Two simulations of turbulent Couette flows were performed at friction Reynolds numbers of 1000 and 2000 in a large box of dimensions $L_x=16{\rm \pi} h$, $L_y=2h$ and $L_z=6{\rm \pi} h$, where h is the semi-height of the channel. The study focuses on the differences in the intensity and scaling of turbulence at these two Reynolds numbers. The 2000 case showed a lack of a clear log layer with a higher value of the Von Kármán constant $\kappa$ than Poiseuille channels. The intensities were well-scaled in the buffer layer and below, with a second maximum of the streamwise intensity at approximately 350 wall units. Contrary to Poiseuille channels, the dissipation scales close to the wall in wall units. This fact can be attributed to the constant value of the derivative of the streamwise intensity in wall units. The intensities of the 2000 case showed remarkable differences compared with those at Reynolds number 1000 at the channel centre, likely due to the organization of large scales of the streamwise fluctuactions, $u$. These large scales were thought to be considered ‘infinite’. However, for the 2000 case, while all the structures have a width of $\ell _z \approx 6/8{\rm \pi} h$, their length varies from $\ell _x \approx 6{\rm \pi} h$ to $\ell _x \approx 16{\rm \pi} h$, which clearly contradicts the trends obtained in the past. This is a new effect that has not been reported for turbulent Couette flow and points to the uncertainty and sensitivity that is observed for certain statistical quantities.

In direct numerical simulations (DNS) of turbulent Couette flow, the observation has been made that the long streamwise rolls increase in length with the Reynolds number (Lee & Moser, J. Fluid Mech., vol. 842, 2018, pp. 128–145). To understand this, we employ both linear stability theory and its extension to resolvent analysis. For this, we emphasise the high Reynolds number ($Re \rightarrow \infty$) and small streamwise wavenumbers ($\alpha \rightarrow 0$) limit, imposing the distinguished limit $Re_{\alpha }=Re \, \alpha = O(1)$. We find that in case of linear stability theory, $Re_{\alpha }$ acts as a global invariant in the resulting eigenvalue problem, while in case of resolvent analysis, $Re_{\alpha }$ acts as a local invariant in the behaviour of the energy of the system characterised through the first singular value $\sigma _1$ of the resolvent operator within the investigated asymptotic limit. In order to obtain constant streamwise structures for increasing Reynolds numbers, the respective streamwise wavenumber has to decrease, which verifies the observations from DNS studies of an increasing length of the streamwise structures with the Reynolds number. In linear stability theory, a parameter reduction is achieved for the above asymptotic limit, resulting in the modified Orr–Sommerfeld and Squire equations being dependent only on $Re_{\alpha }$. The behaviour of both the coherent structures obtained from linear stability theory and resolvent analysis are compared with each other and show similar behaviours over $Re_{\alpha }$.

We use the Lie symmetry theory to derive new scaling laws for passive scalar dynamics under the influence of a constant mean gradient of the scalar in decaying homogeneous isotropic turbulence. For this purpose, we apply symmetry analysis to the equations for two-point correlation of the scalar and velocity fluctuations. It is shown that, in contrast to the classical self-similarity approach, the general invariant solutions, respectively scaling laws, of the two-point functions are constructed using the symmetry approach, without requiring an a priori set of similarity scales to carry on the analysis. In the context of the current analysis also, scaling laws for one-point quantities of the scalar variance $\overline{\unicode[STIX]{x1D703}^{2}}$, the transverse scalar flux $\overline{u_{2}\unicode[STIX]{x1D703}}$ and the variance of the turbulent velocity fluctuations $\overline{u^{2}}$ are established, which are essentially related to the scaling symmetries. A key step to derive the scaling laws is the symmetry breaking induced by the constant mean scalar gradient. We use the results of a highly resolved direct numerical simulation of Gauding et al. (Comput. Fluids, vol. 180, 2019, pp. 206–217) to verify the scaling laws and the self-similarity of the two-point correlation functions. It is shown that the general symmetry solutions obtained from symmetry results provide a very good similarity to these functions.

The present contribution is a significant extension of the work by Kelbin et al. (J. Fluid Mech., vol. 721, 2013, pp. 340–366) as a new time-dependent helical coordinate system has been introduced. For this, Lie symmetry methods have been employed such that the spatial dependence of the originally three independent variables is reduced by one and the remaining variables are: the cylindrical radius $r$ and the time-dependent helical variable $\unicode[STIX]{x1D709}=(z/\unicode[STIX]{x1D6FC}(t))+b\unicode[STIX]{x1D711}$, $b=\text{const.}$ and time $t$. The variables $z$ and $\unicode[STIX]{x1D711}$ are the usual cylindrical coordinates and $\unicode[STIX]{x1D6FC}(t)$ is an arbitrary function of time $t$. Assuming $\unicode[STIX]{x1D6FC}=\text{const.}$, we retain the classical helically symmetric case. Using this, and imposing helical invariance onto the equation of motion, leads to a helically symmetric system of Euler and Navier–Stokes equations with a time-dependent pitch $\unicode[STIX]{x1D6FC}(t)$, which may be varied arbitrarily and which is explicitly contained in all of the latter equations. This has been conducted both for primitive variables as well as for the vorticity formulation. Hence a significantly extended set of helically invariant flows may be considered, which may be altered by an external time-dependent strain along the axis of the helix. Finally, we sought new conservation laws which can be found from the helically invariant Euler and Navier–Stokes equations derived herein. Most of these new conservation laws are considerable extensions of existing conservation laws for helical flows at a constant pitch. Interestingly enough, certain classical conservation laws do not admit extensions in the new time-dependent coordinate system.

Local conservation laws are systematically constructed for three-dimensional time-dependent viscous and inviscid incompressible fluid flows, in primitive variables and vorticity formulation, using the direct construction method. Complete sets of local conservation laws in primitive variables are derived for the case of conservation law multipliers depending on derivatives up to the second order. In the vorticity formulation, there exists an infinite family of vorticity-dependent conservation laws involving an arbitrary differentiable function of space and time, holding for both viscous and inviscid cases. The infinite conservation law family is used to generate further independent hierarchies of conservation laws that essentially involve vorticity and arbitrary flow parameters, which are determined by known evolution equations such as those for momentum, energy or helicity, though not necessarily in the form of a conservation law. The new conservation laws are not restricted to any reduced flow geometry such as planar or axisymmetric limits. Examples are considered.

Helically invariant reductions due to a reduced set of independent variables $(t, r, \xi )$ with $\xi = az+ b\varphi $ emerging from a cylindrical coordinate system of viscous and inviscid time-dependent fluid flow equations, with all three velocity components generally non-zero, are considered in primitive variables and in the vorticity formulation. Full sets of equations are derived. Local conservation laws of helically invariant systems are systematically sought through the direct construction method. Various new sets of conservation laws for both inviscid and viscous flows, including families that involve arbitrary functions, are derived. For both Euler and Navier–Stokes flows, infinite sets of vorticity-related conservation laws are derived. In particular, for Euler flows, we obtain a family of conserved quantities that generalize helicity. The special case of two-component flows, with zero velocity component in the invariant direction, is additionally considered, and special conserved quantities that hold for such flows are computed. In particular, it is shown that the well-known infinite set of generalized enstrophy conservation laws that holds for plane flows also holds for the general two-component helically invariant flows and for axisymmetric two-component flows.

In order to analyse the geometric structure of turbulent flow patterns and their statistics for various scalar fields we adopt the dissipation element (DE) approach and apply it to turbulent channel flow by employing direct numerical simulations (DNS) of the Navier–Stokes equations. Gradient trajectories starting from any point in a scalar field in the directions of ascending and descending scalar gradients will always reach an extremum, i.e. a minimum or a maximum point, where . The set of all points and trajectories belonging to the same pair of extremal points defines a dissipation element. Extending previous DE approaches, which were only applied to homogeneous turbulence, we here focus on exploring the influence of solid walls on the dissipation element distribution. Employing group-theoretical methods and known symmetries of Navier–Stokes equations, we observe for the core region of the flow, i.e. the region beyond the buffer layer, that the probability distribution function (p.d.f.) of the DE length exhibits an invariant functional form, in other words, self-similar behaviour with respect to the wall distance. This is further augmented by the scaling behaviour of the mean DE length scale which shows a linear scaling with the wall distance. The known proportionality of the mean DE length and the Taylor length scale is also revisited. Utilizing a geometric analogy we give the number of DE elements as a function of the wall distance. Further, it is observed that the DE p.d.f. is rather insensitive, i.e. invariant with respect both to the Reynolds number and the actual scalar which has been employed for the analysis. In fact, a very remarkable degree of isotropy is observed for the DE p.d.f. in regions of high shear. This is in stark contrast to classical Kolmogorov scaling laws which usually exhibit a strong dependence on quantities such as shear, anisotropy and Reynolds number. In addition, Kolmogorov’s scaling behaviour is in many cases only visible for very large Reynolds numbers. This is rather different in the present DE approach which applies also for low Reynolds numbers. Moreover, we show that the DE p.d.f. agrees very well with the log-normal distribution and derive a log-normal p.d.f. model taking into account the wall-normal dependence. Finally, the conditional mean scalar differences of the turbulent kinetic energy at the extremal points of DE are examined. We present a power law with scaling exponent of known from Kolmogorov’s hypothesis for the centre of the channel and a logarithmic law near the wall.

The homotopy analysis method (HAM) is applied to a nonlinear ordinary differential equation (ODE) emerging from a closure model of the von Kármán–Howarth equation which models the decay of isotropic turbulence. In the infinite Reynolds number limit, the von Kármán–Howarth equation admits a symmetry reduction leading to the aforementioned one-parameter ODE. Though the latter equation is not fully integrable, it can be integrated once for two particular parameter values and, for one of these values, the relevant boundary conditions can also be satisfied. The key result of this paper is that for the generic case, HAM is employed such that solutions for arbitrary parameter values are derived. We obtain explicit analytical solutions by recursive formulas with constant coefficients, using some transformations of variables in order to express the solutions in polynomial form. We also prove that the Loitsyansky invariant is a conservation law for the asymptotic form of the original equation.

A new theoretical approach for turbulent flows based on Lie-group analysis is presented. It unifies a large set of ‘solutions’ for the mean velocity of stationary parallel turbulent shear flows. These results are not solutions in the classical sense but instead are defined by the maximum number of possible symmetries, only restricted by the flow geometry and other external constraints. The approach is derived from the Reynolds-averaged Navier–Stokes equations, the fluctuation equations, and the velocity product equations, which are the dyad product of the velocity fluctuations with the equations for the velocity fluctuations. The results include the logarithmic law of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couette flow and in the centre of a rotating channel flow, and a new exponential mean velocity profile not previously reported that is found in the mid-wake region of high Reynolds number flat-plate boundary layers. The algebraic scaling law is confirmed in both the centre and the near-wall regions in both experimental and DNS data of turbulent channel flows. In the case of the logarithmic law of the wall, the scaling with the distance from the wall arises as a result of the analysis and has not been assumed in the derivation. All solutions are consistent with the similarity of the velocity product equations to arbitrary order. A method to derive the mean velocity profiles directly from the two-point correlation equations is shown.

The Lie group approach developed by Oberlack (1997) is used to derive new scaling laws for high-Reynolds-number turbulent pipe flows. The scaling laws, or, in the methodology of Lie groups, the invariant solutions, are based on the mean and fluctuation momentum equations. For their derivation no assumptions other than similarity of the Navier–Stokes equations have been introduced where the Reynolds decomposition into the mean and fluctuation quantities has been implemented. The set of solutions for the axial mean velocity includes a logarithmic scaling law, which is distinct from the usual law of the wall, and an algebraic scaling law. Furthermore, an algebraic scaling law for the azimuthal mean velocity is obtained. In all scaling laws the origin of the independent coordinate is located on the pipe axis, which is in contrast to the usual wall-based scaling laws. The present scaling laws show good agreement with both experimental and DNS data. As observed in experiments, it is shown that the axial mean velocity normalized with the mean bulk velocity um has a fixed point where the mean velocity equals the bulk velocity independent of the Reynolds number. An approximate location for the fixed point on the pipe radius is also given. All invariant solutions are consistent with all higher-order correlation equations. A large-Reynolds-number asymptotic expansion of the Navier–Stokes equations on the curved wall has been utilized to show that the near-wall scaling laws for at surfaces also apply to the near-wall regions of the turbulent pipe flow.

On the basis of the two-point velocity correlation equation a new tensor length-scaleequation and in turn a dissipation rate tensor equation and the pressure–strain correlationare derived by means of asymptotic analysis and frame-invariance considerations. The newdissipation rate tensor equation can account for non-isotropy effects of the dissipationrate and streamline curvature. The entire analysis is valid for incompressible as well asfor compressible turbulence in the limit of small Mach numbers. The pressure–straincorrelation is expressed as a functional of the two-point correlation, leading to anextended compressible version of the linear formulation of the pressure–strain correlation.In this turbulence modelling approach the only terms which still need ad hoc closure assumptions are the triple correlation of the fluctuating velocitiesand a tensor relation between the length scale and the dissipation rate tensor. Hence, aconsistent formulation of the return term in the pressure–strain correlation and thedissipation tensor equation is achieved. The model has been integrated numerically forseveral different homogeneous and inhomogeneous test cases and results are compared withDNS, LES and experimental data.