Turbulence Lengthscales and Spectra

Save this PDF as:

Size: px
Start display at page:

Transcription

1 School of Mechanical Aerospace and Civil Engineering TPFE MSc Advanced Turbulence Modelling Introduction RANS based schemes, studied in previous lectures, average the effect of turbulent structures. Turbulence Lengthscales and Spectra T. J. Craft George Begg Building, C41 Reading: S. Pope, Turbulent Flows D. Wilcox, Turbulence Modelling for CFD Closure Strategies for Turbulent and Transitional Flows, (Eds. B.E. Launder, N.D. Sandham) Notes: Blackboard and CFD/TM web server: - People - T. Craft - Online Teaching Material They solve for the mean flow, and introduce models for the Reynolds stresses. None of the turbulent structures are resolved in such approaches. Note, however, that unsteady (URANS) can be performed, allowing large-scale unsteady structures (not strictly turbulence) to be resolved. Examples of such cases include vortex shedding from bluff bodies, and Raleigh-Benard type convection cells in unstably stratified flows. Turbulence Lengthscales and Spectra 211/12 1 / 18 Turbulence Lengthscales and Spectra 211/12 2 / 18 In the following lectures we consider approaches such as Large Eddy Simulation, which resolve some, or all, of the turbulent structures in a flow. To appreciate how these schemes work, and understand how to assess limitations, we need to to know what length and time scales are present in a turbulent flow. We have previously met the idea of the turbulent kinetic energy spectrum, giving information on the range of eddy sizes or lengthscales in the flow. Here we revisit this, to consider it and correlations between turbulent fluctuations in a more quantitative manner. LES and DNS In Direct Numerical Simulation (DNS) the Navier-Stokes equations are solved with sufficient numerical resolution to represent all the turbulent scales accurately. In Large Eddy Simulations (LES) the large scales are resolved, but the effects of the smaller ones (sub-grid-scales, for example) must be modelled. The physical modelling input is thus less than in RANS schemes, and relatively simple, since most of the turbulence energy is contained in the larger scales. However, one does need to ensure the appropriate scales have actually been resolved. One thus needs to know what scales should be present in a flow, and be able to assess what has actually been resolved in a simulation. Recognising the effects of not resolving certain scales can also be important. Turbulence Lengthscales and Spectra 211/12 3 / 18 Turbulence Lengthscales and Spectra 211/12 4 / 18

2 Spatial Correlations For simplicity, we begin by considering a homogeneous turbulence field. An example could be a large tank, initially stirred by running a grid through it. A range of turbulent eddy sizes would be generated, which would gradually decay over time. At any two points x a and x b we can examine how correlated the velocity fluctuations are by averaging their product: R ij (x a,x b,t) = u i (x a,t)u j (x b,t) If the flow is homogeneous, it is only the separation between the two points which is important, so the two-point correlation can be written as R ij (r,t) = u i (x,t)u j (x + r,t) which will be independent of the location x. At very large separation distances the correlation will be zero. where denotes the averaging process. We might equally write this as a correlation between two points x a and x a + r, where r = x b x a : R ij (x a,r,t) = u i (x a,t)u j (x a + r,t) At smaller separation distances there will be some correlation between the velocities (for example, if the points lie in the same eddy). As the separation distance becomes zero the correlation gives the Reynolds stress u i u j. Turbulence Lengthscales and Spectra 211/12 5 / 18 Turbulence Lengthscales and Spectra 211/12 6 / 18 Fourier Transforms The Fourier transform of a function f (t) can be written as where i 2 = 1. g(ω) = FT [f (t)] = 1 2π f (t) exp( iωt) dt (1) The inverse transformation is given by f (t) = FT 1 [g(ω)] = g(ω) exp(iωt) dω (2) By examining the transform g(ω) we can identify what frequencies are present in the original function f, and identify which are the dominant frequencies (or corresponding time periods). Obviously, if f is a function of space the Fourier transform gives information on the wavelengths present in it. One attractive feature of Fourier transforms is that transforms of function derivatives are relatively easy to obtain. The Fourier transform of df (t)/dt, for example, is given by FT [df (t)/dt] = iω g(ω) where g(ω) is the Fourier transform of f, as in equation (1). As an illustration, consider the velocity u(x) in a box of side length L. If we assume this can be decomposed into N sinusoidal functions, then we could write u(x) = a (1) cos(2πx/l) + + a (N) cos(2πnx/l) = a (m) cos(k (m) x) The wavelengths present are (L, L/2,..., L/N), and the corresponding wave numbers, K, are (2π/L,..., 2Nπ/L). Turbulence Lengthscales and Spectra 211/12 7 / 18 Turbulence Lengthscales and Spectra 211/12 8 / 18

3 Extending this to three dimensions, and allowing all eddy sizes, we could write [ [ ] u j (x) = û j (K 1,K 2,K 3 )exp(ik 1 x 1 )dk 1 ]exp(ik 2 x 2 )dk 2 exp(ik 3 x 3 )dk 3 = = û j (K 1,K 2,K 3 )exp(ik 1 x 1 + ik 2 x 2 + ik 3 x 3 )dk 1 dk 2 dk 3 û j (K )exp(ik.x)dk 1 dk 2 dk 3 The vectors x and K give the position in physical space and wave number vector in Fourier space. û(k ) gives the amplitude of sinusoidal oscillations in direction K, with wavelength L = 2π/ K. It can be obtained by a Fourier transform of the velocity signal u(x): û j (K ) = 1 (2π) 2 u j (x)exp( ik.x)dx 1 dx 2 dx 3 Turbulence Energy Spectrum From its definition, the turbulent kinetic energy can be written as k =.5u i u i = u i (x,t)u i (x,t) using the averaging notation employed above. It can be shown that for the two-point correlation R ij (r) = u i (x)u j (x + r) the corresponding Fourier transform is given by R ij (K ) = û i (K )û j ( K ) From the above definition of k, we can thus write k = E 3D (K )dk 1 dk 2 dk 3 where the 3D energy spectrum is given by E 3D (K ) =.5 R ii (K ) =.5 û i (K )û i ( K ) Turbulence Lengthscales and Spectra 211/12 9 / 18 Turbulence Lengthscales and Spectra 211/12 1 / 18 The energy spectrum, E 3D, represents the contribution to k of a sinusoidal form in direction K with wavelength L = 2π/ K. We can define the spectral density of turbulent kinetic energy, E( K ) as the total contribution from E 3D (K ) over a sphere of radius K (ie. all contributions with wavelength L = 2π/ K ), so k = E(K )dk The energy spectrum, E(K ), typically shows a maximum for some wave number K M, corresponding to the larger energy-containing eddies. E(K ) decays at higher wave numbers (smaller eddies). Dissipation Rate Spectrum Recall the dissipation rate is defined as ε = ν u i x j u i x j In the results quoted above on Fourier transforms, we noted that FT [df (t)/dt] = iω g(ω) where g(ω) was the transform of f (t). Applying this in 3-D to obtain the transform of ε leads to ε = ν K 2 û i (K )û i ( K )dk 1 dk 2 dk 3 or ε = D 3D (K )dk 1 dk 2 dk 3 where D 3D (K ) = 2ν K 2 E 3D (K ). Turbulence Lengthscales and Spectra 211/12 11 / 18 Turbulence Lengthscales and Spectra 211/12 12 / 18

4 We can again express this in terms of a spectral density of dissipation, to give ε = D(K )dk = 2νK 2 E(K )dk Comparing to the behaviour of the energy spectrum density, D(K ) only decreases with wave number once E(K ) is decaying faster than K 2, and will thus have its peak at a higher wave number than E(K ). This confirms previous statements that dissipation is mainly associated with the smaller eddies, whilst the turbulence energy is mostly associated with the larger eddies. Kolmogorov Model Based on empirical observations and dimensional analysis, the Kolmogorov model allows us to derive some quantitative relations for the spectral densities of k and ε. The model is based on the following assumptions: Turbulent kinetic energy is carried by the larger eddies, which are unaffected by molecular viscosity, ν. The viscosity, ν, affects only the smaller eddies, responsible for dissipation. The energy dissipated by the small eddies comes from the larger ones, which in turn obtain energy from the mean field. We denote the large, energy-containing, eddies characteristic lengthscale as L t, and that of the small dissipating eddies by η. Turbulence Lengthscales and Spectra 211/12 13 / 18 Turbulence Lengthscales and Spectra 211/12 14 / 18 The above assumptions lead to L t = f (k,ε) and η = f (ν,ε) The first implies that the dissipation rate is determined by the flow of energy down the cascade from the large scales (hence the level of ε depends on what happens in the large scales). The second shows that the lengthscale of the small eddies, η, must then adapt to be consistent with the dissipation rate. Dimensional analysis leads to L t k 3/2 ε and ( ν 3 η ε ) 1/4 η is called the Kolmogorov scale. Eddies smaller than this will be instantly dissipated by viscosity. L t, the scale of the large eddies, is called the integral scale. From the lengthscale expressions above we obtain L t η = k 3/2 ε 3/4 = R3/4 ν3/4 t where R t is the turbulent Reynolds number met earlier. As noted before, as R t increases, the range of scales present in the flow thus increases. Note that the turbulent Reynolds number is typically much smaller than the mean or bulk Reynolds number. For example, in a pipe flow we might have k 1/2.3U m, and L t.1d, giving R t.3u m D/ν. Turbulence Lengthscales and Spectra 211/12 15 / 18 Turbulence Lengthscales and Spectra 211/12 16 / 18

5 At high enough Reynolds numbers there will be a range of wave numbers called the inertial range where eddies simply pass on energy to the smaller structures at the same rate they receive it from the larger ones. In this range the spectral density cannot depend on k or ν, so we must have E = f (ε,k ), and the only dimensionally correct combination is E(K ) = c k ε 2/3 K 5/3 This is often referred to as Kolmogorov s model, with a constant c k = 1.5, and is generally well-verified by measurements. The picture given by the model for high Reynolds number flows, as the Reynolds number is increased (eg. by decreasing viscosity) is as follows: The mean flow and large scale structures remain unchanged. The value of ε also remains unchanged, representing the rate at which energy is extracted from the mean field and transferred across the inertial range of wave numbers. The bandwidth of the inertial range increases with Re, as smaller and smaller eddies are created until the amount of energy cascading from the larger scales is dissipated to heat. The corresponding dissipation spectral density is given by D(k) = 2νc k ε 2/3 K 1/3 Turbulence Lengthscales and Spectra 211/12 17 / 18 Turbulence Lengthscales and Spectra 211/12 18 / 18

11 Navier-Stokes equations and turbulence

11 Navier-Stokes equations and turbulence So far, we have considered ideal gas dynamics governed by the Euler equations, where internal friction in the gas is assumed to be absent. Real fluids have internal

A subgrid-scale model for the scalar dissipation rate in nonpremixed combustion

Center for Turbulence Research Proceedings of the Summer Program 1998 11 A subgrid-scale model for the scalar dissipation rate in nonpremixed combustion By A. W. Cook 1 AND W. K. Bushe A subgrid-scale

Solar Wind Heating by MHD Turbulence

Solar Wind Heating by MHD Turbulence C. S. Ng, A. Bhattacharjee, and D. Munsi Space Science Center University of New Hampshire Acknowledgment: P. A. Isenberg Work partially supported by NSF, NASA CMSO

Lecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2)

Lecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2) In this lecture How does turbulence affect the ensemble-mean equations of fluid motion/transport? Force balance in a quasi-steady turbulent boundary

Lecture 8 - Turbulence. Applied Computational Fluid Dynamics

Lecture 8 - Turbulence Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (2002-2006) Fluent Inc. (2002) 1 Turbulence What is turbulence? Effect of turbulence

Validation of CFD Simulations for Natural Ventilation

Jiang, Y., Allocca, C., and Chen, Q. 4. Validation of CFD simulations for natural ventilation, International Journal of Ventilation, (4), 359-37. Validation of CFD Simulations for Natural Ventilation Yi

Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture No. # 36 Pipe Flow Systems

Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture No. # 36 Pipe Flow Systems Welcome back to the video course on Fluid Mechanics. In today

Lecturer, Department of Engineering, ar45@le.ac.uk, Lecturer, Department of Mathematics, sjg50@le.ac.uk

39 th AIAA Fluid Dynamics Conference, San Antonio, Texas. A selective review of CFD transition models D. Di Pasquale, A. Rona *, S. J. Garrett Marie Curie EST Fellow, Engineering, ddp2@le.ac.uk * Lecturer,

Assessment of Numerical Methods for DNS of Shockwave/Turbulent Boundary Layer Interaction

44th AIAA Aerospace Sciences Meeting and Exhibit, Jan 9 Jan 12, Reno, Nevada Assessment of Numerical Methods for DNS of Shockwave/Turbulent Boundary Layer Interaction M. Wu and M.P. Martin Mechanical and

Civil Engineering Hydraulics Mechanics of Fluids. Flow in Pipes

Civil Engineering Hydraulics Mechanics of Fluids Flow in Pipes 2 Now we will move from the purely theoretical discussion of nondimensional parameters to a topic with a bit more that you can see and feel

Particles in turbulence: the need for a multi-scale approach

Particles in turbulence: the need for a multi-scale approach Jos Derksen Multi-Scale Physics Department Delft University of Technology The Netherlands email: jos@klft.tn.tudelft.nl Motivation Outline solid

Chapter 1. Governing Equations of Fluid Flow and Heat Transfer

Chapter 1 Governing Equations of Fluid Flow and Heat Transfer Following fundamental laws can be used to derive governing differential equations that are solved in a Computational Fluid Dynamics (CFD) study

Introductory FLUENT Training

Chapter 10 Transient Flow Modeling Introductory FLUENT Training www.ptecgroup.ir 10-1 Motivation Nearly all flows in nature are transient! Steady-state assumption is possible if we: Ignore transient fluctuations

Lecture 1: Introduction to Random Walks and Diffusion

Lecture : Introduction to Random Walks and Diffusion Scribe: Chris H. Rycroft (and Martin Z. Bazant) Department of Mathematics, MIT February, 5 History The term random walk was originally proposed by Karl

Conservation Laws of Turbulence Models

Conservation Laws of Turbulence Models Leo G. Rebholz 1 Department of Mathematics University of Pittsburgh, PA 15260 ler6@math.pitt.edu October 17, 2005 Abstract Conservation of mass, momentum, energy,

Comparison of flow regime transitions with interfacial wave transitions

Comparison of flow regime transitions with interfacial wave transitions M. J. McCready & M. R. King Chemical Engineering University of Notre Dame Flow geometry of interest Two-fluid stratified flow gas

High resolution modelling of flow-vegetation interactions: From the stem scale to the reach scale

Septième École Interdisciplinaire de Rennes sur les Systèmes Complexes - 13 th October 2015 High resolution modelling of flow-vegetation interactions: From the stem scale to the reach scale Tim Marjoribanks

Examination paper for Solutions to Matematikk 4M and 4N

Department of Mathematical Sciences Examination paper for Solutions to Matematikk 4M and 4N Academic contact during examination: Trygve K. Karper Phone: 99 63 9 5 Examination date:. mai 04 Examination

Sound. References: L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol. 2, Gas Dynamics, Chapter 8

References: Sound L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol., Gas Dynamics, Chapter 8 1 Speed of sound The phenomenon of sound waves is one that

7. DYNAMIC LIGHT SCATTERING 7.1 First order temporal autocorrelation function.

7. DYNAMIC LIGHT SCATTERING 7. First order temporal autocorrelation function. Dynamic light scattering (DLS) studies the properties of inhomogeneous and dynamic media. A generic situation is illustrated

CFD Based Air Flow and Contamination Modeling of Subway Stations

CFD Based Air Flow and Contamination Modeling of Subway Stations Greg Byrne Center for Nonlinear Science, Georgia Institute of Technology Fernando Camelli Center for Computational Fluid Dynamics, George

CHEMICAL ENGINEERING AND CHEMICAL PROCESS TECHNOLOGY - Vol. I - Interphase Mass Transfer - A. Burghardt

INTERPHASE MASS TRANSFER A. Burghardt Institute of Chemical Engineering, Polish Academy of Sciences, Poland Keywords: Turbulent flow, turbulent mass flux, eddy viscosity, eddy diffusivity, Prandtl mixing

Basic Equations, Boundary Conditions and Dimensionless Parameters

Chapter 2 Basic Equations, Boundary Conditions and Dimensionless Parameters In the foregoing chapter, many basic concepts related to the present investigation and the associated literature survey were

AN EFFECT OF GRID QUALITY ON THE RESULTS OF NUMERICAL SIMULATIONS OF THE FLUID FLOW FIELD IN AN AGITATED VESSEL

14 th European Conference on Mixing Warszawa, 10-13 September 2012 AN EFFECT OF GRID QUALITY ON THE RESULTS OF NUMERICAL SIMULATIONS OF THE FLUID FLOW FIELD IN AN AGITATED VESSEL Joanna Karcz, Lukasz Kacperski

Abaqus/CFD Sample Problems. Abaqus 6.10

Abaqus/CFD Sample Problems Abaqus 6.10 Contents 1. Oscillatory Laminar Plane Poiseuille Flow 2. Flow in Shear Driven Cavities 3. Buoyancy Driven Flow in Cavities 4. Turbulent Flow in a Rectangular Channel

ESSENTIAL COMPUTATIONAL FLUID DYNAMICS

ESSENTIAL COMPUTATIONAL FLUID DYNAMICS Oleg Zikanov WILEY JOHN WILEY & SONS, INC. CONTENTS PREFACE xv 1 What Is CFD? 1 1.1. Introduction / 1 1.2. Brief History of CFD / 4 1.3. Outline of the Book / 6 References

ANALYSIS OF METALLURGICAL MELTING PROCESSES USING NUMERICAL SIMULATION

ANALYSIS OF METALLURGICAL MELTING PROCESSES USING NUMERICAL SIMULATION E. Baake Institute of Electrotechnology, Leibniz University of Hannover 30167 Hannover, Wilhelm Busch Str. 4, Germany baake@etp.uni-hannover.de

Introduction to CFD Basics

Introduction to CFD Basics Rajesh Bhaskaran Lance Collins This is a quick-and-dirty introduction to the basic concepts underlying CFD. The concepts are illustrated by applying them to simple 1D model problems.

Purdue University - School of Mechanical Engineering. Objective: Study and predict fluid dynamics of a bluff body stabilized flame configuration.

Extinction Dynamics of Bluff Body Stabilized Flames Investigator: Steven Frankel Graduate Students: Travis Fisher and John Roach Sponsor: Air Force Research Laboratory and Creare, Inc. Objective: Study

Application of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semi-infinite domain)

Application of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semi-infinite domain) The Fourier Sine Transform pair are F. T. : U = 2/ u x sin x dx, denoted as U

Appendix D Digital Modulation and GMSK

D1 Appendix D Digital Modulation and GMSK A brief introduction to digital modulation schemes is given, showing the logical development of GMSK from simpler schemes. GMSK is of interest since it is used

Direct and large-eddy simulation of rotating turbulence

Direct and large-eddy simulation of rotating turbulence Bernard J. Geurts, Darryl Holm, Arek Kuczaj Multiscale Modeling and Simulation (Twente) Anisotropic Turbulence (Eindhoven) Mathematics Department,

Lecture 1. The nature of electromagnetic radiation.

Lecture 1. The nature of electromagnetic radiation. 1. Basic introduction to the electromagnetic field: Dual nature of electromagnetic radiation Electromagnetic spectrum. Basic radiometric quantities:

Inverse problems. Nikolai Piskunov 2014. Regularization: Example Lecture 4

Inverse problems Nikolai Piskunov 2014 Regularization: Example Lecture 4 Now that we know the theory, let s try an application: Problem: Optimal filtering of 1D and 2D data Solution: formulate an inverse

Part IV. Conclusions

Part IV Conclusions 189 Chapter 9 Conclusions and Future Work CFD studies of premixed laminar and turbulent combustion dynamics have been conducted. These studies were aimed at explaining physical phenomena

Chemistry 223: Maxwell-Boltzmann Distribution David Ronis McGill University

Chemistry 223: Maxwell-Boltzmann Distribution David Ronis McGill University The molecular description of the bulk properties of a gas depends upon our knowing the mathematical form of the velocity distribution;

NUMERICAL SIMULATION OF REGULAR WAVES RUN-UP OVER SLOPPING BEACH BY OPEN FOAM

NUMERICAL SIMULATION OF REGULAR WAVES RUN-UP OVER SLOPPING BEACH BY OPEN FOAM Parviz Ghadimi 1*, Mohammad Ghandali 2, Mohammad Reza Ahmadi Balootaki 3 1*, 2, 3 Department of Marine Technology, Amirkabir

Comparison of the Vertical Velocity used to Calculate the Cloud Droplet Number Concentration in a Cloud-Resolving and a Global Climate Model

Comparison of the Vertical Velocity used to Calculate the Cloud Droplet Number Concentration in a Cloud-Resolving and a Global Climate Model H. Guo, J. E. Penner, M. Herzog, and X. Liu Department of Atmospheric,

Fast Fourier Transforms and Power Spectra in LabVIEW

Application Note 4 Introduction Fast Fourier Transforms and Power Spectra in LabVIEW K. Fahy, E. Pérez Ph.D. The Fourier transform is one of the most powerful signal analysis tools, applicable to a wide

Turbulence and Fluent

Turbulence and Fluent Turbulence Modeling What is Turbulence? We do not really know 3D, unsteady, irregular motion in which transported quantities fluctuate in time and space. Turbulent eddies (spatial

Table 9.1 Types of intrusions of a fluid into another and the corresponding terminology.

Chapter 9 Turbulent Jets SUMMARY: This chapter is concerned with turbulent jets, namely their overall shape and velocity structure. The first jets being considered are those penetrating in homogeneous

Chapter 8. Viscous Flow in Pipes

Chapter 8 Viscous Flow in Pipes 8.1 General Characteristics of Pipe Flow We will consider round pipe flow, which is completely filled with the fluid. 8.1.1 Laminar or Turbulent Flow The flow of a fluid

Aerodynamic Department Institute of Aviation. Adam Dziubiński CFD group FLUENT

Adam Dziubiński CFD group IoA FLUENT Content Fluent CFD software 1. Short description of main features of Fluent 2. Examples of usage in CESAR Analysis of flow around an airfoil with a flap: VZLU + ILL4xx

Introduction to Green s Functions: Lecture notes 1

October 18, 26 Introduction to Green s Functions: Lecture notes 1 Edwin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-16 91 Stockholm, Sweden Abstract In the present notes I try to give a better

Chapter 2. Derivation of the Equations of Open Channel Flow. 2.1 General Considerations

Chapter 2. Derivation of the Equations of Open Channel Flow 2.1 General Considerations Of interest is water flowing in a channel with a free surface, which is usually referred to as open channel flow.

Lecture L22-2D Rigid Body Dynamics: Work and Energy

J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

Use of OpenFoam in a CFD analysis of a finger type slug catcher. Dynaflow Conference 2011 January 13 2011, Rotterdam, the Netherlands

Use of OpenFoam in a CFD analysis of a finger type slug catcher Dynaflow Conference 2011 January 13 2011, Rotterdam, the Netherlands Agenda Project background Analytical analysis of two-phase flow regimes

Basic Principles in Microfluidics

Basic Principles in Microfluidics 1 Newton s Second Law for Fluidics Newton s 2 nd Law (F= ma) : Time rate of change of momentum of a system equal to net force acting on system!f = dp dt Sum of forces

Along-wind self-excited forces of two-dimensional cables under extreme wind speeds

The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 Along-wind self-excited forces of two-dimensional cables under extreme wind

Viscous flow in pipe

Viscous flow in pipe Henryk Kudela Contents 1 Laminar or turbulent flow 1 2 Balance of Momentum - Navier-Stokes Equation 2 3 Laminar flow in pipe 2 3.1 Friction factor for laminar flow...........................

F en = mω 0 2 x. We should regard this as a model of the response of an atom, rather than a classical model of the atom itself.

The Electron Oscillator/Lorentz Atom Consider a simple model of a classical atom, in which the electron is harmonically bound to the nucleus n x e F en = mω 0 2 x origin resonance frequency Note: We should

ME6130 An introduction to CFD 1-1

ME6130 An introduction to CFD 1-1 What is CFD? Computational fluid dynamics (CFD) is the science of predicting fluid flow, heat and mass transfer, chemical reactions, and related phenomena by solving numerically

MATH 110 Spring 2015 Homework 6 Solutions

MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =

Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence

Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence M. R. Overholt a) and S. B. Pope Sibley School of Mechanical and Aerospace Engineering, Cornell University,

Note 5.1 Stress range histories and Rain Flow counting

June 2009/ John Wægter Note 5.1 Stress range histories and Rain Flow counting Introduction...2 Stress range histories...3 General...3 Characterization of irregular fatigue loading...4 Stress range histogram...5

Auto-Tuning Using Fourier Coefficients

Auto-Tuning Using Fourier Coefficients Math 56 Tom Whalen May 20, 2013 The Fourier transform is an integral part of signal processing of any kind. To be able to analyze an input signal as a superposition

Transport of passive and active tracers in turbulent flows

Chapter 3 Transport of passive and active tracers in turbulent flows A property of turbulence is to greatly enhance transport of tracers. For example, a dissolved sugar molecule takes years to diffuse

Probability and Random Variables. Generation of random variables (r.v.)

Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly

5.4 The Heat Equation and Convection-Diffusion

5.4. THE HEAT EQUATION AND CONVECTION-DIFFUSION c 6 Gilbert Strang 5.4 The Heat Equation and Convection-Diffusion The wave equation conserves energy. The heat equation u t = u xx dissipates energy. The

CFD Application on Food Industry; Energy Saving on the Bread Oven

Middle-East Journal of Scientific Research 13 (8): 1095-1100, 2013 ISSN 1990-9233 IDOSI Publications, 2013 DOI: 10.5829/idosi.mejsr.2013.13.8.548 CFD Application on Food Industry; Energy Saving on the

L r = L m /L p. L r = L p /L m

NOTE: In the set of lectures 19/20 I defined the length ratio as L r = L m /L p The textbook by Finnermore & Franzini defines it as L r = L p /L m To avoid confusion let's keep the textbook definition,

Lecture 11 Boundary Layers and Separation. Applied Computational Fluid Dynamics

Lecture 11 Boundary Layers and Separation Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (2002-2006) Fluent Inc. (2002) 1 Overview Drag. The boundary-layer

A C O U S T I C S of W O O D Lecture 3

Jan Tippner, Dep. of Wood Science, FFWT MU Brno jan. tippner@mendelu. cz Content of lecture 3: 1. Damping 2. Internal friction in the wood Content of lecture 3: 1. Damping 2. Internal friction in the wood

Purpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I

Part 3: Time Series I Purpose of Time Series Analysis (Figure from Panofsky and Brier 1968) Autocorrelation Function Harmonic Analysis Spectrum Analysis Data Window Significance Tests Some major purposes

Scalar Transport. Introduction. T. J. Craft George Begg Building, C41. Eddy-Diffusivity Modelling. TPFE MSc Advanced Turbulence Modelling

School of Mechanical Aerospace and Civil Engineering TPFE MSc Advanced Turbulence Modelling Scalar Transport T. J. Craft George Begg Building, C41 Reading: S. Pope, Turbulent Flows D. Wilco, Turbulence

Fundamentals of Plasma Physics Waves in plasmas

Fundamentals of Plasma Physics Waves in plasmas APPLAuSE Instituto Superior Técnico Instituto de Plasmas e Fusão Nuclear Vasco Guerra 1 Waves in plasmas What can we study with the complete description

TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS

TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of

Reynolds Number Dependence of Energy Spectra in the Overlap Region of Isotropic Turbulence

Flow, Turbulence and Combustion 63: 443 477, 1999. 2000 Kluwer Academic Publishers. Printed in the Netherlands. 443 Reynolds Number Dependence of Energy Spectra in the Overlap Region of Isotropic Turbulence

CHAPTER 4 CFD ANALYSIS OF THE MIXER

98 CHAPTER 4 CFD ANALYSIS OF THE MIXER This section presents CFD results for the venturi-jet mixer and compares the predicted mixing pattern with the present experimental results and correlation results

Lecture 18: The Time-Bandwidth Product

WAVELETS AND MULTIRATE DIGITAL SIGNAL PROCESSING Lecture 18: The Time-Bandwih Product Prof.Prof.V.M.Gadre, EE, IIT Bombay 1 Introduction In this lecture, our aim is to define the time Bandwih Product,

4 Microscopic dynamics

4 Microscopic dynamics In this section we will look at the first model that people came up with when they started to model polymers from the microscopic level. It s called the Oldroyd B model. We will

Lecture 6 - Boundary Conditions. Applied Computational Fluid Dynamics

Lecture 6 - Boundary Conditions Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (2002-2006) Fluent Inc. (2002) 1 Outline Overview. Inlet and outlet boundaries.

NUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES

Vol. XX 2012 No. 4 28 34 J. ŠIMIČEK O. HUBOVÁ NUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES Jozef ŠIMIČEK email: jozef.simicek@stuba.sk Research field: Statics and Dynamics Fluids mechanics

NUMERICAL STUDY OF FLOW AND TURBULENCE THROUGH SUBMERGED VEGETATION

NUMERICAL STUDY OF FLOW AND TURBULENCE THROUGH SUBMERGED VEGETATION HYUNG SUK KIM (1), MOONHYEONG PARK (2), MOHAMED NABI (3) & ICHIRO KIMURA (4) (1) Korea Institute of Civil Engineering and Building Technology,

Conceptual similarity to linear algebra

Modern approach to packing more carrier frequencies within agivenfrequencyband orthogonal FDM Conceptual similarity to linear algebra 3-D space: Given two vectors x =(x 1,x 2,x 3 )andy = (y 1,y 2,y 3 ),

Nyquist Sampling Theorem. By: Arnold Evia

Nyquist Sampling Theorem By: Arnold Evia Table of Contents What is the Nyquist Sampling Theorem? Bandwidth Sampling Impulse Response Train Fourier Transform of Impulse Response Train Sampling in the Fourier

CBE 6333, R. Levicky 1 Differential Balance Equations

CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,

Graduate Certificate Program in Energy Conversion & Transport Offered by the Department of Mechanical and Aerospace Engineering

Graduate Certificate Program in Energy Conversion & Transport Offered by the Department of Mechanical and Aerospace Engineering Intended Audience: Main Campus Students Distance (online students) Both Purpose:

CFD Simulation of Reacting Flows

Chapter 2 CFD Simulation of Reacting Flows In this chapter we examine the basic ideas behind the direct numerical solution of differential equations. This approach leads to methods that can handle nonlinear

The nearly-free electron model

Handout 3 The nearly-free electron model 3.1 Introduction Having derived Bloch s theorem we are now at a stage where we can start introducing the concept of bandstructure. When someone refers to the bandstructure

The continuous and discrete Fourier transforms

FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

On the Sensitivity of LES to Numerics and Grid Resolution in the Process of Landing-Gear Noise Prediction

CEAA-2012, Svetlogorsk, Russia On the Sensitivity of LES to Numerics and Grid Resolution in the Process of Landing-Gear Noise Prediction M. Shur, M. Strelets, A. Travin St.-Petersburg State Polytechnic

Sound and stringed instruments

Sound and stringed instruments Lecture 14: Sound and strings Reminders/Updates: HW 6 due Monday, 10pm. Exam 2, a week today! 1 Sound so far: Sound is a pressure or density fluctuation carried (usually)

CFD Simulation of Subcooled Flow Boiling using OpenFOAM

Research Article International Journal of Current Engineering and Technology E-ISSN 2277 4106, P-ISSN 2347-5161 2014 INPRESSCO, All Rights Reserved Available at http://inpressco.com/category/ijcet CFD

Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem

S. Widnall 16.07 Dynamics Fall 008 Version 1.0 Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem The Three-Body Problem In Lecture 15-17, we presented the solution to the two-body

Turbulence Modeling in CFD Simulation of Intake Manifold for a 4 Cylinder Engine

HEFAT2012 9 th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics 16 18 July 2012 Malta Turbulence Modeling in CFD Simulation of Intake Manifold for a 4 Cylinder Engine Dr MK

Electromagnetic Spectrum

Introduction 1 Electromagnetic Spectrum The electromagnetic spectrum is the distribution of electromagnetic radiation according to energy, frequency, or wavelength. The electro-magnetic radiation can be

Notes for AA214, Chapter 7. T. H. Pulliam Stanford University

Notes for AA214, Chapter 7 T. H. Pulliam Stanford University 1 Stability of Linear Systems Stability will be defined in terms of ODE s and O E s ODE: Couples System O E : Matrix form from applying Eq.

Frequency Response and Continuous-time Fourier Transform

Frequency Response and Continuous-time Fourier Transform Goals Signals and Systems in the FD-part II I. (Finite-energy) signals in the Frequency Domain - The Fourier Transform of a signal - Classification

Commercial CFD Software Modelling

Commercial CFD Software Modelling Dr. Nor Azwadi bin Che Sidik Faculty of Mechanical Engineering Universiti Teknologi Malaysia INSPIRING CREATIVE AND INNOVATIVE MINDS 1 CFD Modeling CFD modeling can be

08/19/09 PHYSICS 223 Exam-2 NAME Please write down your name also on the back side of this exam

08/19/09 PHYSICS 3 Exam- NAME Please write down your name also on the back side of this exam 1. A sinusoidal wave of frequency 500 Hz has a speed of 350 m/s. 1A. How far apart (in units of cm) are two

Dimensional Analysis

Dimensional Analysis An Important Example from Fluid Mechanics: Viscous Shear Forces V d t / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Ƭ = F/A = μ V/d More generally, the viscous

Room Acoustic Reproduction by Spatial Room Response

Room Acoustic Reproduction by Spatial Room Response Rendering Hoda Nasereddin 1, Mohammad Asgari 2 and Ayoub Banoushi 3 Audio Engineer, Broadcast engineering department, IRIB university, Tehran, Iran,

NZQA registered unit standard 20431 version 2 Page 1 of 7. Demonstrate and apply fundamental knowledge of a.c. principles for electronics technicians

NZQA registered unit standard 0431 version Page 1 of 7 Title Demonstrate and apply fundamental knowledge of a.c. principles for electronics technicians Level 3 Credits 7 Purpose This unit standard covers

5 Signal Design for Bandlimited Channels

225 5 Signal Design for Bandlimited Channels So far, we have not imposed any bandwidth constraints on the transmitted passband signal, or equivalently, on the transmitted baseband signal s b (t) I[k]g

Acoustic Velocity, Impedance, Reflection, Transmission, Attenuation, and Acoustic Etalons

Acoustic Velocity, Impedance, Reflection, Transmission, Attenuation, and Acoustic Etalons Acoustic Velocity The equation of motion in a solid is (1) T = ρ 2 u t 2 (1) where T is the stress tensor, ρ is

CFD Simulation of the NREL Phase VI Rotor

CFD Simulation of the NREL Phase VI Rotor Y. Song* and J. B. Perot # *Theoretical & Computational Fluid Dynamics Laboratory, Department of Mechanical & Industrial Engineering, University of Massachusetts

Keywords: Heat transfer enhancement; staggered arrangement; Triangular Prism, Reynolds Number. 1. Introduction

Heat transfer augmentation in rectangular channel using four triangular prisms arrange in staggered manner Manoj Kumar 1, Sunil Dhingra 2, Gurjeet Singh 3 1 Student, 2,3 Assistant Professor 1.2 Department