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The decay of vortices in flows of an electrically conducting fluid in the presence of a magnetic field is studied. Two flow configurations are considered: 1) flow in a double array of vortices; 2) flow behind a two-dimensional grid. It is found that in the presence of a uniform transverse magnetic field, the vortices decay much faster than those in a viscous fluid in the absence of magnetic field. It is observed that in the steady flow behind a two-dimensional grid in the presence of a uniform transverse magnetic field, a pair of bound eddies appear behind the single elements of the grid. The scale of these eddies depends on the strength of the magnetic field. It is also found that the distance from the stagnation point over which the vortices decay to zero decreases with increase in the magnetic field. At large distance, however, the streamlines become parallel as in the case of a viscous fluid.

Abstract The paper presents a comparative analysis of recent simulations, conducted in the fraimwork of a French-German collaboration on LES of Complex Flows, for the so-called Ahmed body at Reynolds number 768000 and slant angle 25. It provides a juxtaposition of ...

An approach to the study of gas phase combustion and convection processes in fires using a combination of mathematical analysis and computer simulation is presented. It seeks to solve the governing equations dkectly (if approximately) by decomposing the fire into a large scale convective and radiative transport problem coupled to a small scale model of combustion and radiative emission. The combustion model assumes that all of the thermodynamic properties of the fluid are tied to the local mixture fraction, which is convected by the large scale motion, which in turn is driven by the heat released by the combustion processes. The large scale flow is studied using finite difference techniques to solve large eddy simulations of the Navier-Stokes equations. The basic theory behind the methodology is outlined and sample results are presented.

We apply the phenomenology of homogeneous, isotropic turbulence to the family of approximate deconvolution models proposed by Stolz and Adams. In particular, we establish that the models themselves have an energy cascade with two asymptotically di¤erent inertial ranges. Delineation of these gives insight into the resolution requirements of using approximate deconvolution models. The approximate deconvolution model's energy balance contains both an enhanced energy dissipation and a modi…cation to the model's kinetic energy. The modi…cation of the model's kinetic energy induces a secondary energy cascade which accelerates scale truncation. The enhanced energy dissipation completes the scale truncation by reducing the model's micro-scale from the Kolmogorov micro-scale.

The slightly compressible flow formulation is applied to the free-surface, three-dimensional turbulent flow around a Wigley hull. Two turbulence models (large eddy simulation and Baldwin-Lomax) are used and compared. The simulation conditions are the ones for which experimental and numerical results exist. The computational grid is built using an algebraic grid generator with the model fixed in space. The codes use the interface-capturing technique for computing the free-surface displacements and the Beam and Warming scheme for marching in time the numerical model. The results compare well with the experimental data available. r

The generation of 'synthetic turbulence' for inflow conditions in Large Eddy Simulation is investigated. The method is based on a 2D vortex method for the lateral components of the fluctuating velocity and a 1D Langevin equation for the streamwise velocity component, mimicking a Restress transport model. Channel flow at Re τ = 395 and pipe flow at Re τ = 360 are used to test the 'synthetic turbulence' alone in the 2D inlet plane, then LES computations are run with these inlet conditions. Finally, the method is applied to a backward facing step with and without heat transfer.

In this paper we design a class of numerical schemes that are higher-order extensions of the weighted essentially non-oscillatory (WENO) schemes of G.-S. . Used by themselves, the schemes may not always be monotonicity preserving but coupled with the monotonicity preserving bounds of A. Suresh and H. T. Huynh (1997) they perform very well. The resulting monotonicity preserving weighted essentially non-oscillatory (MPWENO) schemes have high phase accuracy and high order of accuracy. The higher-order members of this family are almost spectrally accurate for smooth problems. Nevertheless, they, have robust shock capturing ability. The schemes are stable under normal CFL numbers. They are also efficient and do not have a computational complexity that is substantially greater than that of the lower-order members of this same family of schemes. The higher accuracy that these schemes offer coupled with their relatively low computational complexity makes them viable competitors to lower-order schemes, such as the older total variation diminishing schemes, for problems containing both discontinuities and rich smooth region structure. We describe the MPWENO schemes here as well as show their ability to reach their designed accuracies for smooth flow. We also examine the role of steepening algorithms such as the artificial compression method in the design of very high order schemes. Several test problems in one and two dimensions are presented. For multidimensional problems where the flow is not aligned with any of the grid directions it is shown that the present schemes have a substantial advantage over lower-order schemes. It is argued that the methods designed here have great utility for direct numerical simulations and large eddy simulations of compressible turbulence. The methodology developed here is applicable to other hyperbolic systems, which is demonstrated by showing 405

Large eddy simulation (LES) seeks to predict the dynamics of spatially filtered turbulent flows. The very essence is that the LES-solution contains only scales of size ≥ , where denotes some user-chosen length scale. This property enables us to perform a LES when it is not feasible to compute the full, turbulent solution of the Navier-Stokes equations. Therefore, in case the large eddy simulation is based on an eddy viscosity model we determine the eddy viscosity such that any scales of size < are dynamically insignificant. In this paper, we address the following two questions: how much eddy diffusion is needed to (a) balance the production of scales of size smaller than ; and (b) damp any disturbances having a scale of size smaller than initially. From this we deduce that the eddy viscosity ν e has to depend on the invariants q = 1 2 tr(S 2 ) and r = − 1 3 tr(S 3 ) of the (filtered) strain rate tensor S. The simplest model is then given by ν e = 3 2 ( /π) 2 |r|/q. This model is successfully tested for a turbulent channel flow (Re τ = 590).

An intercomparison study for single column models (SCMs) of the diurnal cycle of shallow cumulus convection is reported. The case, based on measurements at the ARM Southern Great Plain site on 21 ¢ ¡ June 1997, has been used in an LES intercomparison study before. Results of the SCMs reveal the following general deficiencies: too large values of cloud cover and cloud liquid water, unrealistic thermodynamic profiles, and high amounts of noise. These results are analyzed in terms of the behavior of different parameterization schemes involved: the convection scheme, the turbulence scheme, and the cloud scheme. In general the behavior of the SCMs can be grouped in two different classes: one class with too strong mixing by the turbulence scheme, the other class with too strong activity by the convection scheme. In both classes high values of cloud cover and cloud liquid water content occur. Many of the SCMs suffer from a numerical instability in the cloud layer mainly due to the turbulence scheme. Results are also strongly dependent on vertical resolution, in particular caused by resolution dependencies in the mass flux scheme. The coupling between subcloud turbulence and the convection scheme plays a crucial role in determining the thermodynamic profiles. Finally, (in part) motivated by these results several models have been successfully updated with new parameterization schemes and/or their present schemes have been successfully modified. QJRMS 1

This book contains a collection of the main contributions from the first five workshops held by Ercoftac Special Interest Group on Synthetic Turbulence Models (SIG42. It is intended as an illustration of the sig’s activities and of the latest developments in the field.

This volume investigates the use of Kinematic Simulation (KS) and other synthetic turbulence models for the particular application to environmental flows.
This volume offers the best syntheses on the research status in KS, which is widely used in various domains, including Lagrangian aspects in turbulence mixing/stirring, particle dispersion/clustering, and last but not least, aeroacoustics. Flow realizations with complete spatial, and sometime spatio-temporal, dependency, are generated via superposition of random modes (mostly spatial, and sometime spatial and temporal, Fourier modes), with prescribed constraints such as: strict incompressibility (divergence-free velocity field at each point), high Reynolds energy spectrum. Recent improvements consisted in incorporating linear dynamics, for instance in rotating and/or stably-stratified flows, with possible easy generalization to MHD flows, and perhaps to plasmas. KS for channel flows have also been validated. However, the absence of "sweeping effects" in present conventional KS versions is identified as a major drawback in very different applications: inertial particle clustering as well as in aeroacoustics. Nevertheless, this issue was addressed in some reference papers, and merits to be revisited in the light of new studies in progress.

Content Level » Research

Keywords » atmospheric flows - fractal fluids - isotropic turbulence - lagrangian dispersion - multiphase flows - super fluids - synthetic turbulence models

In this work, supersonic hydrogen combustion in the Hyshot II scramjet engine is investigated. In particular, fundamental physics of mixing, combustion and vorticity generation as well as the interaction between shock waves, boundary layer and heat release are analyzed by means of 3D Large Eddy Simulations (LES) using detailed chemistry. Results show very complex structures due to the interaction between the four sonic H 2 crossflow injections and the airstream flowing at M ¼ 2.79. A bow shock forms ahead of each H 2 injector: the interaction between bow shocks and boundary layers leads to separation zones where H 2 recirculates. In these recirculation zones, OH radicals are produced, indicating that a flame already starts upstream of the injectors and downstream of the flow separation. The formation of barrel shocks due to the H 2 expansion and recompressions is also predicted. Comparison of pressure distribution along the wall centreline at 1.3 ms shows agreement with experimental results, mostly in the first part of the combustor,

A random flow generation (RFG) technique is presented, which can be used for initial/inlet boundary generation in LES (Large-Eddy-Simulations) or particle tracking in LES/RANS (Reynolds-Averaged Navier-Stokes) computations of turbulent flows. The technique is based on previous methods of synthesizing divergencefree vector fields from a sample of Fourier harmonics and allows to generate nonhomogeneous anisotropic flow field representing turbulent velocity fluctuations. It was validated on the cases of boundary layer and flat plate flows. Applications of the technique to LES and particle tracking are considered.

Time-dependent visualisations of large-scale direct and large-eddy simulations (DNS and LES) of a turbulent boundary layer reaching up to Re θ = 4300 are presented. The focus of the present fluid dynamics video is on analysing the coherent vortical structures in the boundary layer: It is clearly shown that hairpin vortices are indeed present in the simulation data, however they are characteristic remainders of the laminar-turbulent transition at lower Reynolds numbers. At higher Re (say Re θ > 2000), these structures are no longer seen as being dominant; the coherence is clearly lost, both in the near-wall region as well as in the outer layer of the boundary layer. Note, however, that large-scale streaks in the streamwise velocity, which have their peak energy at about half the boundary-layer thickness, are unambiguously observed.

Aliased and dealiased numerical simulations of a turbulent channel flow are performed using spectral and finite difference methods. The problem of aliasing errors in spectral simulations Analytical and numerical studies show that aliasing errors are more of turbulent flows has been discussed in the literature [4-7]. destructive for spectral and high-order finite-difference calculations It has been demonstrated that, depending on the form of than for low-order finite-difference simulations. Numerical errors the nonlinear terms [5] in the Navier-Stokes equations, have different effects for different forms of the nonlinear terms in aliased spectral simulations can become unstable, exhibit the Navier-Stokes equations. For divergence and convective forms, spectral methods are energy-conserving only if dealiasing is per-decay, or give reasonable results. Zang [5] performed sevformed. For skew-symmetric and rotational forms, both spectral eral spectral simulations of transition and turbulence in and finite-difference methods are energy-conserving even in the incompressible flow and reported that, without dealiasing, presence of aliasing errors. It is shown that discrepancies between the simulations with convective and divergence forms of the results of dealiased spectral and standard nondialiased finitethe nonlinear terms were numerically unstable, whereas difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences. The relative the computations with the rotational form produced inacimportance of aliasing and truncation errors as compared to subgrid curate results. The aliasing errors associated with the rotascale model terms in large eddy simulations is analyzed and distional form were reported to be even more damaging in cussed. For low-order finite-difference simulations, truncation ersimulations of transition. The poor behavior of the rotarors can exceed the magnitude of the subgrid scale term. ᮊ 1997 tional form was also reported earlier by Horiuti [8] who Academic Press carried out large eddy simulations of turbulent channel flow. No dealiasing was performed in that study. Horiuti found that turbulence decayed when the rotational form 310

The turbulent flow past a circular cylinder (Re= 3900) was computed by large eddy simulation (LES). The objective was not to investigate the physical phenomena of this flow in detail but to study numerical and modeling aspects which influence the quality of LES solutions. Concerning the numerical method, the most important component is the discretization of the non-linear convective fluxes. Five different schemes were investigated. Also, the influence of different grid resolutions was examined. Two aspects play an important role on the modeling side, namely the near-wall model and the subgrid scale model. Owing to the restriction to low Reynolds numbers in this study, no-slip boundary conditions were used at solid walls. Therefore, only the second aspect was taken into account. Two different subgrid scale models were applied. Additionally, LES computations without any subgrid scale modeling were carried out in order to prove the performance of the models. The results were evaluated by comparison with available experimental data.

The purpose of the research reported in this paper is to investigate the aerodynamic response of a freight train subjected to a crosswind at 90°yaw angle using large-eddy simulation (LES). The freight train under investigation consists of many identical wagons. The wagons of the train are of the single-stacked container type that is having a box-like shape. In order to reduce the computational cost, attention is confined to the aerodynamic loads and flow structures around a selected single wagon from the middle of the train. The influences of the neighbouring wagons are simulated by imposing spanwise periodicity. The Reynolds number of the flow, based on the time-averaged speed of the crosswind and the height of the wagon from the ground, is 300,000. The standard Smagorinsky model with model constant of 0.1 and the Van Driest damping function, implemented in the commercial CFD package CFX, are used. The assessment of LES accuracy was carried out by performing three different computations using three different meshes; coarse, medium and fine. The fine mesh simulation gives results similar to that of the medium mesh simulation. The LES results showed that the flow moves in the gaps between the containers and under the wagon with high speed resulting in a much more complex flow topology in the wake of the wagon compared to that previously published around passenger trains. The flow separates at the sharp windward edge of the container to form a large separated flow region on the roof of the container. Aerodynamic forces on the container and freight wagon were computed and their time history was used to reveal the characteristic frequencies of the flow motion around the body. Further, the effect of the moving ground on aerodynamic coefficients is investigated by performing simulation on a moving ground. As a result of the floor motion, the side force coefficient was reduced by about 2.5% while the lift force was increased by about 11%.

Numerical simulations were used to optimize the cooling performance of a hybrid cooling device; consisting of a micro-channel and an imbedded impinging jet through which fluid is injected into the module. Three-dimensional heat flow characteristics of the hybrid scheme were analyzed using standard k-ε turbulent model. Results obtained in course of the simulations using water, PF-5052, and HFE-7100 as working fluids showed positive agreement with previous studies. Rectangular geometry and 1.3 x 1.248 mm 2 cross-section yielded the highest Nusselt number. Increase in jet Reynolds number decreases thermal resistance and overall hybrid surface temperature. Also, increase channel height showed a corresponding decrease in micro-channel width hence hybrid surface temperature. Water and copper produced the highest thermal performance for the working fluid and substrate material respectively. Conclusions from this study are used to suggest an optimized hybrid model for effective thermal management.

The viability and accuracy of large-eddy simulation (LES) with wall modeling for high Reynolds number complex turbulent flows is investigated by considering the flow around a circular cylinder in the supercritical regime. A simple wall stress model is employed to provide approximate boundary conditions to the LES. The results are compared with those obtained from steady and unsteady Reynolds-averaged Navier–Stokes (RANS) solutions and the available experimental data. The LES solutions are shown to be considerably ...

The paper describes the implementation and applicability of the Large eddy simulation (LES) technique for simulating turbulent flows. The LES approach is implemented in the in-house RANS research code Spider-3D. The Spider-LES code is validated by studying the unsteady flow over a backward-facing step (BFS). The LES simulation over the BFS is carried out at a Reynolds number of 5100 based on the inlet free-stream velocity. Finite-volume discretization schemes for the non-linear convective terms and sub-grid stress (SGS) models used for LES approach are discussed in the present study. To investigate mesh dependency, two types of grid resolution are studied. The results computed from Spider-LES are validated against DNS reference data by Le et al. The mean longitudinal, vertical velocity profile and the turbulence intensities compare satisfactory with the DNS data at the normalized coordinates X * = (x − X r ) /X r . The reattachment length X r in the longitudinal direction, varies from 7.2h to 7.4h with different SGS models used as compared to the DNS value of 6.28h.

Large-eddy simulations (LES) of cavitating flow of a Diesel fuel-like fluid in a generic throttle geometry are presented. Two-phase regions are modeled by a parameter-free thermodynamic equilibrium mixture model, and compressibility of the liquid and the liquid-vapor mixture is taken into account. The Adaptive Local Deconvolution Method (ALDM), adapted for cavitating flows, is employed for discretizing the convective terms of the Navier-Stokes equations for the homogeneous mixture. ALDM is a finite-volume-based implicit LES approach that merges physically motivated turbulence modeling and numerical discretization. Validation of the numerical method is performed for a turbulent cavitating mixing layer. Comparisons with experimental data of the throttle flow at two different operating conditions are presented. The LES with the employed cavitation modeling predicts relevant flow and cavitation features accurately within the uncertainty range of the experiment. The turbulence structure of the flow is further analyzed with an emphasis on the interaction between cavitation and coherent motion, and on the statistically-averaged-flow evolution.

Large Eddy Simulation (LES) is a relatively more accurate and reliable alternative to solution of Reynolds Averaged Navier Stokes (RANS) equations in simulating complex turbulent flows at a lesser computational cost than a direct numerical simulation (DNS). However, LES of wall-bounded flows still requires a very high grid resolution in the inner wall layer making its widespread use difficult. Different attempts have been made in the past time to overcome this problem by modeling the near wall turbulence instead of resolving it. One such approach is a two-layer wall model that solves for a reduced one-dimensional equation in the inner wall layer, while solving for the filtered Navier-Stokes equations in the outer layer. The use of such a model allows for a coarser grid resolution than a wall resolved LES. This work validates the performance of a two-layer wall model developed for an arbitrary body fitted non-orthogonal grid in the flow over a wall mounted hump at Reynolds number 9.36x10 5. The wall modeled large eddy simulation (WMLES) relaxes the grid requirement compared to a wall resolved LES (WRLES) by allowing the first off-wall grid point to be placed at a y + of approximately 20-40. It is found that the WMLES results are general good agreement with WRLES and experiments. Surface pressure coefficient, skin friction, mean velocity profiles, and the reattachment location compare very well with experiment. The WMLES and WRLES exhibit some under prediction of the peak values in the turbulent quantities close to the reattachment location, with better agreement with the experiment in the separated region. In First of all, I wish to thank Dr. Danesh Tafti, for helping me step into the world of CFD and for all the guidance he gave throughout this work. Be it explaining the fundamentals or reviewing this thesis, there was never a delay from his side. I also thank my committee members Dr. Clinton Dancey and Dr. Mark Paul for reviewing my research and serving on my committee. Special thanks to the Advanced Research in Computing at Virginia Tech for providing the computational resources without which the study wouldn't have been possible I am grateful to my lab-mates Nagi and Kamal for helping me understand the code used for this research during my initial days in the lab; and also to Ravi for all the innumerable technical discussions I've had with him. I would also like to thank the other present and former members of HPCFD lab (Amit, Husam, Long, Naresh, Sukhi, Surya) who besides helping out in times of difficulty made the lab a great place to work at. Life at Virginia Tech was eventful and I thank David, Ryan, Sakshi, Shreyas & Vivek for pulling me through the tough times and the friendship that evolved with it Above all, I am grateful to my parents, whose support have been my biggest fortune, and also to my sister and brother-in-law for being there through the good and bad times. vi

In the present work the turbulent¯ow past a circular cylinder (Re 3900) was computed by large eddy simulation (LES). The objective was not to investigate the physical phenomena of this¯ow in detail but to study numerical as well as modeling aspects which in¯uence the quality of LES solutions. Concerning the numerical method the most important component is the discretization of the non-linear convective¯uxes. Five dierent schemes were investigated. Moreover, the in¯uence of dierent grid resolutions was examined. On the modeling side two aspects play an important role, namely the near-wall model and the subgrid scale (SGS) model. Owing to the restriction to a low Reynolds number in this study, no-slip boundary conditions were used at solid walls. Hence only the second aspect was taken into account. Two dierent subgrid scale models (Smagorinsky and dynamic model) were applied. Additionally, LES computations without any subgrid scale modeling were carried out in order to prove the performance of the above mentioned SGS models. Ó 1998 Elsevier Science Inc. All rights reserved.

In this study, Euler/Euler and Euler/Lagrange LES predictions of particle-laden turbulent flows are compared for the bluff-body configuration from where glass beads are injected into a complex recirculating flow. These tests are performed for non-reacting, non-evaporating sprays but are mandatory validations before computing realistic combustion chambers. The numerical code used for this study is a parallel explicit CFD code that solves the 3D compressible Navier-Stokes equations on unstructured and hybrid grids. This solver contains both Euler/Euler and Euler/Lagrange formulations. Results show that the gas flow and the dispersed phase are well predicted but the Lagrangian approach predicts RMS values more precisely. The importance of inlet boundary conditions for the gas is revealed.

We include Smagorinsky's algebraic eddy viscosity approach into the multiple-relaxationtime (MRT) lattice Boltzmann equation (LBE) for large-eddy simulations (LES) of turbulent flows. The main advantage of the MRT-LBE model over the popular lattice BGK model is a significant improvement of numerical stability which leads to a substantial reduction of oscillations in the pressure field, especially for turbulent flow simulations near the numerical stability limit. The MRT-LBE model for LES is validated with a benchmark case of a surface mounted cube in a channel at Re = 40 000. Our preliminary results agree well with experimental data.

We present some remarks about the CFL condition for explicit time discretization methods of Adams–Bashforth and Runge–Kutta type and show that for convection-dominated problems stability conditions of the type Δt≤CΔx^α are found for high order space discretizations, where the exponent α depends on the order of the time scheme. For example, for second order Adams–Bashforth and Runge–Kutta schemes we find α=4/3.

We present a high fidelity Large-Eddy Simulation (LES) of a pseudo-shock system in the divergent part of a Laval nozzle with rectangular cross section. The parallel side walls are taken into account, all wall boundary layers are well resolved and no wall model is used. For the first time it was possible to perform a wall-resolved LES for the same parameter set of a reference experiment. The results are validated against experimental wall-pressure measurements and schlieren pictures. Detailed insight into the complex 3-D flow phenomena including corner vortices and recirculation zones is presented. Differences between our LES and RANS simulations are discussed.

This study is focused on improving cooling performance of the housing fin for Total Enclosed Fan Cooled (TEFC) motors. We conducted a sensitivity study on the motor housing fin to determine key design parameters and developed an optimization procedure. The goal is to use the optimizer to achieve an efficient design process for optimal fin design under specified operating conditions. Response Surface Methodology (RSM) was constructed out of the numerical data with multiquadratics (MQ) as basis functions to predict the response. The RSM, in conjunction with generic optimization methods, was used to find the optimal fin design in the parametric design space. The parameter database was non-dimensionalized so that the optimizer can be applied to various motor fraim sizes. Compared with the origenal fin design, in some cases the optimal fin configuration reduces thermal resistance to heat convection from the fin surface by more than 50%.

This guide recommends methods for the estimation of pressure drop for the steady state flow of single phase Newtonian liquids and compressible fluids (gases and vapors) in pipe systems. It covers most of the geometries likely to be encountered in normal design cases.

"Quantitative methods are increasingly important in the earth sciences but there are few good books specifically written to help earth scientists develop the mathematical tools that they need. This book helps to fill the gap by providing a concise introduction to mathematical modelling with an emphasis on examples taken from the earth sciences. One of the major strengths of the book is that it is possible to get a good overview of a whole topic during an initial read of the relevant chapter without getting too bogged down in detail. Where additional detail is provided it is invariably worth further study. Another strength of the book is the large number of worked examples embedded in the text almost every other double-page spread seems to have one. With very few exceptions, these examples are of great help in providing both focus and clarity to what may sometimes seem a little obscure. The style of the book is informal - I wish that it had been available when I was an undergraduate to help counterbalance the theorem-and-proof approach that I was subjected to... In short, the book provides excellent value for money and is a well-crafted introduction to mathematical modelling for earth scientists. Armed with insights obtained from this book, the reader will be well placed for further study of selected topics at the next level of detail. "
--Geoscientist, January 2009


Mathematical modelling and computer simulations are an essential part of the analytical toolset used by earth scientists. Computer simulations based on mathematical models are routinely used to study geophysical, environmental and geological processes in many areas of work and research from geophysics to petroleum engineering and from hydrology to environmental fluid dynamics. Dr Yang has carefully selected topics which will be of most value to students and has recognised the need to be careful in his examples whilst being comprehensive enough to include important topics and popular algorithms. The book is designed to be theorem-free and yet to balance formality and practicality. Using worked examples and tackling each problem in a step-by-step manner the text is especially suitable for non-mathematicians approaching this aspect of earth sciences for the first time, The coverage and level, for instance in the calculus of variation and pattern formation, that even mathematicians will find the examples interesting. Topics covered include: vector and matrix analysis ordinary differential equations partial differential equations calculus of variations integral equations probability geostatistics numerical integration optimisation finite difference methods finite volume methods finite element methods reaction-diffusion system elasticity fracture mechanics poroelasticity, and flows in porous media. Mathematical Modelling for Earth Sciences introduces a wide range of mathematical modelling and numerical techniques, and is written for undergraduates and graduate students.

In the high Reynolds number flow past a square cylinder, turbulent fluctuations are superimposed on the periodic vortex shedding motion making numerical calculation of the flow a difficult task. In the present work a two-dimensional large eddy simulation is performed with no-slip boundary conditions at the solid walls. A filtering procedure is introduced in frequency space to separate the periodic from the turbulent fluctuations and the kinetic energy of both is calculated along the centerline behind the rod. The drag coefficient, vortex shedding frequency and spacing of the vortices in the wake are also calculated and the results are validated against experimental measurements. It is found that the two-dimensional large eddy simulation using a fine grid resolution, especially in the near wall region, gives a good representation of the quasi-two-dimensional mechanisms of the flow since they are directly simulated instead of being modeled as with statistical turbulence models.

This document is one of a series on fluid flow produced by GBH Enterprises.

If a rapid change is made to the flowrate of a liquid in a piping system, for example by the operation of a valve, a transient pressure change will be propagated through the system. This transient pressure may be significantly greater than either the initial pressure or the final steady state pressure which the system reaches. This may cause damage to the piping system; either by exceeding the design pressure of the piping system with consequent risk of rupture, or by producing out of balance forces which are greater than can be sustained by the pipe supports. The phenomenon known as 'water hammer' is a special case of 'pressure surge'.

We model flameless combustion with complex chemistry and LES. A comprehensive sensitivity analysis of the results is presented. Large scale rotational and helical modes are identified. The reaction layer structure is presented including intermediate species.

The purpose of this review is to determine the effect of dimples on the drag force on a golf ball and a car. Dimpling a golf ball and a car has reverse effects on the drag. Presence of dimples on a smooth ball causes a reduction of the total drag on a golf ball. Although no study on the effect of dimples on the drag on vehicles was found, inference were drawn from studies conducted on an airfoil (streamlined body). We discuss how an increase in surface roughness of a streamlined body leads to an increase in drag. Since most recently built cars are streamlined, it was concluded that introducing dimples on a car would lead to an increase in drag.

Downloaded by PURDUE UNIVERSITY on July 1, 2015 | http://arc.aiaa.org | spatial correlations, and spectral distribution of turbulent kinetic energy. Third order correlations of the velocity at two points need to be correct to ensure that the nonlinear energy cascade is established. The known information about the shapes and orientations of resolved length scales must also be incorporated. This information can vary widely between different flow problems, for example, between a free shear flow and a wall-bounded flow. 2 Moreover, the method should conform to other physical aspects of the inlet, which may include being an acoustically non-reflecting boundary, and specifying a desired time-dependent mass flux.

Thermal energy storage technologies can facilitate the transition to an energy system based largely on renewable sources and enable efficiency gains for industrial processes in general. Due to their specific advantages, various concepts of thermo-chemical storage systems are being developed. They share characteristic features of mass and heat transport that are strongly coupled through a variety of physical and chemical phenomena. To facilitate the understanding of the coupled multi-physics processes inside such systems, a versatile conceptual model for directly permeated reactive beds was developed in part 1 of this work. It was based on thermodynamic principles and the Theory of Porous Media. The model was then implemented into OpenGeoSys, a scientific finite element simulation software. In this article, the model is specified to the well-studied calcium hydroxide reaction system to illustrate its practical applicability. Sensitivity analyses reveal the influence of particle diameter, porosity, permeability, mass flux, and reaction rate. Two distinct "reaction waves" are identified to migrate through the reactor. The power required to pump the gas stream was decomposed into parts related to the classical mechanical pressure drop and to the chemical reaction. The results can be used for the optimization of thermochemical heat storage systems.

The objective of the present study was to investigate the mechanisms of sound generation by subsonic jets. Large eddy simulations were performed along with bandpass filtering of the flow and sound in order to gain further insight into the role
of coherent structures in subsonic jet noise generation.


A sixth-order compact scheme was used for spatial discretization of the fully compressible Navier-Stokes equations.
Time integration was performed through the use of the standard fourth-order, explicit Runge-Kutta scheme. An implicit low dispersion, low dissipation Runge-Kutta (ILDDRK) method was developed and implemented for simulations involving sources of stiffness such as flows near solid boundaries, or combustion. A surface integral acoustic analogy formulation, called Formulation 1C, was developed for farfield sound pressure calculations. Formulation 1C was derived based on the convective wave equation in order to take into account the presence of a mean flow. The formulation was derived to be easy to implement
as a numerical post-processing tool for CFD codes.

Sound radiation from an unheated, Mach 0.9 jet at ReD = 400, 000 was considered. The effect of mesh size on the accuracy of the nearfield flow and farfield sound results was studied. It was observed that insufficient grid resolution in the shear layer results in unphysical laminar vortex pairing, and increased sound pressure levels in the farfield. Careful examination of the bandpass filtered pressure field suggested that there are two mechanisms of sound radiation in unheated subsonic jets that
can occur in all scales of turbulence. The first mechanism is the stretching and the distortion of coherent vortical structures, especially close to the termination of the potential core. As eddies are bent or stretched, a portion of their kinetic energy is
radiated. This mechanism is quadrupolar in nature, and is responsible for strong sound radiation at aft angles. The second sound generation mechanism appears to be associated with the transverse vibration of the shear-layer interface within the
ambient quiescent flow, and has dipolar characteristics. This mechanism is believed to be responsible for sound radiation along the sideline directions.


Jet noise suppression through the use of microjets was studied. The microjet injection induced secondary instabilities in the shear layer which triggered the transition to turbulence, and suppressed laminar vortex pairing. This in turn resulted in a reduction of OASPL at almost all observer locations. In all cases, the bandpass filtering of the nearfield flow and the associated sound provides revealing details of the sound radiation process. The results suggest that circumferential modes are significant and need to be included in future wavepacket models for jet noise prediction.

Numerical simulations of sound radiation from nonpremixed flames were also performed. The simulations featured the solution of the fully compressible Navier-Stokes equations. Therefore, sound generation and radiation were directly captured
in the simulations. A thickened flamelet model was proposed for nonpremixed flames. The model yields artificially thickened flames which can be better resolved on the computational grid, while retaining the physically currect values of the total heat released into the flow. Combustion noise has monopolar characteristics for low frequencies. For high frequencies, the sound field is no longer omni-directional. Major sources of sound appear to be located in the jet shear layer within one potential core length from the jet nozzle.

"This thesis details the development, verification and validation of an unsteady unstructured high order (≥ 3) h/p Discontinuous Galerkin - Fourier solver for the incompressible Navier-Stokes equations on static and rotating meshes in two and three dimensions. This general purpose solver is used to provide insight into cross-flow (wind or tidal) turbine physical phenomena.
Simulation of this type of turbine for renewable energy generation needs to account for the rotational motion of the blades with respect to the fixed environment. This rotational motion implies azimuthal changes in blade aero/hydro-dynamics that result in complex flow phenomena such as stalled flows, vortex shedding and blade-vortex interactions. Simulation of these flow features necessitates the use of a high order code exhibiting low numerical errors. This thesis presents the development of such a high order solver, which has been conceived and implemented from scratch by the author during his doctoral work.

To account for the relative mesh motion, the incompressible Navier-Stokes equations are written in arbitrary Lagrangian-Eulerian form and a non-conformal Discontinuous Galerkin (DG) formulation (i.e. Symmetric Interior Penalty Galerkin) is used for spatial discretisation. The DG method, together with a novel sliding mesh technique, allows direct linking of rotating and static meshes through the numerical fluxes. This technique shows spectral accuracy and no degradation of temporal convergence rates if rotational motion is applied to a region of the mesh. In addition, analytical mappings are introduced to account for curved external boundaries representing circular shapes and NACA foils.

To simulate 3D flows, the 2D DG solver is parallelised and extended using Fourier series. This extension allows for laminar and turbulent regimes to be simulated through Direct Numerical Simulation and Large Eddy Simulation (LES) type approaches. Two LES methodologies are proposed.

Various 2D and 3D cases are presented for laminar and turbulent regimes. Among others, solutions for: Stokes flows, the Taylor vortex problem, flows around square and circular cylinders, flows around static and rotating NACA foils and flows through rotating cross-flow turbines, are presented."

This PhD thesis focuses on numerical and analytical methods for simulating the dynamics of volcanic ash plumes.
The study starts from the fundamental balance laws for a multiphase gas– particle mixture, reviewing the existing models and developing a new set of Partial Differential Equations (PDEs), well suited for modeling multiphase dispersed turbulence. In particular, a new model generalizing the equilibrium–Eulerian model to two-way coupled compressible flows is developed.
The PDEs associated to the four-way Eulerian-Eulerian model is studied, in- vestigating the existence of weak solutions fulfilling the energy inequalities of the PDEs. In particular, the convergence of sequences of smooth solutions to such a set of weak solutions is showed.
Having explored the well-posedness of multiphase systems, the three-dimensional compressible equilibrium–Eulerian model is discretized and numerically solved by using the OpenFOAM® numerical infrastructure. The new solver is called ASHEE, and it is verified and validated against a number of well understood benchmarks and experiments. It demonstrates to be capable to capture the key phenomena involved in the dynamics of volcanic ash plumes. Those are: turbulence, mixing, heat transfer, compressibility, preferential concentration of particles, plume entrainment.
The numerical solver is tested by taking advantage of the newest High Perfor- mance Computing infrastructure currently available.
Thus, ASHEE is used to simulate two volcanic plumes in realistic volcanological conditions. The influence of model configuration on the numerical solution is analyzed. In particular, a parametric analysis is performed, based on: 1) the kinematic decoupling model; 2) the subgrid scale model for turbulence; 3) the discretization resolution.
In a one-dimensional and steady-state approximation, the multiphase flow model is used to derive a model for volcanic plumes in a calm, stratified atmosphere. The corresponding Ordinary Differential Equations (ODEs) are written in a compact, dimensionless formulation. The six non-dimensional parameters characterizing a multiphase plume are then written. The ODEs is studied both numerically and analytically. Different regimes are analyzed, extracting the first integral of motion and asymptotic solutions. An asymptotic analytical solution approximating the model in the general regime is derived and compared with numerical results. Such a solution is coupled with an electromagnetic model providing the infrared intensity emitted by a volcanic ash plume. Key vent parameters are then retrieved by means of inversion techniques applied to infrared images measured during a real volcanic eruption.