Preface Part I. Turbulence: 1. Introduction 2. Coherent structures 3. Proper orthogonal decomposition 4. Galerkin projection Part II. Dynamical Systems: 5. Qualitative theory 6. Symmetry 7. One-dimensional 'turbulence' 8. Randomly perturbed systems Part III. 9. Low-dimensional Models: 10. Behaviour of the models Part IV. Other Applications and Related Work: 11. Some other fluid problems 12. Review: prospects for rigor Bibliography.

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Turbulence, Coherent Structures, Dynamical Systems and Symmetry

A new subgrid scale model is proposed for Large Eddy Simulations in complex geometries. This model which is based on the square of the velocity gradient tensor accounts for the effects of both the strain and the rotation rate of the smallest resolved turbulent fluctuations. Moreover it recovers the proper y
3 near-wall scaling for the eddy viscosity without requiring dynamic procedure. It is also shown from a periodic turbulent pipe flow computation that the model can handle transition.

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Subgrid-scale stress modelling based on the square of the velocity gradient tensor

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

▪ Abstract We review the direct numerical simulation (DNS) of turbulent flows. We stress that DNS is a research tool, and not a brute-force solution to the Navier-Stokes equations for engineering problems. The wide range of scales in turbulent flows requires that care be taken in their numerical solution. We discuss related numerical issues such as boundary conditions and spatial and temporal discretization. Significant insight into turbulence physics has been gained from DNS of certain idealized flows that cannot be easily attained in the laboratory. We discuss some examples. Further, we illustrate the complementary nature of experiments and computations in turbulence research. Examples are provided where DNS data has been used to evaluate measurement accuracy. Finally, we consider how DNS has impacted turbulence modeling and provided further insight into the structure of turbulent boundary layers.

/pdf/direct-numerical-simulation-a-tool-in-turbulence-research-2a650vh1lf.pdf

DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research

▪ Abstract Relationships between small and large scales of motion in turbulent flows are of much interest in large-eddy simulation of turbulence, in which small scales are not explicitly resolved and must be modeled. This paper reviews models that are based on scale-invariance properties of high-Reynolds-number turbulence in the inertial range. The review starts with the Smagorinsky model, but the focus is on dynamic and similarity subgrid models and on evaluating how well these models reproduce the true impact of the small scales on large-scale physics and how they perform in numerical simulations. Various criteria to evaluate the model performance are discussed, including the so-called a posteriori and a priori studies based on direct numerical simulation and experimental data. Issues are addressed mainly in the context of canonical, incompressible flows, but extensions to scalar-transport, compressible, and reacting flows are also mentioned. Other recent modeling approaches are briefly introduced.

Scale-Invariance and Turbulence Models for Large-Eddy Simulation

Direct numerical simulations (DNS) and experiments are carried out to study fully developed turbulent pipe flow at Reynolds number Rec ≈ 7000 based on centreline velocity and pipe diameter The agreement between numerical and experimental results is excellent for the lower-order statistics (mean flow and turbulence intensities) and reasonably good for the higher-order statistics (skewness and flatness factors) To investigate the differences between fully developed turbulent flow in an axisymmetric pipe and a plane channel geometry, the present DNS results are compared to those obtained from a channel flow simulation Beside the mean flow properties and turbulence statistics up to fourth order, the energy budgets of the Reynolds-stress components are computed and compared The present results show that the mean velocity profile in the pipe fails to conform to the accepted law of the wall, in contrast to the channel flow This confirms earlier observations reported in the literature The statistics on fluctuating velocities, including the energy budgets of the Reynolds stresses, appear to be less affected by the axisymmetric pipe geometry Only the skewness factor of the normal-to-the-wall velocity fluctuations differs in the pipe flow compared to the channel flow The energy budgets illustrate that the normal-to-the-wall velocity fluctuations in the pipe are altered owing to a different ‘impingement’ or ‘splatting’ mechanism close to the curved wall

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002211209400131X

Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment

A method for large eddy simulation (LES) is presented in which the sub-grid-scale modeling is achieved by filtering procedures alone The procedure derives from a deconvolution model, and provides a mathematically consistent approximation of unresolved terms arising from any type of nonlinearity The formal steps of primary filtering to obtain LES equations, approximate deconvolution to construct the subgrid model term and regularization are combined into an equivalent filter This filter should be an almost perfect low pass filter below a cut-off wavenumber and then fall off smoothly The procedure has been applied to a pressure-velocity-entropy formulation of the Navier–Stokes equations for compressible flow to perform LES of two fully developed, turbulent, supersonic channel flows and has been assessed by comparison against direct numerical simulation (DNS) data Mach numbers are 15 and 30, and Reynolds numbers are 3000 and 6000, respectively Effects of filter cut-off location, choice of differentiation scheme (a fifth-order compact upwind formula and a symmetric sixth-order compact formula were used), and grid refinement are examined The effects are consistent with, and are readily understood by reference to, filtering characteristics of the differentiation and the LES filter All simulations demonstrate a uniform convergence towards their respective DNS solutions

An explicit filtering method for large eddy simulation of compressible flows

Turbulence in supersonic channel flow is studied using direct numerical simulation. The ability of outer and inner scalings to collapse profiles of turbulent stresses onto their incompressible counterparts is investigated. Such collapse is adequate with outer scaling when sufficiently far from the wall, but not with inner scaling. Compressibility effects on the turbulent stresses, their anisotropy, and their balance equations are identified. A reduction in the near-wall pressure-strain, found responsible for the changed Reynolds-stress profiles, is explained using a Green's-function-based analysis of the pressure field

Compressibility effects and turbulence scalings in supersonic channel flow

The effects of compressibility are studied in low Reynolds number turbulent supersonic channel flow via a direct numerical simulation. A pressure-velocity-entropy formulation of the compressible Navier-Stokes equations which is cast in a characteristic, non-conservative form and allows one to specify exact wall boundary conditions, consistent with the field equations, is integrated using a fifth-order compact upwind scheme for the Euler part, a fourth-order Pade scheme for the viscous terms and a third-order low-storage Runge-Kutta time integration method. Coleman et al fully developed supersonic channel flow at M?=?1.5 and Re?=?3000 is used to test the method. The nature of fluctuating variables is investigated in detail for the wall layer and the core region based on scatter plots. Fluctuations conditioned on sweeps and ejections in the wall layer are especially instructive, showing that positive temperature, entropy and total temperature fluctuations are mainly due to sweep events in this specific situ...

Turbulent supersonic channel flow

Direct numerical simulation databases of turbulent channel and pipe flow have been used in order to assess the energy transfer between resolved and unresolved motions in large‐eddy simulations. To this end, the velocity fields are split into three parts: a statistically stationary mean flow, the resolved, and the unresolved turbulent fluctuations. The distinction between the resolved and unresolved motions is based on the application of a cutoff filter in spectral space. Within the buffer layer a backward transfer of averaged kinetic energy from subgrid to grid‐scale turbulent motions has been found to exist, which is primarily caused by subgrid‐scale stresses aligned with the mean rates of strain. Such reverse transfer generally cannot be described by the simple eddy‐viscosity‐type subgrid models usually applied in large‐eddy simulations. The use of a conditional averaging technique revealed that the reverse transfer of energy within the near‐wall flow is strongly enhanced by coherent motions, such as the well‐known bursting events.

Rainer Friedrich

Papers

Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment

An explicit filtering method for large eddy simulation of compressible flows

Compressibility effects and turbulence scalings in supersonic channel flow

Turbulent supersonic channel flow

Subgrid‐scale energy transfer in the near‐wall region of turbulent flows