Progress in Turbulence III

Fluid turbulence determines drag characteristics of land/air/seaborne vehicles, mixing and reaction rates in chemical reactors, industrial mixers, burners, and complex non-equilibrium phenomena such as flame reignition and extinction. It is central to weather and environmental predictions, cloud precipitation and albido, atmospheric and oceanic transport and ocean-atmosphere interactions. Central to three-dimensional turbulent fluid flows is the vortex-stretching mechanism which generates the multitude of eddy sizes that make simulations prohibitive. Studies aiming at understanding the intrinsic dynamics of fully developed turbulence have concentrated on homogeneous isotropic turbulence in the wind tunnel [1, 2, 3], where the turbulence dynamics result purely from the vortex-stretching mechanism in isolation. Here we report the discovery of a new type of homogeneous isotropic three-dimensional fluid turbulence where the average vortex stretching diminishes and the level of small-scale intermittency remains constant as the turbulence intensifies. Hence, the deepest of all turbulence properties, vortex stretching and intermittency, can be tampered with.

Spectral and physical forcing algorithms to generate stationary isotropic turbulence are compared in terms of characteristics of turbulence. Homogeneous isotropic turbulence is first validated with the stochastic forcing algorithm using a different shell of wavenumbers. Linear forcing algorithm is then applied to the generated stochastic field to keep the characteristics of turbulence stationary in time. Linearly forcing is very fast because there is no need to transform from a spectral to physical space. However, levels of fluctuations are very high that very long duration for simulations needed to achieve robust statistics.

Investigating the multi-point correlation (MPC) equations for the velocity and pressure fluctuations in the limit of homogeneous turbulence a new scaling symmetry has been discovered. Interesting enought this property is not shared with the Euler or Navier-Stokes equations from which the MPC equations have orginally emerged. This was first observed for parallel wall-bounded shear flows (see [2]) though there this property only holds true for the two-point equation. Hence, in a strict sense there it is broken for higher order correlation equations. Presently using this extended set of symmetry groups a much wider class of invariant solutions or turbulent scaling laws is derived for homogeneous and homogeneous-isotropic turbulence which is in stark contrast to the classical power law decay arising from Birkhoff’s or Loitsiansky’s integrals. In particular, we show that the experimentally observed specific scaling properties of fractal-generated turbulence (see [1, 4]) fall into this new class of solutions. Due to this specific grid a breaking of the classical scaling symmetries due to a wide range of scales acting on the flow is accomplished. This in particular leads to a constant integral and Taylor length scale downstream of the fractal grid and the exponential decay of the turbulent kinetic energy along the same axis. These particular properties can only be conceived from MPC equations using the new scaling symmetry. The latter new scaling law may have been the first clear indication towards the existence of the extended statistical scaling group. Though the latter is not obvious from the instantaneous Euler or Navier-Stokes equations it is directly implied.

The dependence on the lateral distance

y

to the center-plane and on time

t

of the average of the scalar dissipation rate

$\chi=2D|\nabla Z|^2$

conditioned on the scalar

Z

has been investigated in a temporally-evolving shear layer using a direct numerical simulation. As the inviscid scaling of the unconditional mean dissipation rate is approached, the conditional mean

$\bar{\chi}_Z$

follows also an approximate self-similar behavior within the time interval considered in the simulations and

$\bar{\chi}_Z$

shows two strong peaks at a distance of about one vorticity thickness from the center-plane, with dissipation values about 3 times the one in the central region. It is showed that this spatial variation is introduced by the nonturbulent/turbulent transition regions, which are identified by means of gradient trajectories of the scalar field.

A procedure to construct turbulence models is outlined beginning with the simplest case, a model for weak time-dependent perturbations of homogeneous isotropic turbulence, and ending with some models for inhomogeneous turbulence. The approach combines features of Yoshizawa’s two-scale direct interaction approximation and the Hilbert expansion of kinetic theory.

Kolmogorov’s statistical theory of turbulence is based on the existence of the invariant measure of the Navier-Stokes flow. Recently the existence of the invariant measure was established in the three-dimensional case [2]. It was established for uni-directional flow in [1] and for rivers in [3]. Below we will discuss how one can try to go about approximating the invariant measure in three dimensions.

We examine the spatial Markov properties of all three velocity components in inhomogeneous turbulence. We use measurement data of the axisymmetric far wake behind a disk at

Re

 = 2·10

4

, measured simultaneously with cross hot wire probes at twelve different distances from the flow axis. We show that the velocity components and Reynolds stresses can be approximated by Markov processes for large enough separations perpendicular to the flow direction. Our results indicate that the n-point correlations of the velocity components and the Reynolds-stresses in inhomogeneous turbulence might be approximated by a stochastic process governed by a Fokker-Planck equation, which could be the basis of a stochastic closure of the Reynolds averaged momentum equations.

The multiple-point probability density

f

(

v

1

,

r

1

;

v

2

,

r

2

; ...

v

N

,

r

N

) of velocity increments

v

i

at different length scales

r

i

is investigated in a direct numerical simulation of two-dimensional turbulence. It has been shown for experimental data of three-dimensional turbulence, that this probability density can be represented by conditional probability densities in form of a Markov chain [1]. We have extended this analysis to the case of two-dimensional forced turbulence in the inverse cascade regime.

Two-dimensional turbulence admits two different ranges of scales: a direct enstrophy cascade from the injection scale to the small scales and an inverse energy cascade at large scales. It has already been shown in previous papers that vortical structures are responsible for the transfers of energy upscale while filamentary structures are responsible for the forward transfer of the enstrophy. Here we introduce an original wavelet-based new mathematical tool, the interaction function, for studying the space localization of the enstrophy fluxes. It is based on twodimensional orthogonal wavelet decompositions of the two terms involved in the transport term in the Navier-Stokes equations.

In the context of dissipation element analysis of scalar fields in turbulence [1], the elongation of elements by the velocity difference at the minimum and maximum points was found to increase linearly with the length of an element. To provide a theoretical basis for this finding by analyzing two-point properties along the gradient trajectories, an equation for the mean product of the scalar gradient at two points along the same trajectory is derived. In the inertial range a balance similar to that from which Kolmogorov’s 4/5 law can be derived.While that law leads to a 1/3 scaling for the velocity difference, by conditioning on gradient trajectories we obtain a linear relation between the velocity difference and the two-point’s arclength on the same trajectory. Results from DNS show satisfactory agreement with the theoretical prediction.

We present a stochastic analysis of turbulence data, which provides access to the joint probability of finding velocity increments at several scales. The underlying stochastic process in form of a Fokker-Planck equation can be reconstructed from given data. Intermittency effects are included. The stochastic process is Markovian for scale separations larger than the

Einstein-Markov coherence length

l

EM

, which is closely related to the Taylor microscale

λ

. We extend our analysis to turbulence generated by a fractal square grid. We find that in contrast to other types of turbulence, like free-jet turbulence, the n-scale statistics of the velocity increments and the leading coefficients of the Fokker-Planck equation do not depend strongly on the Reynolds number.

Holographic PIV (HPIV) is a fully three-dimensional technique for measuring flows. A field of small tracer-particles is recorded in double-exposure holograms obtained with two reference waves. Particle displacements are extracted from CCD-based 3D-scans of real images with three-dimensional correlation methods. To cope with the problem of noise from out-of-focus particles the technique of light-in-flight holography has been introduced that utilizes properties of low coherent laser light. This drastically increases the signal-to-noise ratio in deep-field particle holography. Some experimental checks of these concepts are presented and first results from measurements in the wake of an airfoil are shown.

Tomographic Particle Image Velocimetry (Tomo-PIV) is a promising new PIV technique. However, its high computational costs often make time-resolved measurements impractical. In this paper, a new preprocesseing technique is tested experimentally for the first time on a simple vortex ring. The new method produces very similar velocity and vorticity distributions to the standard iterated solution, at a fraction of the computational cost. Therefore, through this new technique, the processing of thousands of vector fields required for turbulent statistics can be made significantly more affordable; an important step in the development of Tomo-PIV.

We present data measured with our new 2D Laser-Cantilever- Anemometer (2D LCA) in comparison to data acquired with a commercial x-wire anemometer.Measurements were taken in the wake of a cylinder with a diameter of

D

 = 1

cm

in a distance of 68D. Stochastic analyses show good agreement between the measured data of both anemometers.

Experiments using stereoscopic high-speed particle image velocimetry (PIV) to take measurements in a cross-stream (i.e. wall-normal/spanwise) plane in a turbulent boundary layer have been used to produce full 3D velocity fields. The 3D fields were constructed from planar 3C fields by using Taylor’s hypothesis to create a pseudo-spatial x-dimension from the temporally resolved measurements. This has produced a 3D viewof the elongated regions of high and low streamwise momentum found in the boundary layer, often referred to as ‘long structures’, and provided information on the arrangement, length and characteristic angle of these structures. Long structures are also seen to be associated with regions of high Reynolds stress. Vortical motions are visualised using swirling strength, and indicate that there is a prevalence for vortices to surround the low speed long structures. The vortices are characterised, and are found to resemble hairpin vortices in some respects.

The main problem of cup anemometry is the different response time for increasing and decreasing wind velocities due to its moment of inertia. This results in an overestimation of wind speed under turbulent wind conditions, the so-called over-speeding. Additionally, routine calibrations are necessary due to the wear of bearings. Motivated by these problems the sphere anemometer, a new simple and robust sensor for wind velocity measurements without moving parts, was developed at the University of Oldenburg. In contrast to other known thrust-based sensors, the sphere anemometer uses the light pointer principle to detect the deflection of a bending tube caused by the drag force acting on a sphere mounted at its top. This technique allows the simultaneous determination of wind speed and direction via a two-dimensional position sensitive detector.

The behaviour of the newly designed sensor under turbulent conditions was investigated by comparative measurements of gusty wind using the sphere anemometer, cup anemometer and a hot-wire as a reference. It turned out that the sphere anemometer provides a better temporal and spatial resolution than cup anemometers do.

High Reynolds number turbulence is ubiquitous in a number of flow of practical interest and crucial to draw conclusions regarding the physics of turbulence. Although recent laboratory experiments, measurements in planetary boundary layer and direct numerical simulations provide a huge amount of information, none of these data sets provide high Reynolds number, high spatial resolution and well converged statistics at the same time. As a response to this problem, an international collaboration between a group of universities and research centers started some years ago to build large scale infrastructures for high Reynolds number experiments. The Center for International Cooperation in Long Pipe Experiments, CICLOPE (www.ciclope.unibo.it) at the University of Bologna, was created for this purpose and will be open to international scientists through different collaboration programs. The laboratory is currently under construction and the first facility, which will be installed there is a large pipe flow experiment that will allow fully resolved turbulence measurements even at high Reynolds number.

A comprehensive experimental study is conducted of the aerodynamic characteristics of an NACA0012 airfoil over a large range of angle (

α

) of attack and low- to ultra-low cord Reynolds numbers, 5.3×10

3

- 5.1×10

4

, which is of both fundamental and practical importance. While the mean and fluctuating lift and drag coefficients were measured using a load cell, the detailed flow structure is documented using particle image velocimetry and laser-induced fluorescence flow visualization.

C

D

increases monotonically from

α

 = 0° to 90°, whilst

C

L

grows from 0 to its maximum at

α

 = 45° and then drops. The near wake characteristics examined were found to be consistent with the force measurements, including the vortex formation length, wake width, spanwise vorticity, wake bubble size and wavelength of K-H vortices.

We propose a new turbulence closure model based on the budget equations for the key second moments: turbulent kinetic energy (TKE), turbulent potential energy (TPE) and vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature). Besides the concept of the turbulent total energy (TTE = TKE + TPE), we take into account the non-gradient correction to the traditional buoyancy flux formulation. The proposed model permits the existence of turbulence at any gradient Richardson number, Ri. For the stationary, homogeneous regime the turbulence closure model yields universal dependencies of the flux Richardson number, turbulent Prandtl number, anisotropy of turbulence, and normalized vertical fluxes of momentum and heat on the gradient Richardson number, Ri.We also take into account an additional vertical flux ofmomentum and additional productions of turbulent kinetic energy, turbulent potential energy and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). Accounting for the internal gravity waves, the Ri-dependencies of the flux Richardson number, turbulent Prandtl number, anisotropy of turbulence, vertical fluxes of momentum and heat lose their universality. In particular, with increasing wave energy, the maximal value of the flux Richardson number (attained at very large Ri) decreases. In contrast to the mean wind shear which generates only the horizontal TKE, IGW generate both horizontal and vertical TKE, and thus lead to a more isotropic turbulence at very large Ri. IGW also increase the share of TPE in the turbulent total energy. A well-known effect of IGW is their direct contribution to the vertical transport ofmomentum.Depending on the direction (downward or upward), it either strengthens of weakens the total vertical flux of momentum. Predictions from the proposed model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulations (DNS) and large-eddy simulations (LES).

Wind turbines are commonly placed in complex terrain meaning that they may work under conditions of e.g. high turbulence level. This and other effects may give rise to that the turbine works under yawed inflow with respect to the turbine disc plane during a large fraction of its operational time. In this paper we show how the inflow towards the rotor disc is affected by yaw through PIV measurements.

This study is about the meandering phenomenon of a wind turbine wake. The atmospheric boundary layer is reproduced in a wind tunnel as well as wind turbines using porous discs. Hot wire anemometry is used to carry out time investigations through the study of space-time correlations. Particle Image Velocimetry is also used to observe the spatial development of the wake and especially the horizontal oscillations characterising the meandering.

In this contribution, we focus on the relevance of effects of atmospheric turbulence on the power output of a wind energy converter system. In particular, we introduce and critically discuss a dynamical approach for an appropriate mapping of the power conversion process, which provides a basis for modelling the transfer of turbulent structures from the wind to the corresponding output of electrical power.

Under certain conditions the first order geometric auto regressive process has statistical properties similar to atmospheric boundary layer wind speed. In this contribution, we investigate this similarity and analyse the extent to which this stochastic process is a suitable model for wind speed simulation.

A study of the turbulent air flow around a wind turbine based on unsteady Reynolds-averared Navier-Stokes (URANS) simulation in order to determine the induced loads on its blades for off-design flow cases is performed. In particular the impact of applied turbulence models on the flow simulation results will be discussed.

Climate change and finite fossil fuel resources make it urgent to turn into electricity generation from mostly renewable energies. One major part will play wind-energy supplied by wind-turbines of rated power up to 10 MW. For their design and development wind field models have to be used. The standard models are based on the empirical spectra, for example by von Karman or Kaimal. From investigation of measured data it is clear that gusts are underrepresented in such models. Based on some fundamental discoveries of the nature of turbulence by Friedrich [1] derived from the Navier-Stokes equation directly, we used the concept of Continuous Time Random Walks to construct three dimensional wind fields obeying non-Gaussian statistics. These wind fields were used to estimate critical fatigue loads necessary within the certification process. Calculations are carried out with an implementation of a beam-model (FLEX5) for two types of state-of-the-art wind turbines The authors considered the edgewise and flapwise blade-root bending moments as well as tilt moment at tower top due to the standard wind field models and our new non-Gaussian wind field model. Clear differences in the loads were found.

Wind energy converters such as wind turbines permanently work in the atmospheric boundary layer. For the modelling of the dynamics and for the optimisation of design and material of wind turbines synthetic models for atmospheric turbulence are applied already for a long time. The main purpose of these models is to provide fast and efficient methods for numerical simulation of random fields, that show some characteristic features of atmospheric turbulence. Typically they only have a partial connection to the fundamental equations of fluid dynamics. After a short overview summarizing widespread models by Veers and Mann, that are based on the simulation of random fields in the Fourier domain, advanced models for the simulation of velocity fields are discussed.

Two unsteady numerical techniques, LES and PANS with different computer requirements were used to predict of the flow around a tall finite cylinder. The well resolved LES was found to predict the flow in agreement with previous experimental observations, while PANS was found to suffer from the combination of the

k

 − 

ε

model and wall function close to the wall of the cylinder and too coarse a resolution.

Large Eddy Simulations of turbulent flows around a segment of the FX-79-W151 rotor blade have been performed for the Reynolds numbers

Re

 = 5·10

3

and

Re

 = 5·10

4

and the angle of attack 12° using a spectral element method with 8th order polynomials. The turbulence statistics obtained in the LES reveal regions of laminar and turbulent flow separation for the lower and higher Reynolds numbers, respectively, which lead to different loads on the blade.

Direct Numerical Simulations (DNS) of pipe flow with and without swirl are carried out and compared. Swirling pipe flow due to pipe rotation has been reported numerous times in the past, e.g. by DNS in [5] or by experiments in [2]. In the present case, a pressure gradient in the azimuthal direction is setting up a swirl in the near wall region. In comparison to the rotating pipe, the swirl is to a greater extent concentrated close to the pipe wall. However, while the rotating pipe has been investigated experimentally several times, e.g. in [2], the present case is not realizable in practice. A 25R long pipeline with diameter

D

 = 2

R

is considered. The pipe is divided into about 6.3 million grid cells in a cylindrical coordinate system (129, 97, and 512 in

θ

, r, and z, respectively). The Navier-Stokes equations are solved by a finite-difference code, developed by Orlandi [4], on a staggered grid, non-uniform in the radial direction. In addition, the flow is driven by an axial pressure gradient sufficient to keep a constant bulk Reynolds number

Re

b

 = 

U

b

D

/

ν

= 4900. Here,

U

b

is the bulk velocity and

ν

is the kinematic viscosity.

U

p

( = 2

U

b

) is the centerline velocity in the laminar Poiseuille profile used as the initial start up profile. Cyclic boundary conditions are used in the axial- and the azimuthal direction.

The objective of this study is to develop models for the subgrid-scale (SGS) stress and the SGS scalar flux by applying the same kind of methodology that leads to the explicit algebraic Reynolds stress model, EARSM [2], and the explicit algebraic scalar flux model, EASFM [3], for RANS. The idea is that these new models can improve the description of the anisotropy compared to eddy viscosity models. Since the new models can include the effect of system rotation in a natural way they have a particular potential for rotating flows.

Flows with favorable pressure gradient (FPG) and adverse pressure gradient (APG) are of great importance practically and theoretically. In practice, many industrial applications, especially aerodynamics, involve flows with pressure gradients and separation. In theory, wall shear stress does not dominate this type of flow[8].Much remains not understood.

When developing turbulence modelling from scratch certain questions arise which inevitably turn into methodological problems regarding this topic

What makes Euclidean transformations in classical continuum mechanics, in particular turbulence modelling, so special ?

Why is frame-dependency in all unclosed terms of existing algebraic models, e.g. in the Reynolds-stress tensor, always only modelled by the mean objective intrinsic spin tensor, i.e. the mean vorticity tensor measured in a rotating frame relative to an inertial frame: 〈

W

ij

〉 = 〈

ω

ij

〉 + 

ε

ijk

Ω

k

?

Why is the mean pressure or one of its gradients never taken along as a closure variable ?

Why does there still not exist a clear-cut mathematical formulation of the material frame-indifference (MFI)-principle in general continuum mechanics (if it applies as a physical approximation for reducing constitutive equations) ? What consequences does a proper mathematical formulation have for modelling turbulence in the limit of a 2D flow state ?

Answers to these questions are given herein, except for the last question which is beyond the scope of this article [1]. From the outset it is clear, that in order to give an unambiguous answer to these interlinked questions one needs a new mathematical framework, or more precisely, a setting of universal form-invariance (UFI) which extends the classical framework of an Euclidean geometry being used so far [1].

A hybrid model based on the unsteady Reynolds averaged Navier-Stokes approach represented by the one-equation Spalart-Allmaras model and the Large Eddy Simulation called Detached Eddy Simulation (DES) was applied for turbulent flow simulations. This turbulent approach was implemented into the flow solver based on the Finite-Element Method with pressure stabilized and streamlines upwind Petrov-Galerkin stabilization techniques. The effectiveness and robustness of this updated solver is successfully demonstrated at benchmark calculation represented by an unsteady turbulent flow past a cylinder at Reynolds number 3900. Results such as velocity fields and the flow periodicity, Reynolds stress tensor and eddy viscosity and pressure coefficient distributions are discussed and relatively good agreement was found to direct numerical simulations and experiments.

We develop an efficient method to generate convecting synthetic turbulence in complex mean flows. In this article we present the extension to anisotropic turbulence, realistic spectra and sweeping effects, i.e. the advection of inertial range structures by the energy containing large scales. We show that due to this formulation sweeping effects can be included and the spatial-temporal field shows the similarity of Kraichnan’s sweeping hypothesis.

The flow in plate-fin-and-tube heat exchangers is featured by interesting dynamics of vortical structures, which, due to close proximity of bounding walls that suppress instabilities, differs significantly from the better-known patterns around long cylinders. Typically, several distinct vortex systems can be identified both in front and behind the pin. Their signature on the pin and end-walls reflects directly in the local heat transfer. The Reynolds numbers is usually moderate and the incoming flow is non-turbulent, transiting to turbulence on or just behind the first or few subsequent pin/tube rows. Upstream from the first pin a sequence of several horseshoe vortices attached to the boundingwall is created, while the unsteady wake contains also multiple vortical systems which control the entrainment of fresh fluid and its mixing with the hot fluid that was in contact with the heated surfaces [1]. The conventional CFD using standard turbulence models, as practiced by heat exchangers industries, falls short in capturing the subtle details of the complex vortex systems. A fine-grid LES can provide accurate solutions, but for more complex configurations and higher Re numbers a hybrid RANS/LES using a coarser grid seems a more rational option, provided it can capture all important flow and vortical features.

In order to shed more light on the flow structures, their role in heat transfer and the capabilities of a simple hybrid model to return the salient flow feature, we conduct in parallel a well-resolved LES, and coarse mesh hybrid and URANS simulations of initially non-turbulent flow over a single heated short cylinder bounded by infinite walls.

The background of three dimensional (3D) hydrodynamic/vortical fluctuations in a stochastically forced, laminar and incompressible plane Couette flow is simulated numerically. It was found that the fluctuating background in the flow has the following characteristics: The hydrodynamic fluctuations show the nonexponential, transient growth; an anisotropy of the fluctuating velocity field increases with the shear rate; existence of the streamwise structural regularities (coherent structures) with the characteristic length-scale of the order of a channel width; appearance of the nonzero velocity cross-correlations; Symmetry breaking of the spanwise reflection of the dynamical processes due to the stochastic forcing.

The well known challenge of turbulence modeling is the inclusion of the effects of the large-scale structure in the one-point model equations. We proposed an approach [1,2] to the Reynolds stresses modeling that is based on an unclosed equation in terms of the spectral tensorr

$\Phi_{i,j}(\vec{k})$

deduced in [3]:

$ \frac{d\Phi_{ij}}{dt} - E_{\alpha \beta} \frac{\partial k_{\alpha} \Phi_{ij}}{\partial k_{\beta}} + E_{i \alpha} \Phi_{\alpha j } + E_{j \alpha} \Phi_{i \alpha} - E_{j \alpha} \overline{u_{i} u_{\alpha}} $

$-2E_{\alpha \beta} \frac{k_\alpha}{k^2} (k_i \Phi_{\beta j} + k_j \Phi_{i \beta}) + \dots = 0 \,, (1)$

where

$E_{i j}(t) = \partial U_i / \partial x_j$

and dots denote missed nonlinear and dissipation terms approximated in our model.

In this paper, we analyze the heat transfer modulation produced by the addition of small particles to the base fluid in particle-laden channel flow. We performed direct numerical simulations without gravity at shear Reynolds number

Re

τ

= 150 and molecular Prandtl number

Pr

 = 3 (liquid-solid flow), using the Eulerian-Lagrangian point-particle approach and considering full (momentum and energy) coupling between the fluid and the particles. For thermally-developing flow, we notice significant changes (up to 10 %) to the fluid heat transfer which can be attributed to particle distribution not yet in equilibrium with turbulence. This transient state promotes the

extra

transfer mechanisms that modify strongly the overall heat fluxes. Results for fully-developed flow show that these changes tend to become negligible once the hydrodynamic and thermal equilibrium is achieved.

Numerical simulations are used to study the vertical dispersion of fluid particles in homogeneous turbulent flows with a stable stratification. The results of direct numerical simulations are in good agreement with the relation for the long time fluid particle dispersion, 〈

δz

2

〉 = 2

ε

P

t

/

N

2

, derived by [6], though with a small dependence on the buoyancy Reynolds number. Here, 〈

δz

2

〉 is the mean square vertical particle displacement,

ε

P

is the dissipation of potential energy,

t

is time and

N

is the Brunt-Väisälä frequency. A simulation with hyperviscosicity is performed to verify the relation

${\langle \delta z^2 \rangle} = (1+\pi C_{PL})2{\epsilon_P} t/N^2$

for shorter times, also derived by [6]. The agreement is reasonable and we find that

C

PL

~3. The onset of a plateau in 〈

δz

2

〉 is observed in the simulations at

t

~

E

P

/

ε

P

which scales as

$4E_P /N^2$

, where

E

P

is the potential energy.

Dynamics of inertial particles has been thoroughly analyzed for statistically homogeneous and isotropic flows where clustering occurs below the Kolmogorov scale. Since anisotropy is strongly depleted through the inertial range, the advecting field anisotropy may be expected in-influential for the small scale features of particle configurations. By addressing direct numerical simulations (DNS) of a statistically steady particle-laden homogeneous shear flow, we find instead that the small scales of the particle distribution are strongly affected by the geometry of velocity fluctuations at large scales even in the range of scales where isotropization of velocity statistics occurs.

The dynamics of small inertial particles transported by a turbulent flow is crucial in many engineering applications. For instance internal combustion engines or rockets involve the interaction between small droplets, chemical kinetics and turbulence. Small, diluted particles, much heavier than the carrier fluid, are essentially forced only by the viscous drag i.e. the Stokes drag (gravity, feedback on fluid and collisions are neglected). The difference between particle velocity V and fluid

U

produces various anomalous phenomena such as small-scale clustering or preferential accumulation at the wall even for incompressible flows. To stress the interaction between wall bounded flows and particle dynamics we have performed a direct numerical simulation of a fully-developed particle-laden pipe flow. Seven different populations of particles are injected at a fixed location on the axis of the pipe and their evolution is analyzed for a streamwise extension of 200R ( with R the pipe radius) to asses the onset of turbophoresis.

The interaction of small, solid particles with turbulent coherent structures near the walls of a vertical channel is numerically investigated using direct numerical simulation (DNS) together with Lagrangian particle tracking (LPT) and a point-force approximation for the feedback effect of the particles on the fluid. Results from conditional sampling show that the diameter and the streamwise extent of the mean vortices are increased due to the momentum exchange between the two phases, depending on the particle inertia, gravitational settling and interparticle collisions.

We report high-resolution local temperature measurements in the upper and the lower boundary layer of turbulent Rayleigh–Bénard (RB) convection. The measurements were undertaken in air (

Pr

 = 0.71) at constant aspect ratio Γ= 1.13 and variable Rayleigh number 5×10

5

 < 

Ra

 < 10

12

. The primary purpose of the work is to preserve a comprehensive data set of the temperature field against which various phenomenological theories and numerical simulations can be tested. In our talk we show the mean temperature profiles

ϑ

h

(

z

) at the heating and

ϑ

c

(

z

) at the cooling plate. We demonstrate, that the corresponding profiles do not collapse even at low Rayleigh numbers and that the measured bulk temperature

ϑ

b

strongly deviates from the mean between the heating and the cooling plate

ϑ

b

,

t

. Normalizing all temperatures to the difference between the plates, the deviation remains constant for all

Ra

 > 2×10

11

.

Numerical investigations of the impact of riblets on natural transition in a temporal evolving channel flow have been performed. A set of riblet geometries are used for two different Reynolds numbers. The results show that transition is accelerated by all riblets compared with a smooth surface. This acceleration is caused by an amplification of the two-dimensional Tollmien-Schlichting (TS) waves by riblets, where the amplification magnitude dependeds on the specific geometry of the riblets.

We present the most relevant recent findings that allow for a rational timeaveraged description of laminar-turbulent transition of an incompressible nominally two-dimensional and steady boundary layer along the impermeable surface of a rigid blunt body. Rigorous application of matched asymptotic expansions for sufficiently high values of both the Reynolds number and a turbulence-level gauge parameter shows that the presence of a leading-edge stagnation point is associated with the generation of a turbulent shear layer that exhibits an asymptotically small streamwise velocity deficit. Remarkably, however, the turbulence intensity never reaches its theoretically possible maximum that conforms to fully developed turbulent flow.

Waves of spanwise velocity traveling along the walls of a plane turbulent channel flow are studied by Direct Numerical Simulations. We consider waves of spanwise velocity which are oscillating in time and modulated in space along the streamwise direction.Waves which slowly travel forward produce a large reduction of drag. Faster waves, with a phase speed corresponding to the convection velocity of the turbulent fluctuations at the wall, lock with the convecting near-wall turbulent structures and yield drag increase. Backward-traveling waves, on the other hand, invariably produce a drag-reducing effect. To explain the physical mechanisms involved, the generalized Stokes layer induced by the traveling waves is studied in the laminar regime.

We address the dynamics of viscoelastic wall turbulence by means of a generalization of a scale-by-scale approach extended to both an inhomogeneous and viscoelastic case. Analysing the results obtained by a series of Direct Numerical Simulations of a dilute polymer solution in a plane channel, we focus our attention on the alteration of the inertial transfer across the scales and the polymer scaledependent term in the budget for the second order velocity structure function. We confirm that both these observables lead to a scenario were the main alteration of turbulence structure in a channel flow occurs in the buffer layer.

Hairpin structures in the outer region of a turbulent boundary layer subjected to a strong adverse pressure gradient have been studied using PIV. The external flow conditions are similar to those found on the suction side of airfoils in trailing-edge post-stall conditions. Even if the flow is very different from zeropressure- gradient turbulent boundary layers, the gross features of the hairpin vortices and hairpin packets remain similar, even as separation is approached. The hairpin vortices are however slightly more inclined with respect to the wall, and their streamwise separation is smaller when scaled with the boundary layer thickness. The upward growth of the hairpin packets in the streamwise direction is also more important. The variations of these properties are consistent with the variations of the mean strain rates, in particular rates of streamwise contraction and wall-normal extension.

Experimental results are presented on the waviness of the wall-shear stress field of a two-dimensional transitional boundary layer in a zero-pressure gradient flow channel. Arrays of flexible micro-pillars are used as a dense grid of sensors to measure the surface distribution of tangential vorticity (longitudinal and transverse component) over time. The results show a quasi-periodic passage of varicose wave-like disturbances with strong anti-correlation of the transverse component. These are related to packets of 3-4 vortex loops representing the near-wall state of secondary varicose instability of the streaks.

We present a comparative experimental study of a turbulent flow developing in clear water and dilute polymer solutions (25 and 50 wppm polyethylene oxide). The flow is forced by a planar grid that oscillates vertically in a square container of initially still fluid. The two-component velocity fields are measured in a vertical plane passing through the center of the tank by using time resolved Particle Image Velocimetry (PIV).We obtain a lower entrainment rate for polymer solutions as compared to clear water. Extending arguments based on similarity and fractal theory to the case of dilute polymer solutions, we derive a relation between the entrainment rate and the fraction of input energy dissipated by the polymers.

We study experimentally and theoretically mixing at the external boundary of a submerged turbulent jet. In the experimental study we use Particle Image Velocimetry and an Image Processing Technique based on the analysis of the intensity of the Mie scattering to determine the spatial distribution of tracer particles. An air jet is seeded with the incense smoke particles which are characterized by large Schmidt number and small Stokes number. We determine the spatial distributions of the jet fluid characterized by a high concentration of the particles and of the ambient fluid characterized by a low concentration of the tracer particles. In the data analysis we use an approach that is based on analysis of the two-point second-order correlation function of the particle number density fluctuations generated by tangling of the gradient of the mean particle number density by the turbulent velocity field. This gradient is formed at the external boundary of a submerged turbulent jet. We demonstrate that the two-point second-order correlation function of the particle number density does not have universal scaling and cannot be described by a power-law function. The theoretical predictions made in this study are in a qualitative agreement with the obtained experimental results.

The turbulent flow driven by rotating and travelling magnetic fields in a closed cylinder is investigated by means of direct numerical simulations (DNS) and large eddy simulations (LES). Our model is based on the low-induction, low-frequency approximation and employs a spectral-element/Fourier method for discretisation. The spectral vanishing viscosity (SVV) technique was adopted for the LES. The study provides first insights into the developed turbulent flow. In the RMF case, Taylor-Görtler vortices remain the dominant turbulence mechanism, as already in the transitional regime. In contrast to previous predictions we found no evidence that the vortices are confined closer to the wall for higher forcing. In the TMF more than 50 percent of the kinetic energy is bound to the turbulent fluctuations, which renders this field an interesting candidate for mixing applications.

It is well accepted that turbulence in pipe flow is transient for

$Re \lesssim 2000$

and, after some time, always comes back to the laminar state. However, there is no current agreement about the behaviour of turbulence for higher

Re

: whereas some experiments indicate that turbulence becomes sustained after certain critical Reynolds number,

Re

C

, other studies show that the turbulent state keeps its transient behaviour, ruling out any critical point. Our experiments show that these different views are not caused by the different ways of generating turbulence or by some experimental noise, and therefore the turbulent state is always well defined. Our data also suggest that, in spite of the different interpretations, all experimental results presented up to date are compatible when some sistematic error is taken into account.

Direct numerical simulations (DNSs) with a spectral method are performed with large and small computational domains to study the effects of spanwise rotation on a turbulent Poiseuille flow at the very low-Reynolds numbers. In the case without system rotation, quasi-laminar and turbulent states appear side by side in the same computational domain, which is coined as laminar-turbulence pattern. However, in the case with system rotation, the pattern disappears and flow is dominated by quasi-laminar region including very long low-speed streaks coiled by chain-like vortical structures. Increasing the Reynolds number can not generate the laminar-turbulence pattern as long as system rotation is imposed.

A large-eddy simulation of the flow over a high-lift airfoil configuration consisting of a slat and a main wing is performed at a freestream Mach number M=0.16 and an angle of attack of 13°. The Reynolds number, based on the clean chord length and the freestream velocity, is Re=1.4·10

6

. The results show similarities between the turbulent structures of the slat cusp shear layer and a free shear layer and an impinging jet. The periodical occurrence of rollers and streamwise orientated rib vortices contributes essentially to the generated sound.

For many years investigations have been conducted in order to understand the flow instabilities that lead to transition in jets. Among the earliest, the inviscid linear stability analysis of Batchelor and Gill [1] showed that immediately at the jet exit, where the velocity profile has a ‘top-hat’ behaviour, all the instability modes are able to be exponentially amplified while in the far field region only the helical mode seems to be unstable. The transition between these two different instability regions is still unclear and the analysis is complicated by the presence of several unstable modes embedded in the turbulence background. Therefore, a large amount of analytical theories, simulations and experiments have been done in order to highlight the role and the dynamics of a single or few modes in the evolution of the flow (cf. [3] and [7]). Investigations

Over the past decades a variety of passive and active flow control mechanisms have been tested and applied in a variety of canonical as well as applied flow cases. An example for the latter is the coaxial jet flow, which has mainly been investigated regarding the receptivity to active flow control strategies (see e.g. [1]), probably due to the multitude of parameters characterising the complex flow field [2].

Physical and numerical experiments (see e.g. [3] and [5]) have established that the vortical motion in coaxial jet flows is dominated by the vortices emerging from the outer shear layer. The frequency of these vortices is related to the Kelvin- Helmholtz instability as predicted by linear stability analysis for single jets. The vortices in the inner shear layer, on the other hand, are trapped in the spaces left free between two consecutive outer shear layer vortices, and are therefore sharing the frequencies of the most amplified modes of the outer shear layer and do not relate to the values one would expect from linear stability analysis. This fact has become known as the “locking phenomenon”, which describes the mutual interaction of both shear layers. Nevertheless it is believed that only the outer shear layer is able to significantly control the evolution of the inner shear layer [7], which may explain the focus of control strategies on the outer shear layer.

In this work, direct numerical simulation of turbulence is coupled to lagrangian tracking to study the behavior of 220

μ

m bubbles in a vertical turbulent channel flow. Both one-way and two-way coupling approaches and both upward and downward flows are considered. For each simulation, the same external imposed pressure gradient is considered. In one-way simulations, this leads to a shear Reynolds number of Re = 150. In the coupled cases, the presence of bubbles increase/decrease the driving pressure gradient, respectively in upward/downward flow, thus yielding to an increase/decrease of the wall shear stress and of the shear Reynold number. For the considered bubble average volume fraction (

α

= 104 ), the corresponding shear Reynolds number are about

Re

τ

,2

U

= 174 for the upflow case and

Re

τ

,2

D

= 121 for the downflow case. Statistics of the fluid and of the bubble phase are presented. The interactions between bubble and the near-wall turbulence structures is also investigated and a preferential bubble segregation in high-speed/low-speed zones is observed for the upflow/downflow cases respectively. An attempt to describe the transfer rate between the gas and the liquid will be included with some preliminary results.

This article describes experimental and numerical investigations of the stability of a water jet with a rectangular cross section aligned with the gravity field. This work focuses on the individual physical phenomena influencing the stability of the free surface jet flow like the contraction of the planar jet as well as the generation of capillary waves. In order to investigate the interaction between these two effects a series of experiments using water as model fluid are conducted in the Karlsruhe Liquid metal Laboratory KALLA at the research centre Karlsruhe. For the measurement of the cross-sectional shape of free water jet the Laser Doppler Anemometer (LDA) has been applied. Complementary to the experiments the numerical simulations have been performed using the commercially available code STAR-CD. The simulations show a reasonable agreement with the experimental measurements within the investigated parameter range.

The separated flow around inclined airfoils can be controlled by unsteady actuation near the leading edge (LE), increasing the maximum lift coefficient without the need for heavy and complex high lift devices such as flaps [3]. Zero net mass flow devices (ZNMF) are often used for this purpose. While certainly favourable for industrial application, actuation via ZNMF faces some problems. In particular, to independently control both amplitude and frequency of the excitation is considered a “great challenge” [4]. This is even more severe when the wave form of the actuation is non-sinusoidal, i. e., contains more than one frequency component.

The effects of the wall-normal rotation on the turbulence channel flow have been studied. A series of direct numerical simulations have been performed with various rotation rates for Reynolds number 180 based on the friction velocity in the non-rotating case. All remarkable changes are discussed.

The vorticity field of fully developed turbulence displays a complex spatial structure consisting of a large number of entangled filamentary vortices (see illustration). As a consequence, the PDF of the vorticity shows a highly non-Gaussian shape with pronounced tails. In the present work a kinetic theory for the turbulent vorticity is presented. Under certain conditions the arising equation may be interpreted as a Fokker- Planck equation giving rise to a Langevin model. The appearing unknown conditional averages are estimated from direct numerical simulations. The Langevin model is shown to reproduce the single point vorticity PDF.

The paper presents results of numerical analysis for helical features of velocity field to investigate the process of tropical cyclone formation, namely, the downward helicity flux through the upper boundary of the viscous atmospheric turbulent boundary layer has been calculated. The simulation was carried out by use of the regional atmospheric ETA model and NCEP reanalysis global data. Calculations were performed for two tropical cyclones - Wilma (Atlantic basin, 2005) and Man-Yi (North-West Pacific, 2007). It has been found, that the chosen helical characteristic reveals an adequate response to basic trends in variation of such important meteorological fields as pressure and wind velocity during the hurricane vortex evolution. The analysis carried out in the paper shows that the helicity flux can be used as an illustrative characteristic to describe the intensity and destructive power of tropical cyclones.

The phenomenon of vortex rings can be observed in some aquatic animals producing propulsion, in oil drilling and gaseous pollution releases. Vortex rings in this laboratory study are generated by pushing a piston through a tube with an orifice opening in water. In this paper, turbulent vortex rings are studied by means of Stereoscopic Particle Image Velocimetry (PIV). Typical turbulence quantities e.g. the turbulent stress distribution can be visualized after some mathematical processes.