Martin Oberlack

Similarity in non-rotating and rotating turbulent pipe flows

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 symmetries and averaging of the G-equation for premixed combustion

It is demonstrated that the G-equation for premixed combustion admits a diversity of symmetries properties, i.e. invariance characteristics under certain transformations. Included are those of classical mechanics such as Galilean invariance, rotation invariance and others. Also a new generalized scaling symmetry has been established. It is shown that the generalized scaling symmetry defines the physical property of the G-equation precisely. That is to say the value of G at a given flame front is arbitrary. It is proven that beside the symmetries of classical mechanics, particularly the generalized scaling symmetry uniquely defines the basic structure of the G-equation. It is also proven that the generalized scaling symmetry precludes the application of classical Reynolds ensemble averaging usually employed in statistical turbulence theory in order to avoid non-unique statistical quantities such as for the mean flame position. Finally, a new averaging scheme of the G-field is presented which is fully consistent with all symmetries of the G-equation. Equations for the mean G-field and flame brush thickness are derived and a route to consistent invariant modelling of other quantities derived from the G-field is illustrated. Examples of statistical quantities derived from the G-field both in the context of Reynolds-averaged models as well as subgrid-scale models for large-eddy simulations taken from the literature are investigated as to whether they are compatible with the important generalized scaling symmetry.

Non-isotropic dissipation in non-homogeneous turbulence

On the basis of the two-point velocity correlation equation a new tensor length-scale equation and in turn a dissipation rate tensor equation and the pressure–strain correlation are derived by means of asymptotic analysis and frame-invariance considerations. The new dissipation rate tensor equation can account for non-isotropy effects of the dissipation rate and streamline curvature. The entire analysis is valid for incompressible as well as for compressible turbulence in the limit of small Mach numbers. The pressure–strain correlation is expressed as a functional of the two-point correlation, leading to an extended 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 velocities and a tensor relation between the length scale and the dissipation rate tensor. Hence, a consistent formulation of the return term in the pressure–strain correlation and the dissipation tensor equation is achieved. The model has been integrated numerically for several different homogeneous and inhomogeneous test cases and results are compared with DNS, LES and experimental data.

On stochastic Damköhler number variations in a homogeneous flow reactor

For combustion in a homogeneous flow reactor stochastic fluctuations of the Damköhler number are analysed. The chemistry is described by a one-step first-order reaction with Arrhenius kinetics. The case of an adiabatic system is studied allowing for a reduction to a single equation for the temperature. A parameter of that equation is the Damköhler number which is separated into a mean and a stochastic term carrying white noise characteristics. This allows for an exact treatment of the probability density function in terms of a Fokker-Planck equation. In the case of steady statistics the Fokker–Planck equation is solved exactly in terms of exponential integrals. It is shown that the probability density function usually admits mono- and bi-modal behaviour but even tri-modal behaviour is observed depending on the activation energy, the heat of combustion, the mean Damköhler number and the variance of the Damköhler number. The most common features are mono-modal characteristics, which indicate that the system is strongly attracted either to the unburned or the burned state. Two cases of Damköhler number variations are studied. The first one corresponds to the classical combustion situation, where two steady states exist and transitions between them can be triggered by Damköhler number fluctuations. The other corresponds to a new mode of combustion, called ‘mild’ combustion, where the steady-state solution depends monotonically on the Damköhler number, such that ignition and extinction events are suppressed.

Extensive strain along gradient trajectories in the turbulent kinetic energy field

Based on direct numerical simulations of forced turbulence, shear turbulence, decaying turbulence, a turbulent channel flow as well as a Kolmogorov flow with Taylor-based Reynolds numbers Re? between 69 and 295, the normalized probability density function of the length distribution of dissipation elements, the conditional mean scalar difference ?kl at the extreme points as well as the scaling of the two-point velocity difference along gradient trajectories ?un are studied. Using the field of the instantaneous turbulent kinetic energy k as a scalar, we find good agreement between the model equation for as proposed by Wang and Peters (2008 J. Fluid Mech. 608 113?38) and the results obtained in the different direct numerical simulation cases. This confirms the independence of the model solution from both the Reynolds number and the type of turbulent flow, so that it can be considered universally valid. In addition, we show a 2/3 scaling for the mean conditional scalar difference. In the second part of the paper, we examine the scaling of the conditional two-point velocity difference along gradient trajectories. In particular, we compare the linear s/? scaling, where ? denotes an integral time scale and s the separation arclength along a gradient trajectory in the inertial range as derived by Wang (2009 Phys. Rev. E 79 046325) with the s?a? scaling, where a? denotes the asymptotic value of the conditional mean strain rate of large dissipation elements.

A SIMPLE based discontinuous Galerkin solver for steady incompressible flows

In this paper we present how the well-known SIMPLE algorithm can be extended to solve the steady incompressible Navier-Stokes equations discretized by the discontinuous Galerkin method. The convective part is discretized by the local Lax-Friedrichs fluxes and the viscous part by the symmetric interior penalty method. Within the SIMPLE algorithm, the equations are solved in an iterative process. The discretized equations are linearized and an equation for the pressure is derived on the discrete level. The equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially, leading to an efficient solution procedure. The extension of the proposed scheme to the unsteady case is straightforward, where fully implicit time schemes can be used. Various test cases are carried out: the Poiseuille flow, the channel flow with constant transpiration, the Kovasznay flow, the flow into a corner and the backward-facing step flow. Using a mixed-order formulation, i.e. order k for the velocity and order k-1 for the pressure, the scheme is numerically stable for all test cases. Convergence rates of k+1 and k in the L^2-norm are observed for velocity and pressure, respectively. A study of the convergence behavior of the SIMPLE algorithm shows that no under-relaxation for the pressure is needed, which is in strong contrast to the application of the SIMPLE algorithm in the context of the finite volume method or the continuous finite element method. We conclude that the proposed scheme is efficient to solve the steady incompressible Navier-Stokes equations in the context of the discontinuous Galerkin method comprising hp-accuracy.

LIE ALGEBRA OF THE SYMMETRIES OF THE MULTI-POINT EQUATIONS IN STATISTICAL TURBULENCE THEORY

We briefly derive the infinite set of multi-point correlation equations based on the Navier–Stokes equations for an incompressible fluid. From this we reconsider the previously derived set of Lie symmetries, i.e. those directly induced by the ones from classical mechanics and also new symmetries. The latter are denoted statistical symmetries and have no direct counterpart in classical mechanics. Finally, we considerably extend the set of symmetries by Lie algebra methods and give the corresponding commutator tables. Due to the infinite dimensionality of the multi-point correlation equations completeness of its symmetries is not proven yet and is still an open question.

Slip Effects on the Peristaltic Flow of a Third Grade Fluid in a Circular Cylindrical Tube

Peristaltic flow of a third grade fluid in a circular cylindrical tube is undertaken when the no-slip condition at the tube wall is no longer valid. The governing nonlinear equation together with nonlinear boundary conditions is solved analytically by means of the perturbation method for small values of the non-Newtonian parameter, the Debroah number. A numerical solution is also obtained for which no restriction is imposed on the non-Newtonian parameter involved in the governing equation and the boundary conditions. A comparison of the series solution and the numerical solution is presented. Furthermore, the effects of slip and non-Newtonian parameters on the axial velocity and stream function are discussed in detail. The salient features of pumping and trapping are discussed with particular focus on the effects of slip and non-Newtonian parameters. It is observed that an increase in the slip parameter decreases the peristaltic pumping rate for a given pressure rise. On the contrary, the peristaltic pumping rate increases with an increase in the slip parameter for a given pressure drop (copumping). The size of the trapped bolus decreases and finally vanishes for large values of the slip parameter.

Symmetries, Invariance and Scaling-Laws in Inhomogeneous Turbulent Shear Flows

An approach to derive turbulent scaling laws based on symmetry analysis is presented. It unifies a large set of scaling laws for the mean velocity of stationary parallel turbulent shear flows. 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. For the plane case 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 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. For a non-rotating and a moderately rotating pipe about its axis an algebraic law was found for the axial and the azimuthal velocity near the pipe-axis with both laws having equal scaling exponents. In case of a rapidly rotating pipe, a new logarithmic scaling law for the axial velocity is developed. The key elements of the entire analysis are two scaling symmetries and Galilean invariance. Combining the scaling symmetries leads to the variety of different scaling laws. Galilean invariance is crucial for all of them. It has been demonstrated that two-equation models such as the k–∈ model are not consistent with most of the new turbulent scaling laws.

Long wavelength approximation to peristaltic motion of an Oldroyd 4-constant fluid in a planar channel

An analysis is carried out to study the peristaltic motion of an incompressible Oldroyd 4-constant fluid in a planar channel. The flow modeling is first developed in dimensionless form and the governing problem is then simplified by adopting a long wavelength assumption. It is found that unlike Oldroyd 3-constant fluid, the governing problem of an Oldroyd 4-constant fluid under a long wavelength approximation contains the rheological parameters. The resulting non-linear problem has been solved numerically using a combination of the finite difference scheme and the iterative method. In addition, an analytical solution is presented for the domain near the channel center. The effect of material parameters on the pumping and trapping is discussed. A comparison with the corresponding results for a viscous Newtonian fluid is also made.