Numerical and Experimental Methods in Fluid Dynamics

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

Softening at drinking water treatment plants is often realised by fluidised bed pellet reactors. Generally, sand is used as seeding material and pellets are produced as a by-product. To improve to sustainability, research has been carried out to replace the seeding material by re-using grained and sieved calcite pellets as seeding material. An explicit fluidisation model is developed to predict the fluid bed state in fluid bed pellet softening reactors with calcite as seeding material. The fluidisation theory is extended in a model whereby soft sensors are derived and experimentally tested for a wide range of seeding material and pellets. With the soft sensors porosity, particle size and pressure drop can explicitly be calculated. Pilot research has been carried out to calibrate and full-scale experiments to validate the fluidisation models. Four different fluidisation models were reviewed from which the original Richardson-Zaki fluid bed model has been selected as the best explicit fluidisation model to predict the porosity, particle size and pressure drop. Applying a discretisation model for the fluid bed pellet reactor, the current operation of the treatment softening can be improved by estimating the fluidisation, pressure drop behaviour and particle profile. Waternet can apply the Richardson-Zaki fluid bed model in practice for building a soft sensor to achieve optimal bed fluid conditions for the softening process.

The effect of freestream turbulence intensities ranging from Tu = 1.26% to Tu = 3.2% is studied. Skin friction measurements made on the surface of the airfoil using oil film interferometry (OFI) show that, in general, the effect of the increased Tu is to inhibit separation of the laminar boundary layer. With increased Tu, the near-wall flow experiences
strong deceleration in the adverse pressure gradient, but does not reverse as it does in the baseline case where Tu = 0.05%. The Cp distribution resulting from this decelerated fluid is similar in appearance to that of a laminar separation bubble. OFI results also show that laminar separation initiates a more rapid transition process than
does higher turbulence intensity: transition of the boundary layer occurs over a shorter distance with Tu = 1.26% than it does with Tu = 2.19% due to the presence of a LSB at the lower turbulence intensity.

The current project is a study of the 1-D linear advection equation that characterize the waves and represent the transport of substance by bulk motion.The system corresponds to an hyperbolic partial differential equation and takes a big interest when it comes to the study of the normal shock wave. To solve this equation, numerical methods are used and different schemes are introduced to better control the solution through stability, accuracy and consistency. In what follows, a numerical study and investigation of the 1-D Linear Advection Equation.

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.

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the
field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is
a three-dimensional phenomenon.

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.

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'.

Prinsloo, G.J. 2011. Inlet Air Flow Meter for an Internal Combustion Engine. Bachelor Engineering Thesis, Mechatronic Engineering, Stellenbosch University, South Africa. doi 10.13140/2.1.3010.4649.
This project focus on the design an Inlet Air Flow Meter for accurate measurement of mass flow rate of atmospheric air into an engine during engine dynamometer tests. The meter will be used to determine the air-fuel ratio and volumetric efficiency of the engine. To determine if an Inlet Air Flow Meter can be designed and constructed to be sufficiently accurate for dynamometer applications, while evaluating the potential for commercial use in high performance engine management systems. The Inlet Air Flow Meter will be used to evaluate new Biofuel formulations and their performance in internal combustion engines, for example non-edible vegetable oils extracted for fatty acid compositions and biodiesel production, and to evaluate the air flow characteristics, engine performance and emissions production of non-edible oils.

Calibration of subsonic wind tunnel.

Pressure distribution over smooth and rough cylinder.

Pressure distribution over symmetric airfoil

Pressure distribution over cambered airfoil & thin airfoils

Force measurement using wind tunnel balance.

Flow over a flat plate at different angles of incidence

Flow visualization over cylinder

Flow visualization studies in low-speed flow over airfoil with different angle of incidence

The possibilities of studying flows and heat transfer in various power installations using dedicated computer programs developed at the Institute of Thermal Physics, Siberian Branch of the Russian Academy of Sciences are shown. The results obtained from studies of processes in furnace chambers carried out on the experimental models of E-500 and P-67 boilers are presented.

The project is about studying the whole rocket. We will work on comparing between analytical and theory results on the converging diverging (C-D) nozzle, also called de Laval nozzle for an isentropic flow, a normal shock wave in the diverging section, an oblique shock wave and an expansion at the exit. Then we will simulate the flow at the upper surface of the rocket by CFD, considering the upper area as diamond shaped. The aim of the simulation is to show the oblique shock wave created by supersonic speed and doesn’t intersect with the rest of the rocket, reducing overall the drag.

The validation of a high-fidelity uncertainty quantification of a high-speed catamaran is presented, with focus on irregular wave analysis and approximation methods used in design optimization studies, namely a deterministic regular wave approximation, and a stochastic regular wave UQ model. The approach includes a priori CFD simulations, followed by the ex post facto EFD campaign. The validation variables are the wave elevation, x-force, heave and pitch motions, vertical acceleration of the bridge and vertical velocity of the flight deck. Time series values are addressed as primary variables, whereas mean-crossing amplitudes, height, and period are indicated as secondary variables. The subseries and bootstrap methods are applied in order to estimate the validation values and 95% confidence intervals for expected value (EV), standard deviation (SD), mode, and quantile function. Additionally, validation values and confidence intervals for time series EV and SD are evaluated by time series theory, based on the sample variance and size, and the autocovariance function. The deterministic regular wave model assesses the EV of the x-force, whereas the stochastic regular wave UQ focuses on SSAs of pitch, acceleration, and velocity, as relevant merit factors for design optimization. On average, all variables are validated, but the validation uncertainty is quite large, due to limited number of inlet wave components of CFD and short run length for both CFD and EFD. For the evaluation of confidence intervals, the subseries method is found on average more conservative for EV and less conservative for SD, compared to time series theory. Primary variables are validated with an average 5% error and 37% confidence interval. Secondary variables are validated with an average 21% error and 38% confidence interval. The regular wave model is validated with nearly 9% error and 15% confidence interval, whereas the UQ model is validated with an average 1% error and 30% confidence interval.

"This paper describes a Chebyshev collocation method for solving the eigenvalue problem that governs the linear stability of fluid film flow down an inclined plane in presence or absence of surfactant at arbitrary Reynolds numbers. The method is based on the expansion of the eigenfunctions in terms of Chebyshev polynomials, point collocation, and the subsequent solution of the resulting generalized eigenvalue problem with the QZ-algorithm. This method is developed to solve Orr-Sommerfeld equation. The results in the absence of surfactant are compared with those presented by Yih. Finally the effect of inertia and Marangoni mode which appears when insoluble surfactant is present is investigated. The effect of instability modes, the Yih mode which is associated with perturbation on fluid film surface and Marangoni mode which is associated with variation of surfactant surface concentration, are studied for flow instability at arbitrary Reynolds numbers. The agreement between the obtained results at limited conditions and also results from other researchers shows the effectiveness of Collocation Method and suggested profiles.
Key words: Collocation, linear stability; Orr-Sommerfeld; Marangoni; Surfactant; Perturbation analysis"

Cavitation is one of the most challenging fluid flow abnormalities leading to detrimental effects on both the centrifugal pump flow behaviors and physical characteristics. Centrifugal pumps’ most low pressure zones are the first cavitation victims, where cavitation manifests itself in form of pitting on the pump internal solid walls, accompanied by noise and vibration, all leading to the pump hydraulic performance degradation. In the present article, a general description of centrifugal pump performance and related parameters is presented. Based on the literature survey, some light were shed on fundamental cavitation features; where different aspects relating to cavitation in centrifugal pumps were briefly discussed.

In this thesis, we deal with a new framework for the numerical approximation of partial differential equations which employs main ideas and tools from compressed sensing in a Petrov-Galerkin setting. The goal is to compute an s-sparse approximation with respect to a trial basis of dimension N (with s ≪ N) by picking m ≪ N randomly chosen test functions, and to employ sparse optimization techniques to solve the resulting m × N underdetermined linear system. This approach has been named COmpRessed SolvING (in short, CORSING).
First, we carry out an extensive numerical assessment of CORSING on advection-diffusion-reaction equations, both in a one- and a two-dimensional setting, showing that the proposed strategy is able to reduce the computational burden associated with a standard Petrov-Galerkin formulation.
Successively, we focus on the theoretical analysis of the method. In particular, we prove recovery error estimates both in expectation and in probability, comparing the error associated with the CORSING solution with the best s-term approximation error. With this aim, we propose a new theoretical framework based on a variant of the classical inf-sup property for sparse vectors, that is named Restricted Inf-Sup Property, and on the concept of local a-coherence, that generalizes the notion of local coherence to bilinear forms in Hilbert spaces. The recovery results and the corresponding hypotheses are then theoretically assessed on one-dimensional advection-diffusion-reaction problems, while in the two-dimensional setting the verification is carried out through numerical tests.
Finally, a preliminary application of CORSING to three-dimensional advection-diffusion-reaction equations and to the two-dimensional Stokes problem is also provided.

An investigation of the flow field around a double-delta wing was carried out at the Aerodynamics Analysis and Design
Laboratory’s (AADL) water tunnel facility. The model has a sharp leading edge and 76=40
configuration, which is representative of current
fighter aircraft. The experiments were conducted at Reynolds numbers of 10,000 and 15,000. The dye injection technique was utilized to analyze
the flow.Acomparison of the vortical flow structure and the bursting point locations was done for a Reynolds number of 15,000 and found to be
in reasonable agreement. The major objective of this experiment was to study the effect of the angle of attack (AOA) and Reynolds number on
the vortical flow structure. In this flow visualization result, the dominant feature of the AOA and Reynolds number effect on the vortical flow
structure over the double-delta wing has been observed. With increasing Reynolds number, the distance between the vortex trajectory and the
model surface becomes smaller. As the AOA increases, the intertwining or coiling up features cannot be seen and the vortex bursting point
locations move upstream. The distance between the vortical streakline and the model surface becomes larger with increasing AOA. The
crossover point of the main and strake wing vortices also depends on the AOA.

Turbulent jet flows with multiple nozzle inlets are investigated computationally using OpenFOAM. The configurations vary from single to five axisymmetric nozzles. First order closure is used with Reynolds Averaged Navier-Stokes equations. Computed results are compared with the available experimental data. The effect of nozzle configuration on the jet flow field is discussed with predicted mean flow and turbulent flow data. Near field of multiple jets shows non-linear behavior. Multiple jets show better performance in the near field based on entrainment, secondary flows and area averaged turbulent kinetic energy. The downstream evolution of multiple jets differs for configurations with and without central jet. The shape parameter confirms the evolution of multiple jets towards an axisymmetric jet.

A residual-based lubrication method is used in this paper to find the flow rate and pressure field in converging-diverging rigid tubes for the flow of time-independent category of non-Newtonian fluids. Five converging-diverging prototype geometries were used in this investigation in conjunction with two fluid models: Ellis and Herschel-Bulkley. The method was validated by convergence behavior sensibility tests, convergence to analytical solutions for the straight tubes as special cases for the converging-diverging tubes, convergence to analytical solutions found earlier for the flow in converging-diverging tubes of Newtonian fluids as special cases for non-Newtonian, and convergence to analytical solutions found earlier for the flow of power-law fluids in converging-diverging tubes. A brief investigation was also conducted on a sample of diverging-converging geometries. The method can in principle be extended to the flow of viscoelastic and thixotropic/rheopectic fluid categories. The method can also be extended to geometries varying in size and shape in the flow direction, other than the perfect cylindrically-symmetric converging-diverging ones, as long as characteristic flow relations correlating the flow rate to the pressure drop on the discretized elements of the lubrication approximation can be found. These relations can be analytical, empirical and even numerical and hence the method has a wide applicability range.

The negative influence of refrigerants on the climate and the immediate environment in terms of their higher global warming potential (GWP) and ozone depletion potential (ODP) has prompted this study. Currently, natural refrigerants are the preferred alternative refrigerants and hydrocarbon is numbered among these natural refrigerants with zero ODP and negligible GWP. In order to improve and enhance the performance of the refrigeration system, the performance characteristics of the system were investigated experimentally using eco-friendly refrigerant HC600a as alternative to HFC134a. In addition, comparisons were made using refrigerant mass charge of 46 g of isobutane (HC600a) and 70 g of conventional refrigerant (HFC134a). Thermodynamic parametric analysis was conducted using electric power consumption, coefficient of performance (COP), cooling load and pull-down time (PDT) for the used mass charges. REFPROP software was applied to capture the thermodynamic properties of the vapou...

Softening at drinking water treatment plants is often realised by fluidised bed pellet reactors. Generally, sand is used as seeding material and pellets are produced as a by-product. To improve to sustainability, research has been carried out to replace the seeding material by re-using grained and sieved calcite pellets as seeding material. An explicit fluidisation model is developed to predict the fluid bed state in fluid bed pellet softening reactors with calcite as seeding material. The fluidisation theory is extended in a model whereby soft sensors are derived and experimentally tested for a wide range of seeding material and pellets. With the soft sensors porosity, particle size and pressure drop can explicitly be calculated. Pilot research has been carried out to calibrate and full-scale experiments to validate the fluidisation models. Four different fluidisation models were reviewed from which the original Richardson-Zaki fluid bed model has been selected as the best explicit...

Effects of the sideslip angle on the wing-kinematics and the aerodynamic forces and moments generated by the locusts were investigated in a low-speed wind tunnel. Three locusts (Schistocerca americana) were tested at tunnel speeds of 0,
2, and 4 m/s at three body angles of attack (0, 3 and 7 degrees) and three sideslip angles (0, -10 and -20 degrees) at each angle of attack. A sensitive custom-built microbalance was used to measure the forces and moments. Simultaneous high -
speed videos were taken to record the wing kinematics. Results in the form of normalized stroke-averaged forces and moments are presented and a correlation with the wing kinematics has been investigated.

Glass-forming liquids exhibit a rich phenomenology upon onfinement. This is often related to the effects arising from wall–fluid interactions. Here we focus on the interesting limit where the separation of the confining walls becomes of the order of a few particle diameters. For a moderately polydisperse, densely packed hard-sphere fluid confined between two smooth hard walls, we show via event-driven molecular dynamics simulations the emergence of a multiple reentrant glass transition scenario upon a variation of the wall separation. Using thermodynamic relations, this reentrant phenomenon is shown to persist also under constant chemical potential. This allows straightforward experimental investigation and opens the way to a variety of applications in micro- and nanotechnology, where channel dimensions are comparable to the size of the contained particles. The results are in line with theoretical predictions obtained by a combination of density functional theory and the mode-coupling theory of the glass transition.

This volume consists of thirteen selected papers that derive from the second ERCOFTAC conference on Mixing in Geophysical Flow held in Vilanova i la Geltru, Barcelona, Spain in March 1997. Some of the papers were presented at the conference while others were written specifically for this volume. Mixing in geophysical flows presents special challenges to fluid dynamicists and modelers, as it includes effects of stratification, the rotation of the Earth, and chemical and biological processes. Significant mixing is associated with turbulence, which in itself is a proorly understood area of fluid dynamics. In the presence of these complicating effects, the understanding of turbulence in natural flows is even more difficult. The papers presented here provide an example of the different techniques used in laboratory, numerical or field work that help to understand the nonhomogeneous and non-isotropic flows that produce Turbulent Mixing in geophysical Flows

Numerical results are compared with experimental results for z/d=0 and x/d=0.125, 0.5, 1.25. Abstract The flow characteristics over a sphere are predicted numerically. Experimental studies have been done in a low speed wind tunnel. In order to verify numerical results velocity and turbulence intensity measurements were done in centerline of sphere. Drag forces were measured for high Reynolds numbers of Re≈30000-140000. The wake region was explored in detail and turbulence and velocity values have been performed numerically at 53000. Reynolds number, at three different locations (z/d=0, 0.125, 0.25, 0.625 and 0.875) downstream of the sphere. Especially, stations which were critical turbulence levels were determined. Numerical predictions are carried out by using the commercial CFD package Fluent. The results of the presented CFD predictions are shown to be in good agreement with the experimental data.

In this paper, analytical expressions correlating the volumetric flow rate to the pressure drop are derived for the flow of Carreau and Cross fluids through straight rigid circular uniform pipes and long thin slits. The derivation is based on the application of Weissenberg-Rabinowitsch-Mooney-Schofield method to obtain flow solutions for generalized Newtonian fluids through pipes and our adaptation of this method to the flow through slits. The derived expressions are validated by comparing their solutions to the solutions obtained from direct numerical integration. They are also validated by comparison to the solutions obtained from the variational method which we proposed previously. In all the investigated cases, the three methods agree very well. The agreement with the variational method also lends more support to this method and to the variational principle which the method is based upon.

Turbulent flows are often treated as a noisy environment by control algorithms of underwater robots. However, aquatic animals such as fish have learned to take advantage of certain unsteady flow. Periodic complex flow, such as that found in the wake of cylinders has been shown to offer energy saving opportunities to fish. We built a fish-like robot with an integrated pressure sensor array housed in the head. The robot can control its tail beat synchronization with respect to the periodic oscillations in the flow behind a cylinder. We show that vortices, represented here by pressure maxima, can be detected and exploited to increase the swimming efficiency of the robot fish while it remains rigidly mounted to a force plate. Force measurements show an efficiency gain of 23% when the tail beat of the robotic fish is synchronized at a particular phase lag.