end{eqnarray}

(3.8d) \begin{eqnarray} \xi^{x^{(1)}_3} &=& \frac{a_{pr}}{2} x^{(1)}_3 t + a_{s,x} x^{(1)}_3 + a_{g,3} t + a_{r,2} x^{(1)}_1 + a_{r,1} x^{(1)}_2 + a_{x,3}, \end{eqnarray}

(3.8e) \begin{eqnarray} \xi^{{}^{(1)}\varrho} &=& \frac{a_{pr}}{\gamma - 1} {}^{(1)}\varrho t \>- \> a_{s, \rho} {}^{(1)}\varrho, \end{eqnarray}

(3.8f) \begin{eqnarray} \xi^{{}^{(1)}v_1} &=& \frac{a_{pr}}{2} {}^{(1)}v_1 t - a_{s,x} {}^{(1)}v_1 + a_{s,t} {}^{(1)}v_1 - a_{r,3} {}^{(1)}v_2 - a_{r,2} {}^{(1)}v_3, \end{eqnarray}

(3.8g) \begin{eqnarray} \xi^{{}^{(1)}v_2} &=& \frac{a_{pr}}{2} {}^{(1)}v_2 t - a_{s,x} {}^{(1)}v_2 + a_{s,t} {}^{(1)}v_2 - a_{r,1} {}^{(1)}v_3 + a_{r,3} {}^{(1)}v_1, \end{eqnarray}

(3.8h) \begin{eqnarray} \xi^{{}^{(1)}v_3} &=& \frac{a_{pr}}{2} {}^{(1)}v_3 t - a_{s,x} {}^{(1)}v_3 + a_{s,t} {}^{(1)}v_3 + a_{r,2} {}^{(1)}v_1 + a_{r,1} {}^{(1)}v_2, \end{eqnarray}

(3.8i) \begin{eqnarray} \xi^{{}^{(1)}\theta} &=& a_{pr} {}^{(1)}\theta t + 2 (a_{s,t} - a_{s,x}) {}^{(1)}\theta, \end{eqnarray}

and

(3.9) \begin{equation}\begin{aligned} \eta^{\varphi_1} = & F_1\!\left(t, {\boldsymbol{x}}^{(1)}, {}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta\right) + \mathrm{i} \varphi_1 \bigg[\frac{a_{pr}}{2} {}^{(1)}v_i x^{(1)}_i + a_{g,1} {}^{(1)}v_1 + a_{g,2} {}^{(1)}v_2 \\ &\left. + a_{g,3} {}^{(1)}v_3 + \frac{a_{\varphi}}{\mathrm{i}} + \frac{5-3\gamma}{2\mathrm{i} (\gamma - 1)} a_{pr} t \right].\end{aligned}\end{equation}

In the above infinitesimals for independent (3.8) and dependent (3.9) variables, it can be seen that the Euler equation symmetries are replicated. These are represented by the same group parameters as Section 3.1, that is, $a_t$ for time translation, $a_{x,i}$ for space translation, $a_{g,i}$ for Galilean symmetry, $a_{s,t}$ for scaling of time, $a_{s,x}$ for scaling of space and $a_{s,\rho}$ for scaling of density and $a_{pr}$ for projective symmetries. We also notice two additional groups, denoted by $F_1$ and $a_{\varphi}$ , that appear in the statistical CF equations. These are as a result named, ‘statistical symmetries’, a convention that is adopted in incompressible turbulence as well [Reference Oberlack and Rosteck12, Reference Wacławczyk, Staffolani, Oberlack, Rosteck, Wilczek and Friedrich21]. The function $F_1$ appears in the infinitesimal, $\eta^{\varphi_1}$ due to principle of superposition, as single-point CF equation (3.4) is a linear differential equation. $F_1$ is an arbitrary function of the independent variables (in the CF formulation), which could lead to a translation type of symmetry in the single-point CF, $\varphi_1$ . On the other hand, $a_{\varphi}$ is a dilation/scaling symmetry group, which will lead to the scaling of $\varphi_1$ .

It is interesting to note that the single-point CF equation (3.4) admits the projection symmetry group for all values of $\gamma$ . However, as seen in the previous section, the CFs, including the transformed ones ( $\varphi_1^*$ ), need to satisfy additional constraints (2.6a)–(2.9). In the following section, the side conditions are enforced on the transformed CFs ( $\varphi_n^*$ ) by imposing restrictions on their infinitesimals ( $\eta^{\varphi_n}$ ).

3.2.1 Imposing side conditions on the infinitesimals

Let us now consider the restrictions on the infinitesimals that are imposed as a result of side conditions. We apply the evolutionary form of the generalised symmetry operator on the reduction side condition (2.6b). This results in:

(3.10a) \begin{equation} \left.\tilde{\eta}^{\varphi_1}\right|_{{}^{(1)}\varrho = 0, {}^{(1)}{\boldsymbol{v}}=\textbf{0}, {}^{(1)}\theta = 0} = 0,\end{equation}

implying,

(3.10b) \begin{equation} \begin{aligned} &\left[ \eta^{\varphi_1} - \xi^t \frac{\partial \varphi_1}{\partial t} - \xi^{x^{(1)}_1} \frac{\partial \varphi_1}{\partial x^{(1)}_1} - \xi^{x^{(1)}_2} \frac{\partial \varphi_1}{\partial x^{(1)}_2} - \xi^{x^{(1)}_3} \frac{\partial \varphi_1}{\partial x^{(1)}_3} - \xi^{{}^{(1)}\varrho} \frac{\partial \varphi_1}{\partial \> {}^{(1)}\varrho} \right. \\ &\qquad \left.- \xi^{{}^{(1)}v_1} \frac{\partial \varphi_1}{\partial \> {}^{(1)}v_1} - \xi^{{}^{(1)}v_2} \frac{\partial \varphi_1}{\partial \> {}^{(1)}v_2} - \xi^{{}^{(1)}v_3} \frac{\partial \varphi_1}{\partial \> {}^{(1)}v_3} - \xi^{{}^{(1)}\theta} \frac{\partial \varphi_1}{\partial \> {}^{(1)}\theta} \right]_{{}^{(1)}\varrho = 0, {}^{(1)}{\boldsymbol{v}}=\textbf{0}, {}^{(1)}\theta = 0} = 0 \end{aligned}\end{equation}

that is,

(3.10c) \begin{equation}\left[ \eta^{\varphi_1} - \mathrm{i} \!\left( \left\langle \rho \right\rangle \xi^{{}^{(1)}\varrho} + \left\langle u_1 \right\rangle \xi^{{}^{(1)}v_1} + \left\langle u_2 \right\rangle \xi^{{}^{(1)}v_2}+ \left\langle u_3 \right\rangle \xi^{{}^{(1)}v_3} + \left\langle T \right\rangle \xi^{{}^{(1)}\theta} \right) \right]_{{}^{(1)}\varrho = 0, {}^{(1)}{\boldsymbol{v}}=\textbf{0}, {}^{(1)}\theta = 0} = 0.\end{equation}

As $\eta^{\varphi_1}, \xi^{{}^{(1)}\varrho}, \xi^{{}^{(1)}v_i}$ and $\xi^{{}^{(1)}\theta}$ do not depend on $\langle \rho \rangle, \langle {\boldsymbol{u}} \rangle$ and $\langle T \rangle$ , this implies

(3.11) \begin{equation}\begin{aligned} &\left. \eta^{\varphi_1} \right|_{{}^{(1)}\varrho={}^{(1)}{\boldsymbol{v}} = {}^{(1)}\theta = 0} = 0, \quad \left. \xi^{{}^{(1)}\varrho} \right|_{{}^{(1)}\varrho = 0, {}^{(1)}{\boldsymbol{v}}=\textbf{0}, {}^{(1)}\theta = 0} = 0, \\ &\left. \xi^{{}^{(1)}v_i} \right|_{{}^{(1)}\varrho = 0, {}^{(1)}{\boldsymbol{v}}=\textbf{0}, {}^{(1)}\theta = 0} = 0, \quad \left. \xi^{{}^{(1)}\theta} \right|_{{}^{(1)}\varrho = 0, {}^{(1)}{\boldsymbol{v}}=\textbf{0}, {}^{(1)}\theta = 0} = 0, \quad i = 1 - 3.\end{aligned}\end{equation}

Applying similar techniques on the coincidence condition (2.7), we obtain

(3.12) \begin{equation} \lim_{|{\boldsymbol{x}}^{(n+1)}-{\boldsymbol{x}}^{(n)}|\to 0} \tilde{\eta}^{\varphi_{n+1}} = \tilde{\eta}^{\varphi_n} \!\left(.., {}^{(n+1)}\varrho+{}^{(n)}\varrho, {}^{(n+1)}{\boldsymbol{v}}+{}^{(n)}{\boldsymbol{v}}, {}^{(n+1)}\theta+{}^{(n)}\theta, ... \right).\end{equation}

Substituting the expressions for $\tilde{\eta}^{\varphi_n}$ and $\tilde{\eta}^{\varphi_{n+1}}$ in the above equation (2.7) in terms of $\eta^{\varphi_n}, \xi^t, \xi^{x^{(k)}_i}, \xi^{{}^{(k)}\varrho}, \xi^{{}^{(k)}v_i}$ and $\xi^{{}^{(k)}\theta}$ , we obtain:

(3.13) \begin{equation}\begin{aligned} \lim_{|{\boldsymbol{x}}^{(n+1)}-{\boldsymbol{x}}^{(n)}|\to 0} &\eta^{\varphi_{n+1}} \!\left(t, {\boldsymbol{x}}^{(1)}, {}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta, ..., {\boldsymbol{x}}^{(n+1)}, {}^{(n+1)}\varrho, {}^{(n+1)}{\boldsymbol{v}}, {}^{(n+1)}\theta, ...\right) \\ &= \eta^{\varphi_n} \!\left(.., {}^{(n+1)}\varrho+{}^{(n)}\varrho, {}^{(n+1)}{\boldsymbol{v}}+{}^{(n)}{\boldsymbol{v}}, {}^{(n+1)}\theta+{}^{(n)}\theta, ... \right).\end{aligned}\end{equation}

The permutation side condition (2.8) results in:

(3.14) \begin{equation}\begin{aligned} \tilde{\eta}^{\varphi_n} &\left( ..., {}^{(l)}\varrho, {}^{(l)}{\boldsymbol{v}}, {}^{(l)}\theta, ..., {}^{(k)}\varrho, {}^{(k)}{\boldsymbol{v}}, {}^{(k)}\theta, ... \right) = \tilde{\eta}^{\varphi_n} \!\left( ..., {}^{(k)}\varrho, {}^{(k)}{\boldsymbol{v}}, {}^{(k)}\theta, ..., {}^{(l)}\varrho, {}^{(l)}{\boldsymbol{v}}, {}^{(l)}\theta, ... \right),\end{aligned}\end{equation}

implying that $\tilde{\eta}^{\varphi_n}$ does not change regardless of order in which spatial locations and their corresponding spectral parameters are considered. The above equation, upon simplification, results in:

(3.15) \begin{equation}\begin{aligned} \eta^{\varphi_n} &\left( ..., {}^{(l)}\varrho, {}^{(l)}{\boldsymbol{v}}, {}^{(l)}\theta, ..., {}^{(k)}\varrho, {}^{(k)}{\boldsymbol{v}}, {}^{(k)}\theta, ... \right) = \eta^{\varphi_n} \!\left( ..., {}^{(k)}\varrho, {}^{(k)}{\boldsymbol{v}}, {}^{(k)}\theta, ..., {}^{(l)}\varrho, {}^{(l)}{\boldsymbol{v}}, {}^{(l)}\theta, ... \right).\end{aligned}\end{equation}

The separation side condition (2.9) leads to the following restrictions on the infinitesimals:

(3.16) \begin{equation} \lim_{|{\boldsymbol{x}}^{(n+1)}-{\boldsymbol{x}}^{(k)}|\to \infty} \tilde{\eta}^{\varphi_{n+1}} = \varphi_1 \tilde{\eta}^{\varphi_n} + \varphi_n \tilde{\eta}^{\varphi_1}.\end{equation}

The separation side condition is not applied in the previous studies on incompressible turbulence [Reference Oberlack and Rosteck12, Reference Wacławczyk, Staffolani, Oberlack, Rosteck, Wilczek and Friedrich21], and the current study. We consider a arbitrarily large domain that is not infinite in extent, in this study. As a result, the above separation condition does not need to be strictly enforced. In addition, in most practical turbulence applications, where scaling laws are derived using symmetry principles, infinite domains are not considered.

Let us now impose the reduction side condition (3.11) on the infinitesimals in (3.8) and (3.9). It is seen that the infinitesimals $\xi^{{}^{(1)}\varrho}, \xi^{{}^{(1)}v_i}$ and $\xi^{{}^{(1)}\theta}$ automatically satisfy the reduction side condition (3.11), as $\xi^{{}^{(1)}\varrho} = \xi^{{}^{(1)}v_i} = \xi^{{}^{(1)}\theta} = 0$ when ${}^{(1)}\varrho = {}^{(1)}{\boldsymbol{v}} = {}^{(1)}\theta = 0$ . Hence, we turn our attention to $\eta^{\varphi_1}$ :

(3.17) \begin{equation} \left. \eta^{\varphi_1} \right|_{{}^{(1)}\varrho = 0, {}^{(1)}{\boldsymbol{v}}=\textbf{0}, {}^{(1)}\theta = 0} = 0 \Rightarrow F_1 \!\left(t, {\boldsymbol{x}}^{(1)}, 0, \textbf{0}, 0\right) + \mathrm{i} \left[ \frac{a_{\varphi}}{\mathrm{i}} + \frac{5- 3\gamma}{2\mathrm{i}(\gamma - 1)} a_{pr} t \right] = 0.\end{equation}

It can be seen that the only symmetry groups that remain after imposing ${}^{(1)}\varrho = {}^{(1)}v_i = {}^{(1)}\theta = 0$ are the projection ( $a_{pr}$ ) group, superposition group ( $F_1$ ) and the dilation group ( $a_{\varphi}$ ). In the above equation, we consider one group at a time, as they are single-parameter symmetry transformations. We begin with the projective group, $a_{pr}$ :

(3.18) \begin{equation} \left. \eta^{\varphi_1} \right|_{{}^{(1)}\varrho = 0, {}^{(1)}{\boldsymbol{v}}=\textbf{0}, {}^{(1)}\theta = 0} = 0 \Rightarrow \mathrm{i} \left[ \frac{5 - 3\gamma}{2 \mathrm{i} (\gamma - 1)} a_{pr} t \right] = 0 \Rightarrow \gamma = \frac{5}{3},\end{equation}

implying that the reduction side condition is satisfied by the projective group only for the case of monatomic gases. This is consistent with what is seen in the Euler equations. In CF formulation, however, the restriction to monatomic gases is imposed by the reduction side condition. The imposition of reduction side condition on the $F_1$ group results in

(3.19) \begin{equation}F_1 \!\left(t, {\boldsymbol{x}}^{(1)}, {}^{(1)}\varrho = 0, {}^{(1)}{\boldsymbol{v}} = \textbf{0}, {}^{(1)}\theta = 0\right) = 0,\end{equation}

implying that $F_1$ must reduce to zero, when spectral parameters are zero. This also implies that $F_1$ cannot be a constant (other than zero), as it would violate reduction condition. Finally, the reduction condition on the dilation group, $a_{\varphi}$ , leads to the restriction that $a_{\varphi} = 0$ . However, one could conceive the following transformation for $\varphi_1$ , by combining $F_1$ and $a_{\varphi}$ groups:

(3.20) \begin{equation} \eta^{\varphi_1} = - a_{\varphi} + \varphi_1 a_{\varphi}.\end{equation}

In this case, we consider $F_1 = - a_{\varphi}$ and as a result, the reduction condition is automatically satisfied. The groups, $a_{\varphi}$ and $F_1 = -a_{\varphi}$ (as seen above, that $F_1$ cannot be a constant), each considered separately do not satisfy the reduction side condition. However, when considered together as shown in equation (3.20), the reduction constraint is upheld. The final form for the single-point CF infinitesimal is

(3.21) \begin{align} \eta^{\varphi_1} = F_1\!\left(t, {\boldsymbol{x}}^{(1)}, {}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta\right) - a_{\varphi} + \mathrm{i} \varphi_1 \left[\frac{a_{pr}}{2} {}^{(1)}v_i x^{(1)}_i + a_{g,1} {}^{(1)}v_1 + a_{g,2} {}^{(1)}v_2 + a_{g,3} {}^{(1)}v_3 + \frac{a_{\varphi}}{\mathrm{i}} \right] ,\nonumber\\[5pt]\end{align}

Similarly, utilising MAPLE [Reference Maple10] for part of the analysis of the two-point CF (upon imposing the reduction side condition) results in:

(3.22) \begin{align} \eta^{\varphi_2} =\, & F_2\!\left(t, {\boldsymbol{x}}^{(1)}, {\boldsymbol{x}}^{(2)}, {}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta, {}^{(2)}\varrho, {}^{(2)}{\boldsymbol{v}}, {}^{(2)}\theta\right) - a_{\varphi}\nonumber\\[4pt]& + \mathrm{i} \varphi_2 \left[ \frac{a_{\varphi}}{\mathrm{i}} + \frac{a_{pr}}{2} \!\left( {}^{(1)}v_i x^{(1)}_i + {}^{(2)}v_i x^{(2}_i \right) + a_{g,1} \!\left( {}^{(1)}v_1 + {}^{(2)}v_1 \right)\right. \\[4pt] & + a_{g,2} \!\left( {}^{(1)}v_2 + {}^{(2)}v_2 \right) + a_{g,3} \!\left( {}^{(1)}v_3 + {}^{(2)}v_3 \right) \Big].\nonumber\end{align}

Finally, using one- and two-point CF infinitesimals, we generalise the expression for an $n-$ point CF infinitesimal, which is given by:

(3.23) \begin{equation} \begin{aligned} \eta^{\varphi_n} =\, & F_n\!\left(t, {\boldsymbol{x}}^{(1)},..., {\boldsymbol{x}}^{(n)}, {}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta,..., {}^{(n)}\varrho, {}^{(n)}{\boldsymbol{v}}, {}^{(n)}\theta\right) - a_{\varphi} \\[4pt] & + \mathrm{i} \varphi_n \left[ \frac{a_{\varphi}}{\mathrm{i}} +\frac{a_{pr}}{2} \sum_{k=1}^n {}^{(k)}v_i x^{(k)}_i+ a_{g,1} \sum_{k=1}^n {}^{(k)}v_1 + a_{g,2} \sum_{k=1}^n {}^{(k)}v_2 + a_{g,3} \sum_{k=1}^n {}^{(k)}v_3 \right]. \end{aligned}\end{equation}

In the above equations (3.22) and (3.23), the functions $F_2$ and $F_n$ appear as a result of the superposition principle. It is important to note that in equations (3.8), (3.21), (3.22) and (3.23), $a_{pr}$ is non-zero only for monatomic gases, that is, when $\gamma \neq 5/3$ , $a_{pr} = 0$ . All the other Euler symmetry groups uphold the side conditions without any restrictions. We impose the coincidence (2.7) and permutation (2.8) side conditions on statistical groups in the next section.

4.1 Projective group

We begin with the projective transformation represented by the group parameter, $a_{pr}$ . This transformation is a symmetry of CF hierarchy only for monatomic gases, $\gamma = 5/3$ . Under this group, the infinitesimals for the dependent and independent variables are given by:

(4.1) \begin{equation} \begin{aligned} &\xi^t = t^2, \quad \xi^{x^{(1)}_1} = x^{(1)}_1 t , \quad \xi^{x^{(1)}_2} = x^{(1)}_2 t , \quad \xi^{x^{(1)}_3} = x^{(1)}_3 t, \quad \xi^{{}^{(1)}\varrho} = 3{}^{(1)}\varrho t, \\ &\xi^{{}^{(1)}v_1} = {}^{(1)}v_1 t, \quad \xi^{{}^{(1)}v_2} = {}^{(1)}v_2 t, \quad \xi^{{}^{(1)}v_3} = {}^{(1)}v_3 t, \quad \xi^{{}^{(1)}\theta} = 2 \> {}^{(1)}\theta t \\ &\eta^{\varphi_1} = \varphi_1 \left[ \mathrm{i} x^{(1)}_i \> {}^{(1)}v_i \right]. \end{aligned}\end{equation}

Below is the global transformation, that is obtained utilising Lie’s first theorem [Reference Bluman and Kumei2]:

(4.2a) \begin{equation} t^* = \frac{t}{1-a_{pr} t}, \qquad x^{(1)^*}_i = \frac{x^{(1)}_i}{1-a_{pr} t},\end{equation}

(4.2b) \begin{equation} {}^{(1)}\varrho^* = \frac{{}^{(1)}\varrho}{\left(1-a_{pr} t\right)^3}, \qquad {}^{(1)}\theta^* = \frac{{}^{(1)}\theta}{\left(1-a_{pr} t\right)^2}, \qquad {}^{(1)}v_i^* = \frac{{}^{(1)}v_i}{1-a_{pr} t}\end{equation}

(4.2c) \begin{equation} \varphi_1^* = \varphi_1 \exp \!\left( \mathrm{i} a_{pr} \frac{ {}^{(1)}v_i x^{(1)}_i}{1-a_{pr} t} \right).\end{equation}

For a general multi-point CF, the infinitesimal and the global transformation take the form:

(4.3) \begin{equation} \eta^{\varphi_n} = \mathrm{i} \varphi_n \left[ x^{(1)}_i \> {}^{(1)}v_i + .. + x^{(n)}_i \> {}^{(n)}v_i \right] , \varphi_n^* = \varphi_n \exp \!\left(\mathrm{i} a_{pr} \frac{ {}^{(1)}v_i x^{(1)}_i + ... + {}^{(n)}v_i x^{(n)}_i}{1-a_{pr} t} \right).\end{equation}

4.2 Dilation groups

Let us now investigate the dilation group, $a_{s,t}$ , also known as ‘scaling in time’. The non-trivial infinitesimals resulting from this symmetry transformation are

(4.4a) \begin{equation} \xi^t = t, \quad \xi^{{}^{(1)}v_i} = {}^{(1)}v_i, \quad \xi^{{}^{(1)}\theta} = 2 \> {}^{(1)}\theta.\end{equation}

All other dependent and independent variables do not transform under this symmetry group. The global form of the transformation is given by:

(4.4b) \begin{equation} t^* = e^{a_{s,t}} t, \quad {}^{(1)}v_i^* = e^{a_{s,t}} \> {}^{(1)}v_i , \quad {}^{(1)}\theta^* = e^{2 a_{s,t}} \> {}^{(1)}\theta.\end{equation}

For ‘scaling in space’ group, $a_{s,x}$ , the infinitesimals and global form of the symmetry transformation are given below:

(4.5a) \begin{equation} \xi^{x^{(1)}}_i = x^{(1)}_i, \quad \xi^{{}^{(1)}v_i} = -{}^{(1)}v_i, \quad \xi^{{}^{(1)}\theta} = -2 \> {}^{(1)}\theta.\end{equation}

(4.5b) \begin{equation} x^{(1)^*}_i = e^{a_{s,x}} x^{(1)}_i, \quad {}^{(1)}v_i^* = e^{-a_{s,x}} \> {}^{(1)}v_i , \quad {}^{(1)}\theta^* = e^{-2 a_{s,x}} \> {}^{(1)}\theta.\end{equation}

The third dilation group that exists only in compressible fluids is ‘scaling of density’, represented by the parameter, $a_{s,\rho}$ . The only non-trivial infinitesimal in the CF formulation is that of density spectral parameter, that is, $\xi^{{}^{(1)}\varrho}$ . The expressions for the infinitesimal and the corresponding global transformation are given below:

(4.6) \begin{equation} \xi^{{}^{(1)}\varrho} = -{}^{(1)}\varrho, \qquad {}^{(1)}\varrho^* = e^{-a_{s,\rho}} \> {}^{(1)}\varrho.\end{equation}

4.3 Galilean invariance

For the case of the Galilean invariance groups, $a_{g,\alpha}$ , where $\alpha=1 - 3$ , the non-zero infinitesimals are

(4.7) \begin{equation} \xi^{x^{(1)}_{\alpha}} = t, \qquad \eta^{\varphi_1} = \mathrm{i} {}^{(1)}v_{\alpha} \varphi_1.\end{equation}

The global transformation takes the form:

(4.8) \begin{equation} x^{(1)*}_{\alpha} = x^{(1)}_{\alpha} + a_{g,\alpha} t, \quad \varphi_1^* = e^{\mathrm{i} \> {}^{(1)}v_{\alpha} a_{g, \alpha}} \> \varphi_1.\end{equation}

In the above equation, repeated $\alpha$ indices do not imply summation.

4.4 Statistical groups

We now turn our attention to the new symmetry groups in the CF formulation that are not seen in the Euler equations. We begin with the function, $F_1$ , seen in the $\eta^{\varphi_1}$ equation (3.21):

(4.9) \begin{equation} \eta^{\varphi_1} = F_1\!\left(t, {\boldsymbol{x}}^{(1)}, {}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta\right),\end{equation}

and all other infinitesimals are trivial in nature. As a result, the global form of this symmetry takes the form:

(4.10) \begin{equation} \varphi_1^* = \varphi_1 + a F_1\!\left(t, {\boldsymbol{x}}^{(1)}, {}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta\right).\end{equation}

For the multi-point CF, the global transformation due to superposition principle (from equation (3.23)), takes the form:

(4.11) \begin{equation} \varphi_n^* = \varphi_n + a F_n \!\left(t, {\boldsymbol{x}}^{(1)}, {}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta, ... , {\boldsymbol{x}}^{(n)}, {}^{(n)}\varrho, {}^{(n)}{\boldsymbol{v}}, {}^{(n)}\theta\right).\end{equation}

To establish how the infinite hierarchy of functions $F_1, ..., F_n$ are related to one another, we impose the reduction side condition, that is, we set the spectral parameters at ${\boldsymbol{x}}^{(n)}$ to zero. The equation then becomes

(4.12) \begin{equation} \varphi_{n-1}^* = \varphi_{n-1} + a F_n \!\left(t, {\boldsymbol{x}}^{(1)}, {}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta, ... , {\boldsymbol{x}}^{(n)}, 0, \textbf{0}, 0\right).\end{equation}

As the dependence on ${\boldsymbol{x}}^{(n)}$ is not exhibited by $\varphi_{n-1}^*$ and $\varphi_{n-1}$ , the reduction condition implies that $F_n$ at ${}^{(n)}\varrho = {}^{(n)}{\boldsymbol{v}} = {}^{(n)}\theta = 0$ does not depend on ${\boldsymbol{x}}^{(n)}$ too. In addition $F_n (... {}^{(n)}\varrho = {}^{(n)}{\boldsymbol{v}} = {}^{(n)}\theta = 0)$ should also identically satisfy the governing equation for $(n-1)-$ point CF. Therefore, $F_n (... {}^{(n)}\varrho = {}^{(n)}{\boldsymbol{v}} = {}^{(n)}\theta = 0)$ reduces to the same family of functions that are represented by $F_{n-1}$ . The set of functions, $F_n$ , to satisfy the permutation side condition (2.8), when:

(4.13) \begin{equation} F_n \!\left( ..., {}^{(l)}\varrho, {}^{(l)}{\boldsymbol{v}}, {}^{(l)}\theta, ..., {}^{(k)}\varrho, {}^{(k)}{\boldsymbol{v}}, {}^{(k)}\theta, ... \right) = F_n \!\left( ..., {}^{(k)}\varrho, {}^{(k)}{\boldsymbol{v}}, {}^{(k)}\theta, ..., {}^{(l)}\varrho, {}^{(l)}{\boldsymbol{v}}, {}^{(l)}\theta, ... \right), \end{equation}

that is, the function value remains unchanged regardless of the order in which the spatial points are considered. Coincidence side condition (2.7) is upheld when:

(4.14) \begin{equation} \lim_{|{\boldsymbol{x}}^{(n+1)}-{\boldsymbol{x}}^{(n)}|\to 0} F_{n+1} = F_n \!\left(.., {}^{(n+1)}\varrho+{}^{(n)}\varrho, {}^{(n+1)}{\boldsymbol{v}}+{}^{(n)}{\boldsymbol{v}}, {}^{(n+1)}\theta+{}^{(n)}\theta, ... \right).\end{equation}

In the case of separation side condition (2.9),

(4.15) \begin{equation}\begin{aligned} \lim_{|{\boldsymbol{x}}^{(n+1)}-{\boldsymbol{x}}^{(k)}| \to \infty} F_{n+1} &= \varphi_1\!\left({}^{(n+1)}\varrho, {}^{(n+1)}{\boldsymbol{v}}, {}^{(n+1)}\theta; t, {\boldsymbol{x}}^{(n+1)}\right) F_n \\ & \quad + \varphi_n F_1 \!\left( t, {\boldsymbol{x}}^{(n+1)}, {}^{(n+1)}\varrho, {}^{(n+1)}{\boldsymbol{v}}, {}^{(n+1)}\theta\right)\end{aligned}\end{equation}

needs to be satisfied. As discussed previously, the separation side condition (2.9) is not imposed in this study.

As an example, let us consider a set of functions $F_n$ that do not have any spatial and temporal dependence, that is,

(4.16) \begin{equation} \eta^{\varphi_n} = F_n \!\left({}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta, ... , {}^{(n)}\varrho, {}^{(n)}{\boldsymbol{v}}, {}^{(n)}\theta\right).\end{equation}

The transformed CFs, $\varphi_n^*$ , given by:

(4.17) \begin{equation} \varphi_n^* = \varphi_n + a F_n \!\left({}^{(1)}\varrho, {}^{(1)}{\boldsymbol{v}}, {}^{(1)}\theta, ... , {}^{(n)}\varrho, {}^{(n)}{\boldsymbol{v}}, {}^{(n)}\theta\right).\end{equation}

automatically satisfy the CF hierarchy. This symmetry is a more general compressible analogue of the ‘shape’ symmetry demonstrated in [Reference Wacławczyk, Staffolani, Oberlack, Rosteck, Wilczek and Friedrich21]. As established in the previous section, the functions $F_n$ cannot be constants as they would then violate the reduction side condition. As evidenced in the case of shape symmetry in incompressible flows [Reference Wacławczyk, Staffolani, Oberlack, Rosteck, Wilczek and Friedrich21], the separation side condition is not satisfied. The discussion pertinent to not strictly enforcing this side condition is presented in Section 3.2.1.

Finally, we investigate the symmetry group $a_{\varphi}$ . The resulting infinitesimals and the global transformation are

(4.18a) \begin{equation} \eta^{\varphi_1} = \varphi_1 - 1, \quad \varphi_1^* = 1+ e^{a_{\varphi}} (\varphi_1 - 1),\end{equation}

(4.18b) \begin{equation} \eta^{\varphi_n} = \varphi_n - 1, \quad \varphi_n^* = 1+ e^{a_{\varphi}} (\varphi_n - 1).\end{equation}

The above global form of the transformation satisfies all the required conditions, except for the separation side condition (2.9). This symmetry is the compressible counterpart of the ‘intermittency’ symmetry, representing the external intermittency seen in turbulent flows [Reference Wacławczyk, Staffolani, Oberlack, Rosteck, Wilczek and Friedrich21]. The implications of the both of the above statistical symmetries is investigated in detail in the next section.

5.1 Averaged kinetic energy

We begin our investigation with averaged kinetic energy (k), which in compressible turbulence is defined as:

(5.1) \begin{equation} k \equiv \left\langle \frac{1}{2} \rho u_i u_i \right\rangle.\end{equation}

As discussed in the introduction, the moments can be obtained from the CF, upon taking derivatives. In the case of kinetic energy, this would imply that we compute

(5.2) \begin{equation} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle = \frac{\mathrm{i}}{2} \left.\frac{\partial^3 \varphi_1}{ \partial \> {}^{(1)}\varrho \partial \> {}^{(1)}v_i \partial \> {}^{(1)}v_i} \right|_{{}^{(1)}\varrho = {}^{(1)}v_i = {}^{(1)}\theta = 0} .\end{equation}

Computing the transformation of k in equation (5.1) under the projective group (4.2) is a bit involved. We first look at the above triple derivative of CF in transformed variables, that is,

(5.3) \begin{equation} \frac{\partial^3 \varphi_1^*}{ \partial \> {}^{(1)}\varrho^* \partial \> {}^{(1)}v_i^* \partial \> {}^{(1)}v_i^*} = \frac{\partial^3}{ \partial \> {}^{(1)}\varrho^* \partial \> {}^{(1)}v_i^* \partial \> {}^{(1)}v_i^*} \left[ \varphi_1 \exp \!\left( \mathrm{i} a_{pr}\frac{ {}^{(1)}v_i x^{(1)}_i}{1-a_{pr} t} \right) \right].\end{equation}

We now obtain the expressions for each of the derivatives with respect to the transformed variables in terms of original variables:

(5.4a) \begin{equation}\begin{aligned} \frac{\partial}{\partial \> {}^{(1)}\varrho^*} &= \frac{\partial t}{\partial \> {}^{(1)}\varrho^*} \frac{\partial}{\partial t} + \frac{\partial x^{(1)}_i}{\partial \> {}^{(1)}\varrho^*} \frac{\partial}{\partial x^{(i)}_i} + \frac{\partial \>{}^{(1)}\varrho }{\partial \> {}^{(1)}\varrho^*} \frac{\partial}{\partial \>{}^{(1)}\varrho} + \frac{\partial \>{}^{(1)}v_i }{\partial \> {}^{(1)}\varrho^*} \frac{\partial}{\partial \>{}^{(1)}v_i} + \frac{\partial \>{}^{(1)}\theta }{\partial \> {}^{(1)}\varrho^*} \frac{\partial}{\partial \>{}^{(1)}\theta} \\ &= \frac{1}{\left(1+a_{pr} t^*\right)^3} \frac{\partial}{\partial \>{}^{(1)}\varrho} = \left(1 - a_{pr} t\right)^3 \frac{\partial}{\partial \>{}^{(1)}\varrho}\end{aligned}\end{equation}

and

(5.4b) \begin{equation}\begin{aligned} \frac{\partial}{\partial \> {}^{(1)}v_j^*} &= \frac{\partial t}{\partial \> {}^{(1)}v_j^*} \frac{\partial}{\partial t} + \frac{\partial x^{(1)}_i}{\partial \> {}^{(1)}v_j^*} \frac{\partial}{\partial x^{(i)}_i} + \frac{\partial \>{}^{(1)}\varrho }{\partial \> {}^{(1)}v_j^*} \frac{\partial}{\partial \>{}^{(1)}\varrho} + \frac{\partial \>{}^{(1)}v_i }{\partial \> {}^{(1)}v_j^*} \frac{\partial}{\partial \>{}^{(1)}v_i} + \frac{\partial \>{}^{(1)}\theta }{\partial \> {}^{(1)}v_j^*} \frac{\partial}{\partial \>{}^{(1)}\theta} \\ &= \frac{1}{1+ a_{pr} t^*} \frac{\partial}{\partial \>{}^{(1)}v_j} = \left(1 - a_{pr} t\right) \frac{\partial}{\partial \>{}^{(1)}v_j}.\end{aligned}\end{equation}

As a result, we obtain

(5.4c) \begin{equation} \begin{aligned} \frac{\partial^3 \varphi_1^*}{ \partial \> {}^{(1)}\varrho^* \partial \> {}^{(1)}v_i^* \partial \> {}^{(1)}v_i^*} & = \left(1-a_{pr} t\right)^5 \frac{\partial^3 \varphi_1^*}{ \partial \> {}^{(1)}\varrho \partial \> {}^{(1)}v_i \partial \> {}^{(1)}v_i} \\[3pt] &= \left(1- a_{pr} t\right)^5 \frac{\partial^3 }{ \partial \> {}^{(1)}\varrho \partial \> {}^{(1)}v_i \partial \> {}^{(1)}v_i} \left[ \varphi_1 \exp \!\left( \mathrm{i} a_{pr} \frac{ {}^{(1)}v_i x^{(1)}_i}{1- a_{pr} t} \right) \right] \\[3pt] &= \left(1- a_{pr} t\right)^5 \left[ \frac{\partial^3 \varphi_1}{\partial \>{}^{(1)}\varrho \partial \> {}^{(1)}v_i \partial \> {}^{(1)}v_i} + \frac{2\mathrm{i} a_{pr} \> x^{(1)}_i}{1- a_{pr} t} \frac{\partial^2 \varphi_1}{\partial \>{}^{(1)}\varrho \partial \> {}^{(1)}v_i } \right. \\[3pt] &\left. - \frac{x^{(1)}_i x^{(1)}_i a_{pr}^2 }{\left(1- a_{pr} t\right)^2} \frac{\partial \varphi_1}{\partial \>{}^{(1)}\varrho } \right] \exp \!\left(\mathrm{i} a_{pr} \frac{ {}^{(1)}v_i x^{(1)}_i}{1- a_{pr} t} \right). \end{aligned}\end{equation}

Finally, the transformed kinetic energy under the projective group is given by:

(5.4d) \begin{equation} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle^* = \left(1 - a_{pr} t\right)^5 \left\langle \frac{1}{2} \rho u_i u_i \right\rangle + a_{pr} x^{(1)}_i \!\left(1 - a_{pr} t\right)^4 \left\langle \rho u_i \right\rangle + \frac{a_{pr}^2}{2} x^{(1)}_i x^{(1)}_i \!\left(1 - a_{pr} t\right)^3 \langle \rho \rangle.\end{equation}

In all of the scaling groups, the global transformation for k is straightforward, as these groups lead to the scaling of kinetic energy. For ‘scaling of time’ group (4.4), the global symmetry transformation of k looks like:

(5.5) \begin{equation} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle^* = e^{-2 a_{s,t}} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle.\end{equation}

For the ‘scaling of space’ group (4.5):

(5.6) \begin{equation} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle^* = e^{2 a_{s,x}} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle.\end{equation}

Finally, the ‘scaling of density’ group (4.6):

(5.7) \begin{equation} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle^* = e^{a_{s,\rho}} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle.\end{equation}

We now shift our focus to two statistical symmetries that are present specifically in the case of CF hierarchy and not seen in Euler equations: shape and intermittency groups ((4.17) and (4.18)). For the shape symmetry (4.17), the global transformation of k, takes the form:

(5.8) \begin{equation}\begin{aligned} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle^* &= \left\langle \frac{1}{2} \rho u_i u_i \right\rangle + \frac{\mathrm{i}}{2} a \left. \frac{\partial^3 F_1}{\partial \> {}^{(1)}\varrho \partial \> {}^{(1)}v_i \partial \> {}^{(1)}v_i} \right|_{{}^{(1)}\varrho = {}^{(1)}v_i = {}^{(1)}\theta = 0} \\ &= \left\langle \frac{1}{2} \rho u_i u_i \right\rangle + a_k,\end{aligned}\end{equation}

where $a_k$ is a constant. Therefore, the addition of a constant to averaged kinetic energy in compressible turbulence leaves its governing equation invariant. In the case of the intermittency group (4.18), the kinetic energy transforms as:

(5.9) \begin{equation} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle^* = e^{a_{\varphi}} \left\langle \frac{1}{2} \rho u_i u_i \right\rangle.\end{equation}

5.2 Averaged internal energy

The second measure of considerable importance in compressible turbulence is the averaged internal energy (e), defined as:

(5.10) \begin{equation} e = \frac{\langle p \rangle}{\gamma - 1}.\end{equation}

As the ratio, $1/(\gamma - 1)$ remains a constant for a given gas, we focus on the average pressure, $\langle p \rangle$ in the remainder of the paper, when we are investigating averaged internal energy. Based on our definition of CF, $\langle p \rangle$ can be obtained by:

(5.11) \begin{equation} \langle p \rangle = - \left. \frac{\partial^2 \varphi_1}{\partial \> {}^{(1)}\varrho \partial \> {}^{(1)}\theta} \right|_{{}^{(1)}\varrho = {}^{(1)}v_i = {}^{(1)}\theta = 0},\end{equation}

where, we have used ideal gas law, $p = \rho T$ , in its non-dimensional form. As a result, $\langle p \rangle$ is obtained by taking derivatives of $\varphi_1$ with respect to ${}^{(1)}\varrho$ and ${}^{(1)}\theta$ .

Let us now begin our investigation of transformation of internal energy under symmetries of CF. For the projective group (4.2), we obtain

(5.12) \begin{equation} \langle p \rangle^* = - \left. \frac{\partial^2 \varphi_1^*}{\partial \> {}^{(1)}\varrho^* \partial \> {}^{(1)}\theta^*} \right|_{{}^{(1)}\varrho^* = {}^{(1)}v_i^* = {}^{(1)}\theta^* = 0}.\end{equation}

We perform similar calculations as shown for k in the previous Subsection 5.1, to obtain the transformed e:

(5.13a) \begin{equation}\begin{aligned} \frac{\partial}{\partial \> {}^{(1)}\theta^*} &= \frac{\partial t}{\partial \> {}^{(1)}\theta^*} \frac{\partial}{\partial t} + \frac{\partial x^{(1)}_i}{\partial \> {}^{(1)}\theta^*} \frac{\partial}{\partial x^{(1)}_i} + \frac{\partial \> {}^{(1)}\varrho}{\partial \> {}^{(1)}\theta^*} \frac{\partial}{\partial \> {}^{(1)}\varrho} + \frac{\partial \> {}^{(1)}v_i}{\partial \> {}^{(1)}\theta^*} \frac{\partial}{\partial \> {}^{(1)}v_i} + \frac{\partial \> {}^{(1)}\theta}{\partial \> {}^{(1)}\theta^*} \frac{\partial}{\partial \> {}^{(1)}\theta} \\ &= \left(1- a_{pr} t\right)^2 \frac{\partial}{\partial \> {}^{(1)}\theta},\end{aligned}\end{equation}

leading to,

(5.13b) \begin{equation} \begin{aligned} \frac{\partial^2 \varphi_1^*}{\partial \> {}^{(1)}\varrho^* \partial \> {}^{(1)}\theta^*} &= \left(1- a_{pr} t\right)^5 \frac{\partial^2 }{\partial \> {}^{(1)}\varrho \partial \> {}^{(1)}\theta} \left[ \varphi_1 \exp \!\left( \mathrm{i} a_{pr} \frac{{}^{(1)}v_i x^{(1)}_i}{1- a_{pr} t} \right) \right] \\ &= \left(1- a_{pr} t\right)^5 \frac{\partial^2 \varphi_1}{\partial \> {}^{(1)}\varrho \partial \> {}^{(1)}\theta} \exp \!\left( \mathrm{i} a_{pr} \frac{{}^{(1)}v_i x^{(1)}_i}{1- a_{pr} t} \right), \end{aligned}\end{equation}

and finally,

(5.13c) \begin{equation} \langle p \rangle^* = \left(1- a_{pr} t\right)^5 \langle p \rangle.\end{equation}

In the case of ‘scaling of time’ symmetry (4.4), the global transformation of averaged internal energy is

(5.14) \begin{equation} \langle p \rangle^* = e^{-2 a_{s,t}} \langle p \rangle.\end{equation}

In ‘scaling of space’ (4.5),

(5.15) \begin{equation} \langle p \rangle^* = e^{2 a_{s,x}} \langle p \rangle.\end{equation}

‘Scaling of density’ (4.6) leads to:

(5.16) \begin{equation} \langle p \rangle^* = e^{a_{s,\rho}} \langle p \rangle.\end{equation}

The global transformation of e, in the case of shape symmetry (4.17), is given by:

(5.17) \begin{equation} \begin{aligned} \langle p \rangle^* = \langle p \rangle - a \left. \frac{\partial^2 F_1}{\partial \> {}^{(1)}\varrho \partial \> {}^{(1)}\theta} \right|_{{}^{(1)}\varrho = {}^{(1)}v_i = {}^{(1)}\theta = 0} = \langle p \rangle + a_{e}. \end{aligned}\end{equation}

Finally, the intermittency group (4.18) leads to:

(5.18) \begin{equation} \langle p \rangle^* = e^{a_{\varphi}} \langle p \rangle.\end{equation}

5.3 Higher-order velocity and thermodynamic moments

We now investigate the higher-order moments of density, temperature, pressure and velocity. Since we are interested in multi-point moments, we need to consider the derivatives of the multi-point CF. For example, the multi-point density moments are obtained from the CF as follows:

(5.19) \begin{equation} \left\langle \prod_{l = 1}^n \rho({\boldsymbol{x}}^{(l)}, t) \right\rangle = \frac{1}{\mathrm{i}^n} \left[ \prod_{l = 1}^n \frac{\partial}{\partial \> {}^{(l)}\varrho} \varphi_n \right]_{{}^{(1)}\varrho = {}^{(1)}v_i = {}^{(1)}\theta = ... {}^{(n)}\varrho = {}^{(n)}v_i = {}^{(n)}\theta = 0 }\end{equation}

We now write down the global transformations of each of the higher-order moments with respect to the symmetry transformations of the CF hierarchy given in Section 3.2, starting with the nth-order moment of density:

(5.20a) \begin{equation} a_{pr}\,: \quad \left\langle \prod_{l = 1}^n \rho({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = \left(1- a_{pr} t\right)^{3n } \left\langle \prod_{l = 1}^n \rho({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.20b) \begin{equation} a_{s,\rho}\,: \quad \left\langle \prod_{l = 1}^n \rho({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{ n a_{s,\rho}} \left\langle \prod_{l = 1}^n \rho({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.20c) \begin{equation} F_n\,: \quad \left\langle \prod_{l = 1}^n \rho({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = \left\langle \prod_{l = 1}^n \rho({\boldsymbol{x}}^{(l)}, t) \right\rangle + a_{\rho, n}\end{equation}

(5.20d) \begin{equation} a_{\varphi}\,: \left\langle \prod_{l = 1}^n \rho({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{a_{\varphi}} \left\langle \prod_{l = 1}^n \rho({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

Now the multi-point temperature moments transform as:

(5.21a) \begin{equation} a_{pr}\,: \quad \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = \left(1- a_{pr} t\right)^{2n } \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.21b) \begin{equation} a_{s,t}\,: \quad \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{-2 n a_{s,t}} \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.21c) \begin{equation} a_{s,x}\,: \quad \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{2 n a_{s,x}} \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.21d) \begin{equation} F_n \,: \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle + a_{T, n}\end{equation}

(5.21e) \begin{equation} a_{\varphi}\,: \quad \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{a_{\varphi}} \left\langle \prod_{l = 1}^n T({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

In the case of pressure,

(5.22a) \begin{equation} a_{pr}\,: \quad \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = \left(1- a_{pr} t\right)^{5n } \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.22b) \begin{equation} a_{s,t}\,: \quad \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{-2 n a_{s,t}} \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.22c) \begin{equation} a_{s,x}\,: \quad \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{2 n a_{s,x}} \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.22d) \begin{equation} a_{s,\rho}\,: \quad \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{ n a_{s,\rho}} \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.22e) \begin{equation} F_n\,: \quad \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle + a_{p, n}\end{equation}

(5.22f) \begin{equation} a_{\varphi}\,: \quad \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{a_{\varphi}} \left\langle \prod_{l = 1}^n p({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

The multi-point velocity moments constitute an nth-order tensor that can be obtained from the CF as:

(5.23) \begin{equation} \left\langle u_{i_1} \left({\boldsymbol{x}}^{(1)}, t\right) u_{i_2} \left({\boldsymbol{x}}^{(2)}, t\right) ... u_{i_n} \left({\boldsymbol{x}}^{(n)}, t\right) \right\rangle = \frac{1}{\mathrm{i}^n} \left. \frac{\partial^n \varphi_n}{\partial \> {}^{(1)}v_{i_1} \> \partial \> {}^{(2)}v_{i_2} \> ... \partial \> {}^{(n)}v_{i_n} } \right|_{{}^{(1)}\varrho = .. = {}^{(n)}\theta = 0 },\end{equation}

where $i_1, i_2, ... , i_n$ are indices of the multi-point velocity moment tensor taking values from 1 to 3. The global transformations of the multi-point velocity moment tensor under the CF symmetries are given below:

(5.24a) \begin{equation}\begin{aligned} a_{pr}\,: \left\langle \prod_{l = 1}^n u_{i_l} \left({\boldsymbol{x}}^{(l)}, t\right) \right\rangle^* =& \left(1-a_{pr} t\right)^n \left\langle \prod_{l = 1}^n u_{i_l} \left({\boldsymbol{x}}^{(l)}, t\right) \right\rangle \\ & + \left(1- a_{pr} t\right)^{n-1} \sum_{k = 1}^n \left[ a_{pr} \> x^{(k)}_{i_k} \left\langle \prod_{l = 1, l \neq k}^n u_{i_l} \left({\boldsymbol{x}}^{(l)}, t\right) \right\rangle \right] \\ & + \cdots + \left(1- a_{pr} t\right) \sum_{k = 1}^n \left[ a_{pr}^{n-1} \left\langle u_{i_k} \left({\boldsymbol{x}}^{(k)}, t\right) \right\rangle \prod_{l = 1, l \neq k}^n x^{(l)}_{i_l} \right] \\ &+ a_{pr}^n x^{(1)}_{i_1} x^{(2)}_{i_2} ... x^{(n)}_{i_{n}}\end{aligned}\end{equation}

(5.24b) \begin{equation} a_{s,t}\,: \quad \left\langle \prod_{l = 1}^n u_{i_l} ({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{-n a_{s,t}} \left\langle \prod_{l = 1}^n u_{i_l} ({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.24c) \begin{equation} a_{s,x}\,: \quad \left\langle \prod_{l = 1}^n u_{i_l} ({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{n a_{s,x}} \left\langle \prod_{l = 1}^n u_{i_l} ({\boldsymbol{x}}^{(l)}, t) \right\rangle\end{equation}

(5.24d) \begin{equation} F_n\,: \quad \left\langle \prod_{l = 1}^n u_{i_l} ({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = \left\langle \prod_{l = 1}^n u_{i_l} ({\boldsymbol{x}}^{(l)}, t) \right\rangle + a_{i_1 i_2 .. i_n}\end{equation}

(5.24e) \begin{equation} a_{\varphi}\,: \quad \left\langle \prod_{l = 1}^n u_{i_l} ({\boldsymbol{x}}^{(l)}, t) \right\rangle^* = e^{a_{\varphi}} \left\langle \prod_{l = 1}^n u_{i_l} ({\boldsymbol{x}}^{(l)}, t) \right\rangle.\end{equation}

Thus, all the flow and thermodynamics moments can be obtained from the CF hierarchy and as a result, the process of finding scaling laws via Lie group analysis is considerably simplified.