1 

Interaction between flood defences and pipelines subject to induced earthquake loads in Groningen
In the province of Groningen in recent years strong induced earthquakes are observed, during which a lot of energy is released in the form of seismic waves. These ultimately lead to horizontal ground accelerations at the ground surface, which affect amongst other things the stability of embankments. The main question to be answered in this report is whether the presence of a pipeline imposes an additional risk to the embankment stability, as a pipeline may fail before the embankment does. For this purpose use is made of various empirical and semianalytical methods and the outcomes of these methods are compared with the outcomes found in the finite element software PLAXIS 2D.
In this report a method is proposed in which first the stability of the embankment is checked before the earthquake begins. Ground accelerations may cause loosely packed sandy soils to rearrange to a compacter state. This is only possible if the water in between the soil particles can dissipate. As the shaking of the subsoil during an earthquake is so rapid, this is not the case and the pressure of the water increases during the earthquake. This causes the effective stresses between the soil particles to decrease, thus reducing the soil’s shear strength. When the shear strength reductions are highest at the end of an earthquake, the reduced embankment stability is checked again. The gradual reduction of the embankment stability during the earthquake is determined by coupling this to the intensity of accelerations during the earthquake. By using the accelerations higher than the acceleration at which stability is no longer guaranteed, the sliding plane displacement of the embankment is calculated. By applying this method to a case study soil profile consisting of sandy soil layers it is found that the calculated displacement increases exponentially with increasing values of the peak ground acceleration. When a soil profile of clayey soil layers is considered, the calculated deformations increase linearly with increasing values of the peak ground accelerations.
The response of a continuous pipeline is determined by imposing the sliding plane displacement on a springsupported elastic beam that represents the pipeline. The springs represent the interface stiffness of a pipeline that moves relative to the soil in fourdirections: axial, lateral, upward and downward. Plasticity of the pipeline is accounted for by looking at the pipeline crosssection at the intersection between the sliding plane and the pipeline and by using an iterative procedure between this crosssection and the springsupported beam analysis. It is found that a continuous pipeline can indeed fail before the embankment does, but for failure to occur the displacements have to be rather higher, the soil surrounding the pipeline rather strong and the limit state to indicate failure rather strict. The response of a segmented pipeline is determined by stating that the sliding plane displacement is accommodated by a maximum of two pipeline segments, of which the displacements depend on the relative stiffness of the upward and downward soilpipeline interfaces. It is found that the segmented pipelines can fail on the criteria of axial pullout and joint rotation before the embankment fails. As the axial pullout magnitude does not depend on the soil stiffness, it is expected that the probability of failure is higher than for the continuous pipeline at the same sliding plane displacement magnitudes.
The attempt to model the pipeline in PLAXIS 2D did not succeed as there are yet limitations to how the structural element can be applied. The ground deformations for the clayey soil profile are found to be in good agreement with the hand calculations. The ground deformations of the sandy soil profile are not. Depicting what causes the differences is very difficult, but it appears that the model used to determine the soil behaviour is very sensitive to sloped ground conditions. Insight can be given in mechanisms that are neglected in the hand calculations, but properly interpreting the results requires a lot of experience.

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2 

Wind and boundary layers in RayleighBénard convection. II: Boundary layer character and scaling

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3 

Wind and boundary layers in RayleighBénard convection. I: Analysis and modeling

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4 

Stability of offshore risers conveying fluid

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5 

A gradient plasticity approach to finite element predictions of soil instability

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6 

Channel and shoal development in a short tidal embayment: an idealized model study
In many tidal embayments, complex patterns of channels and shoals are observed. To gain a better understanding of these features, an idealized model, that describes the interaction of water motion, sediment transport and bed evolution in a semienclosed, rectangular basin, is developed and analysed. To explain the initial formation of channels and shoals, twodimensional perturbations superposed on a laterally uniform equilibrium bottom are studied. These perturbations evolve due to convergences of various residual suspended sediment fluxes: a diffusive flux, a flux related to the
bed topography, an advective flux resulting from internally generated overtides and an advective flux due to externally prescribed overtides. For most combinations of these fluxes, perturbations start to grow if the bottom friction is strong enough. Their growth is mainly a result of convergences of diffusive and topographically induced sediment fluxes. Advective contributions due to internally generated overtides enhance this growth. If only diffusive sediment fluxes are considered, the underlying equilibrium is always unstable. This can be traced back to the depth dependence of the deposition parameter. Contrary to the results of previous idealized models, the channels and shoals always initiate in the shallow, landward areas. This is explained by the enhanced generation (compared to that in previous models) of frictional torques in shallow regions. The resulting initial channel–shoal formation compares well with results found in complex numerical model studies. The instability mechanism and the location of the initial formation of bottom patterns do not change qualitatively when varying parameters. Changes are mainly related to differences in the underlying
equilibrium profile due to parameter variations.

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7 

Effectiveness of sensors in flood defences
Introduction
The popularity of monitoring dikes with sensor techniques is rising. It is claimed that sensor techniques lead to significant cost savings and can predict an upcoming dike collapse. But a technical foundation to use the sensor monitoring information in flood safety assessment is lacking. This research investigates the contribution of sensor monitoring information to flood safety and the costeffectiveness of sensor monitoring. Sensor techniques have been tested in fullscale dike failure experiments at the IJkdijk, trying to predict an upcoming dike collapse. The sensor techniques are capable of monitoring deformation, temperature, water pressure, vibrations and moisture. The state of the sensor techniques is doubtful due to subjective analyses, controlled test conditions and a wide variety in failure prediction times: from 1,5 to 102 hours.
Implementation of sensor information
Water pressure is the only variable that constitutes an input for dike safety assessment models. Monitoring water pressures affects the epistemic uncertainty of the water pressure schematization which is caused by the translation from the hydraulic load to water pressures. One must be aware that sensor monitoring either leads to an increased assessment of flood safety if the prior schematization turns out to be done conservatively or a decrease in flood safety if the prior schematization turns out too optimistic. One would expect an increased assessed flood safety due the intended conservative approach. But prior schematization mistakes imply a decreased assessed flood safety. Moreover, monitoring water pressures has minimum impact on the flood safety assessment if other uncertainty aspects dominate the stability assessment. A case study of the canal of Nauerna denotes that the water pressure has resulted in a higher assessed flood safety, but the uncertainties regarding the soil conditions dominate the stability assessment. However, sensor monitoring on itself does not affect the real flood safety: only physical measures affect the real risk of flooding. Important information is obtained from monitoring high water events such that water pressure models can be calibrated to determine design loading conditions for the periodic safety assessment. Also, an additional application is to identify unforeseen risks.
Costbenefit analysis
Conceptual costbenefit models have been set up to determine the costeffectiveness of sensor monitoring over the longterm. The monitoring costs consist of installation, maintenance and operational costs. The benefit from permanent dike reinforcements is gained from specifying the longterm optimal investment strategy based on the minimum sum of flood risk and reinforcements costs. The monitoring information from relevant high water events affects the assessed flooding probability. If the sensor monitoring reduces the assessed flooding probability, the assessed flood risk lowers and savings on permanent dike reinforcements. If a higher flooding probability is obtained by sensor monitoring, this financially leads to additional investments and negative benefits. But the value of knowing this higher flood risk is rationally beneficial. The benefit from temporary measures is gained from timely execution of emergency measures based on the early warning of the sensor system. This benefit depends on the prediction time of the sensor monitoring system and the reaction time to execute the emergency measure. Additional costs for executing the emergency measure must be incorporated. The costbenefit models have been worked out in case studies for dikering 48 and 14.
Conclusion
The conclusion of this research is that sensor monitoring can be implemented in the flood safety, by specifying dike reinforcements in both the periodic safety assessment, as well as the operational situation. However, the investments in sensor monitoring have to be made while a long waiting time is expected before benefits turn out. Then these benefits can financially be disappointing, but do have a certain value of information.

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8 

Cost optimal river dike design using probabilistic methods
This research follows a fully probabilistic approach in order to estimate the optimal design for a river dike crosssection, taking into account the investment costs.
From the theory studied, the failure mechanisms that contribute most to the failure of river dikes are identified. These are overflowing, wave overtopping, piping and inner slope stability. The most important design variables of the dike crosssection dimensions are set and following probabilistic design methods, the probability of failure of many different dike crosssections is estimated based on the abovementioned failure mechanisms.
The aim of the study is to develop a generic method that automatically estimates the failure probabilities of many river dike crosssections and gives the one with the least cost, taking into account the boundary conditions and the requirements that are set by the user.

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9 

Aging Mitigation Schemes for Embedded Memories
With the continuous miniaturization of CMOS technology into the nanometer regime, the reliability of SRAM memories is threatened by accelerated transistor aging such as Bias Temperature Instability (BTI), Hot Carrier Injection (HCI) and gate oxide breakdown. Among these mechanisms BTI is known to be the primary aging mechanism in nanoscale devices. The overall effect of BTI is a gradual increase of threshold voltage (Vth). BTI significantly reduces the Static Noise Margin (SNM) of an SRAM cell and makes it more susceptible to failures. To address the impact of BTI in memory array a variety of bit flipping techniques has been proposed. However, all the proposed bit flipping techniques require at least an additional column to store the inversion flag which imposes considerably large area overhead. In this thesis, we propose two techniques to mitigate BTI induced aging in embedded memory: agingaware instruction encoding and selfcontrolled bit flipping; both schemes take the workload into consideration.
The agingaware instruction encoding technique is based on changing the encoding of the Instruction Set Architecture (ISA) in order to balance the occurrence probabilities of 1s and 0s and therefore minimize the impact of BTI in the embedded memory. To evaluate this scheme, we used the Leon2 processor for course case study. A C++ based simulation environment was used to exhaustively search for an encoding resulting in a balanced occurrence probabilities of 0s and 1s. The simulation results revels that on average up to 30% SNM degradation improvement.
Selfcontrolled bit flipping is based on inverting the content of the memory array during write operation with respect to a specific bit of the written word referred to as flip bit; this bit is left untouched during the write and used as a reference bit to indicate either the written data is inverted or not. Simulation results show that up to 33% SNM degradation improvement can be achieved. The area overhead of the proposed technique is the flip circuitry only. For this reason, our technique saves 64% of the area overhead induced by the periodic flipping techniques.

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10 

Spectral analysis of boundary layers in RayleighBénard convection

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11 

Field evidences and theoretical analysis of the gravitydriven wetting front instability of water runoffs on concrete structures
A series of field observations of the evolution of water runoffs over several vertical concrete walls directly exposed to rain falls is reported in this note. In all the cases, the main water flow originated from the top horizontal surface of the walls. The observations show that the gravitydriven wetting front may propagate in a very unstable way by developing well defined and quite regularly spaced vertical fingerlike features. The mean width < d > and the mean growth's velocity < v> of the fingers appear locally constant, but may vary from a wall surface to another. A simple relationship between
< d > and < v > is deduced from the field data and the narrower the fingers the higher the growth's velocity.
The fingering process is tentatively interpreted by using the theoretical analysis developped by Glass et al (1989b) for the wetting front instability of infitration in unsaturated homogeneous layered soils. It is shown the model accounts qualitatively well for our observations. The variation in the geometry and kinematics of the instabilities from a wall surface to another may therefore be related to variations of the concrete structure at the microscopic scale. The relationship between < d > and < v > reflects the effects of the microstructure. The gravity drivenwetting front instability provides a powerful echanism for a fast and over large distance moisture transfer along concrete constructions. It also leads to an heterogeneous distribution of the moisture content along the wall surface, which may eventually result in large spatial variations of the moistureinduced damages of the building structures.

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12 

Dynamics of prolate ellipsoidal particles in a turbulent channel flow
The dynamical behavior of tiny elongated particles in a directly simulated turbulent flow field is investigated. The ellipsoidal particles are affected both by inertia and hydrodynamic forces and torques. The time evolution of the particle orientation and translational and rotational motions in a statistically steady channel flow is obtained for six different particle classes. The focus is on the influence of particle aspect ratio λ and the particle response time on the particle dynamics, i.e., distribution, orientation, translation, and rotation. Both ellipsoidal and spherical particles tend to accumulate in the viscous sublayer and preferentially concentrate in regions of lowspeed fluid velocity. The translational motion is practically unaffected by the aspect ratio, whereas both mean and fluctuating spin components depend crucially on λ. The ellipsoids tend to align themselves with the mean flow direction and this tendency becomes more pronounced in the wall proximity when the lateral tilting of the elongated particles is suppressed.

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13 

Instability of timedependent winddriven ocean gyres
The winddriven ocean circulation at midlatitudes is susceptible to several types of instabilities. One of the simplest models of these flows is the quasigeostrophic barotropic potential vorticity equation in an idealized ocean basin. In this model, the route to complex spatio/temporal flows is through successive bifurcations. The aim of this study is to describe the physics of the destabilization process of a periodic winddriven flow associated with a secondary bifurcation. Although bifurcation theory has proven to be a valuable tool to determine the physical mechanisms of destabilization of fluid flows, the analysis of the stability of timedependent (for example, periodic) flows, using this methodology, is computationally unpractical, due to the large number of degreesoffreedom involved. The approach followed here is to construct a loworder model using numerical Galerkin projection of the full model equations onto the dynamically active eigenmodes. The resulting reduced model is shown to capture the local dynamics of the full model. The physical mechanism of the destabilization of the periodic winddriven flow is deduced from the reduced model. While there are several stabilizing processes, notably rectification, the destabilization occurs due to timedependent increase of the background horizontal shear in the flow.

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14 

Drivers of residual estuarine circulation in tidally energetic estuaries: straight and irrotational channels with parabolic cross section
The generation of residual circulation in a tidally energetic estuary with constant longitudinal salinity gradient and parabolic cross section is examined by means of a twodimensional crosssectional numerical model, neglecting river runoff and Stokes drift. It is shown how the longitudinal and lateral residual circulation can be decomposed into contributions from various processes such as tidal straining circulation, gravitational circulation, advectively driven circulation, and horizontal mixing circulation. The sensitivity of the residual circulation and its components from various processes to changes in forcing is investigated by varying the Simpson number (nondimensional longitudinal buoyancy gradient) and the unsteadiness parameter (nondimensional tidal frequency), as well as the bed roughness and the width of the estuary. For relatively weak salinity gradient forcing, the tidal straining circulation dominates the residual exchange circulation in support of classical estuarine circulation (upestuary flow near the bed and downestuary flow near the surface). The strength of the longitudinal estuarine circulation clearly increases with increased salinity gradient forcing. However, when the Simpson number exceeds 0.15, the relative contributions of both gravitational circulation and advectively driven circulation to estuarine circulation increase substantially.
Lateral residual circulation is relatively weak for small Simpson numbers and becomes flood oriented (divergent flow near the bed and convergent flow near the surface) for larger Simpson numbers because of increasing contributions from gravitational and advectively driven circulation. Increasing the unsteadiness number leads to decreased longitudinal and lateral residual circulation. Although changes in bed roughness result in relatively small changes in residual circulation, results are sensitive to the width of the estuary, mainly because of changes in residual exchange circulation driven by tidal straining.

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15 

Intermediate frequency band digitized high dynamic range radiometer system for plasma diagnostics and realtime Tokamak control
Article/Letter to the Editor 
Applied Sciences
20110624

Author: 
Bongers, WA.
·
Van Beveren, V.
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Thoen, D.J.
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Nuij, P.J.W.M.
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De Baar, M.R.
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Donné, A.J.H.
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Westerhof, E.
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Goede, A.P.H.
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Krijger, B.
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Van den Berg, M.A.
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Kantor, M.
·
Graswinckel, M.F.
·
Hennen, B.A.
·
Schüller, F.C.

Keywords: 
analoguedigital conversion · fast Fourier transforms · logic arrays · plasma diagnostics · plasma inertial confinement · radiometers · sawtooth instability · tearing instability · Tokamak devices

An intermediate frequency (IF) band digitizing radiometer system in the 100–200 GHz frequency range has been developed for Tokamak diagnostics and control, and other fields of research which require a high flexibility in frequency resolution combined with a large bandwidth and the retrieval of the full wave information of the mmwave signals under investigation. The system is based on directly digitizing the IF band after down conversion. The enabling technology consists of a fast multigiga sample analog to digital converter that has recently become available. Field programmable gate arrays (FPGA) are implemented to accomplish versatile realtime data analysis. A prototype system has been developed and tested and its performance has been compared with conventional electron cyclotron emission (ECE) spectrometer systems. On the TEXTOR Tokamak a proof of principle shows that ECE, together with high power injected and scattered radiation, becomes amenable to measurement by this device. In particular, its capability to measure the phase of coherent signals in the spectrum offers important advantages in diagnostics and control. One case developed in detail employs the FPGA in realtime fast Fourier transform (FFT) and additional signal processing. The major benefit of such a FFTbased system is the realtime tradeoff that can be made between frequency and time resolution. For ECE diagnostics this corresponds to a flexible spatial resolution in the plasma, with potential application in smart sensing of plasma instabilities such as the neoclassical tearing mode (NTM) and sawtooth instabilities. The flexible resolution would allow for the measurement of the full mode content of plasma instabilities contained within the system bandwidth.

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16 

Laplacian Instability of Planar Streamer Ionization Fronts: An Example of Pulled Front Analysis
Streamer ionization fronts are pulled fronts that propagate into a linearly unstable state; the spatial decay of the initial condition of a planar front selects dynamically one specific longtime attractor out of a continuous family. A stability analysis for perturbations in the transverse direction has to take these features into account. In this paper we show how to apply the Evans function in a weighted space for this stability analysis. Zeros of the Evans function indicate the intersection of the stable and unstable manifolds; they are used to determine the eigenvalues. Within this Evans function framework, we define a numerical dynamical systems method for the calculation of the dispersion relation as an eigenvalue problem. We also derive dispersion curves for different values of the electron diffusion constant and of the electric field ahead of the front. Numerical solutions of the initial value problem confirm the eigenvalue calculations. The numerical work is complemented with an analysis of the Evans function leading to analytical expressions for the dispersion relation in the limit of small and large wave numbers. The paper concludes with a fit formula for intermediate wave numbers. This empirical fit supports the conjecture that the smallest unstable wave length of the Laplacian instability is proportional to the diffusion length that characterizes the leading edge of the pulled ionization front.

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17 

Instability of a vehicle moving on an elastic structure
Vibrations of a vehicle that moves on a long elastic structure can become unstable because of elastic waves that the vehicle generates in the structure. A typical example of the vehicle that can experience such instability is a highspeed train. Moving with a sufficiently high speed, this train could generate in the railway track elastic waves, whose reaction might destabilise vibrations of the train. Such instability could increase the level of vibrations of both the train and the railway track, significantly worsening the comfort of passengers and increasing the probability of the track deterioration and the train derailment.
Instability of a moving vehicle on an elastic structure can be classified as one of the "moving load problems". This class of problems has been drawing attention of researches for more than a century being a fundamental issue in dynamics of bridges and railway tracks. Recently, the classical "moving load problem" has attracted researches once again because of the rapid development of highspeed railways. The necessity to take a fresh look at this old problem is based on the fact that in earlier studies it was usually assumed that the load speed is much smaller than the wave velocity in the elastic structure, which the load moves on. Nowadays, this assumption is no longer acceptable, since modern highspeed trains are able to move with a speed that is comparable with the wave velocity in a railway track.
The main objective of this thesis is to study the stability of the traintrack system at high speeds. The practical aim behind this objective is to develop an accurate and efficient method that would allow for choosing parameters of the traintrack system so that the stability is guaranteed at operational train speeds. Having such a method developed, this thesis aims to
study the effect of physical parameters of a moving train bogie on stability of the traintrack system;
analyse the effect of periodical inhomogeneity of the track that is caused by sleepers and rail corrugation on stability of the traintrack system;
investigate the effect of waves in the track subsoil on stability of the traintrack system;
To investigate the influence of the physical parameters of a vehicle, a simplified model for a railway track, namely a onedimensional homogeneous elastically supported Timoshenko beam is used. Since instability depends on the reaction of elastic system, a socalled equivalent stiffness of the Timoshenko beam (a complexvalued function that depends on the frequency of vibrations of the contact point, its velocity and parameters of the beam and foundation) in a moving contact point is introduced and studied. For this development, the most important is the dependence of the equivalent stiffness on the velocity of the contact point. Therefore, this dependence is investigated thoroughly and then compared to that of an EulerBernoulli beam. Then, a twomass oscillator, moving uniformly along such an elastic system is considered. It has been shown that vertical vibrations of this oscillator as it moves along the beam may become unstable if the oscillator's velocity exceeds the minimum phase velocity of waves in the beam. In this case, the equivalent dynamic stiffness of the beam has a negative imaginary part, which may be referred to as a "negative radiation damping" that is caused by radiation of anomalous Doppler waves. Instability domains in the parameter space of the system are found with the help of the Ddecomposition method. The effect of various parameters of the system on its stability is studied. Then, a more realistic model for the vehicle is considered, namely a bogie that has two contact points with the beam. The bogie is modelled by a rigid bar of a finite length on two identical supports. The parametric analysis of the instability domain is performed with the emphasis on the effect of the bogie and the beam parameters and comparative analysis with simpler models (twomass oscillator and simplified bogie) is carried out.
Then, influence of the periodic inhomogeneity of elastic structure (actually, a railway track is periodically inhomogeneous due to the sleepers and also can become inhomogeneous due to corrugation of the rails) on the instability is studied. To this end, a simplistic model for the vehicle is utilised, namely a moving mass. The structure is modelled as an EulerBernoulli beam on viscoelastic foundation. The inhomogeneity is introduced by assuming that either the foundation stiffness or the beam crosssection is a periodic function of the coordinate. By moving on such a structure, the vehicle could experience parametric instability. It is found out that for highspeed trains, the zones of parametric instability are very narrow and, therefore should not be of concern.
What could be a practically important threat is the instability that occurs when the minimum phase velocity of waves in the railway track is exceeded by the train. How large is this velocity? To answer this question, it is not enough to consider onedimensional models of the railway track. The phase velocity of waves in a railway track is strongly influenced by the track subsoil. Therefore, to make a plausible estimation of train velocities at which the instability may arise, a threedimensional model that includes the track subsoil should be employed. To this end, a railway track has been modelled by a beam resting on a viscoelastic halfspace. The instability domain in the space of physical parameters of the system is found and parametrically studied with the help of the Ddecomposition method. The main attention has been paid to the effect of the halfspace parameters, especially to that of the material damping. It has been proved, that instability can occur at the velocities that are reachable for modern highspeed trains.

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18 

Monitoring Travel Time Reliability on Freeways
Travel time and travel time reliability are important attributes of a trip. The current measures of reliability have in common that in general they all relate to the variability of travel times. However, travel time reliability does not only rely on variability but also on the stability of travel times. This paper clarifies the attributes of reliability and proposes a new analytical formula to express travel time unreliability in terms of these elements, in which the travel time (un)reliability is computed as the sum over the products of the consequences (variability or uncertainty) and corresponding probabilities of traffic breakdown (instability). In this Conceptual Travel Time Reliability (CTTR) model, the probability of breakdown on a section is categorized into spontaneous breakdown and induced breakdown, which are independent; the probability of breakdown of a route is formulated as the product of the probability of breakdown of adjacent sections along the route.

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19 

Turbulence characteristics in sharp openchannel bends
In spite of its importance, little is known about the turbulence characteristics in openchannel bends. This paper reports on an experimental investigation of turbulence in one cross section of an openchannel bend. Typical flow features are a bicellular pattern of crossstream circulation (secondary flow) and a turbulence activity in the outer bend that is significantly less than in the equivalent straight uniform shear flow. Measured distributions are given of the turbulent kinetic energy, its production, the mixing coefficients, some parameters characterizing the turbulence structure, and the fourthorder correlations of the turbulent velocity fluctuations. The transport equation for the turbulent kinetic energy is evaluated term by term, on the basis of the measured data. The results show that the turbulence structure is different from straight uniform flow, in that the Reynolds stress tensor is more diagonally dominant. This is shown to be the main cause of the observed reduction of turbulence activity in the outer bend. The usual twoequation turbulence closure models include a transport equation for the turbulent kinetic energy, but they do not account for this modified turbulence structure. The departures of the measured turbulence structure from its equivalent in straight uniform shear flow are related to a curvaturefluxRichardson number Rf which includes the streamline curvature. Such a relation may be useful to improve simple turbulence closure models for curved openchannel flow.

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20 

A linear approach for the evolution of coherent structures in shallow mixing layers
The development of large coherent structures in a shallow mixing layer is analyzed. The results are validated with experimental data obtained from particle tracking velocimetry. The mean flow field is modeled using the selfsimilarity of the velocity profiles. The characteristic features of the downstream development of a shallow mixing layer flow, like the decrease of the velocity difference over the mixing layer, the decreasing growth of the mixing layer width, and the transverse shift of the center of the mixing layer layer are fairly well represented. It turned out that the entrainment coefficient could be taken constant, equal to a value obtained for unbounded mixing layers: α = 0.085. Linearization of the shallow water equations leads to a modified Orr–Sommerfeld equation, with turbulence viscosity and bottom friction as dissipative terms. Growth rates are obtained for each position downstream, using the model for the mean flow field. For a given energy density spectrum at the inflow boundary, integration of the growth rates along the downstream direction yields the spectra at various downstream positions. These spectra provide a measure for the intensity and the length scale of the coherent structures (the dominant mode). The length scales found are in good agreement with the measured ones. The length scale of the most unstable mode appears much larger than the length scale of the dominant mode. Obviously, the longevity of the coherent structures plays a significant role. Three growth regimes can be distinguished: in the first regime the dominant mode is growing, in the second regime the dominant mode is dissipating, but other modes are still growing, and in the third regime all modes are dissipating. It is concluded that the development of the coherent structures in a shallow mixing layer can fairly well be described and interpreted by the proposed linear analysis.

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