DOCSLIB.ORG

A Physical Introduction to Fluid Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Physical Introduction to Fluid Mechanics A Physical Introduction to Fluid Mechanics Study Guide and Practice Problems Spring 2014 A Physical Introduction to Fluid Mechanics Study Guide and Practice Problems Spring 2014 by Alexander J. Smits Professor of Mechanical and Aerospace Engineering Princeton University and Professorial Fellow Department of Mechanical and Aerospace Engineering Monash University December 19, 2013 Copyright A.J. Smits c 2014 Contents 1 Introduction 1 1.1 Study Guide . .1 1.2 Worked Examples . .1 Problems . .9 2 Fluid Statics 13 2.1 Study Guide . 13 2.2 Worked Examples . 14 Problems . 22 3 Equations of Motion in Integral Form 47 3.1 Study Guide . 47 3.2 Worked Examples . 47 Problems . 60 4 Kinematics and Bernoulli's Equation 81 4.1 Study Guide . 81 4.2 Worked Examples . 81 Problems . 84 5 Differential Equations of Motion 99 5.1 Study Guide . 99 5.2 Worked Examples . 99 Problems . 104 6 Irrotational, Incompressible Flows 111 6.1 Study Guide . 111 6.2 Worked Examples . 111 Problems . 113 7 Dimensional Analysis 117 7.1 Study Guide . 117 7.2 Worked Examples . 117 Problems . 122 8 Viscous Internal Flows 133 8.1 Study Guide . 133 8.2 Worked Examples . 133 Problems . 139 v vi CONTENTS 9 Viscous External Flows 151 9.1 Study Guide . 151 9.2 Worked Examples . 152 Problems . 154 10 Open Channel Flow 163 10.1 Study Guide . 163 10.2 Worked Examples . 164 Problems . 167 11 Compressible Flow 185 11.1 Study Guide . 185 11.2 Worked Examples . 185 Problems . 190 12 Turbomachines 193 12.1 Worked Examples . 193 Problems . 197 13 Environmental Fluid Mechanics 201 Problems . 201 Answers to Selected Problems 203 Chapter 1 Introduction 1.1 Study Guide • A fluid is defined as a material that deforms continuously and permanently under the application of a shearing stress. • The pressure at a point in a fluid is independent of the orientation of the surface passing through the point; the pressure is isotropic. • The force due to a pressure p acting on one side of a small element of surface dA defined by a unit normal vector n is given by −pndA. • Pressure is transmitted through a fluid at the speed of sound. • The units we use depend on whatever system we have chosen, and they include quan- tities like feet, seconds, newtons and pascals. In contrast, a dimension is a more abstract notion, and it is the term used to describe concepts such as mass, length and time. • The specific gravity (SG) of a solid or liquid is the ratio of its density to that of water at the same temperature. • A Newtonian fluid is one where the viscous stress is proportional to the rate of strain (velocity gradient). The constant of proportionality is the viscosity, µ, which is a property of the fluid, and depends on temperature. • At the boundary between a solid and a fluid, the fluid and solid velocities are equal; this is called the \no-slip condition." As a consequence, for large Reynolds numbers (>> 1), boundary layers form close to the solid boundary. In the boundary layer, large velocity gradients are found, and so viscous effects are important. • At the interface between two fluids, surface tension may become important. Surface tension leads to the formation of a meniscus, drops and bubbles, and the capillary rise observed in small tubes, because surface tension can resist pressure differences across the interface. 1.2 Worked Examples Example 1.1: Units, and converting between units Consider a rectangular block with dimensions 300 mm × 100 mm × 25 mm, of mass 10 kg, resting on a surface (Figure 1.1). The pressure acting over the area of contact can be found 1 2 CHAPTER 1. INTRODUCTION as follows. Solution: Since pressure is a stress, it has dimensions of force per unit area. When in position (a), the force exerted on the table is equal to the weight of the block (= mass × gravitational acceleration = 98 N), and the average pressure over the surface in contact with the table is given by 98 N=m2 = 39; 200 P a 100 × 25 × 10−6 In position (b), the force exerted on the table is still equal to 98 N, but the average pres- sure over the surface in contact with the table is reduced to 98300 × 25 × 10−6 N=m2, that is, 13; 067 P a. We can repeat this example using engineering units. Let's take a rectangular block, made of a different material with dimensions 12 in: × 4 in: × 1 in:, of mass 20 lbm (this is similar to the case shown in Figure 1.1). Find the pressure acting over the area in the BG system. Solution: The unit lbm is not part of the engineering system of units (see Table 1.1), so we first convert it to slugs, where mass in lb 20 mass in slugs = m = slug = 0:622 slug 32:2 32:2 So 20 lbm = 0:622 slug: When in position (a), the force exerted on the table is equal to the weight of the block (= mass × gravitational acceleration = 20 lbf ), and the average pressure over the surface 2 in contact with the table is 20/(4 × 1) lbf =in: , that is, 5 psi. In position (b), the force exerted on the table is still equal to 20 lbf , but now the average pressure over the surface 2 in contact with the table is 20/(12 × 1) lbf =in: , that is, 1:67 psi. Example 1.2: Hydraulic presses and hoists A hydraulic press uses the transmissibility of fluid pressure to produce large forces. A simple press consists of two connected cylinders of significantly different size, each fitted with a piston and filled with either oil or water (see Figure 1.2). The pressures produced by hydraulic devices are typically hundreds or thousands of psi, so that the weight of the fluid can usually be neglected. If there is an unobstructed passage connecting the two cylinders, then p1 ≈ p2: Since pressure = (magnitude of force)/area, F f A = ; so that F = f A a a Figure 1.1: Pressure exerted by a weight resting on a surface. All dimensions in millimeters. 1.2. WORKED EXAMPLES 3 Figure 1.2: Hydraulic press. Figure 1.3: Hydraulic drum brake system. The pressure transmission amplifies the applied force; a hydraulic press is simply a hydraulic lever. A hydraulic hoist is basically a hydraulic press that is turned around. In a typical garage hoist, compressed air is used (instead of an actuating piston) to force oil through the connecting pipe into the cylinder under a large piston, which supports the car. A lock-valve is usually placed in the connecting pipe, and when the hoist is at the right height the valve is closed, holding the pressure under the cylinder and maintaining the hoist at a constant height. A similar application of the transmissibility of pressure is used in hydraulic brake systems. Here, the force is applied by a foot pedal, increasing the pressure in a \master" cylinder, which in turn transmits the pressure to each brake or \slave" cylinder (Figure 1.3). A brake cylinder has two opposing pistons, so that when the pressure inside the cylinder rises the two pistons move in opposite directions. In a drum brake system, each brake shoe is pivoted at one end, and attached to one of the pistons of the slave cylinder at the other end. As the piston moves out, it forces the brake shoe into contact with the brake drum. Similarly, in a disk brake system, there are two brake pads, one on each side of the disk, and the brake cylinder pushes the two brake pads into contact with the disk. Example 1.3: Finding the pressure in a fluid Consider the piston and cylinder illustrated in Figure 1.4. If the piston had a mass m = 1 kg, and an area A = 0:01 m2, what is the pressure p of the gas in the cylinder? Atmospheric pressure pa acts on the outside of the container. 4 CHAPTER 1. INTRODUCTION Solution: The piston is not moving so that it is in equilibrium under the force due to its own weight, the force due to the gas pressure inside of the piston (acting up), and the force due to air pressure on the outside of the piston (acting down). The weight of the piston = mg, where g is the acceleration due to gravity (9:8 m=s2, or 32:1 ft=s2). The force due to pressure = pressure × area of piston. Therefore 2 pA − paA = mg = 1 kg × 9:8 m =s = 9:8 N That is, 9:8 N p − p = = 980 P a a 0:01 m2 where P a = pascal = N=m2. What is this excess pressure in psi (pounds per square inch)? A standard atmosphere 2 has a pressure of 14:7 lbf =in: , or 101; 325 P a (see Table 2.1). Therefore, 980 P a is equal to 14:7 × 980=101325 psi = 0:142 psi. Example 1.4: Force due to pressure In the center of a hurricane, the pressure can be very low. Find the force acting on the wall of a house, measuring 10 ft by 20 ft, when the pressure inside the house is 30:0 in of mercury, and the pressure outside is 26:3 in: of mercury. Express the answer in lbf and N. Solution: A mercury barometer measures the local atmospheric pressure. A standard 2 2 atmosphere has a pressure of 14:7 lbf =in: , or 101; 325 P a or N=m Table 2.1).
Load more

Recommended publications
  • Che 253M Experiment No. 2 COMPRESSIBLE GAS FLOW
    Rev. 8/15 AD/GW ChE 253M Experiment No. 2 COMPRESSIBLE GAS FLOW The objective of this experiment is to familiarize the student with methods for measurement of compressible gas flow and to study gas flow under subsonic and supersonic flow conditions. The experiment is divided into three distinct parts: (1) Calibration and determination of the critical pressure ratio for a critical flow nozzle under supersonic flow conditions (2) Calculation of the discharge coefficient and Reynolds number for an orifice under subsonic (non- choked) flow conditions and (3) Determination of the orifice constants and mass discharge from a pressurized tank in a dynamic bleed down experiment under (choked) flow conditions. The experimental set up consists of a 100 psig air source branched into two manifolds: the first used for parts (1) and (2) and the second for part (3). The first manifold contains a critical flow nozzle, a NIST-calibrated in-line digital mass flow meter, and an orifice meter, all connected in series with copper piping. The second manifold contains a strain-gauge pressure transducer and a stainless steel tank, which can be pressurized and subsequently bled via a number of attached orifices. A number of NIST-calibrated digital hand held manometers are also used for measuring pressure in all 3 parts of this experiment. Assorted pressure regulators, manual valves, and pressure gauges are present on both manifolds and you are expected to familiarize yourself with the process flow, and know how to operate them to carry out the experiment. A process flow diagram plus handouts outlining the theory of operation of these devices are attached.
    [Show full text]
  • Laws of Similarity in Fluid Mechanics 21
    Laws of similarity in fluid mechanics B. Weigand1 & V. Simon2 1Institut für Thermodynamik der Luft- und Raumfahrt (ITLR), Universität Stuttgart, Germany. 2Isringhausen GmbH & Co KG, Lemgo, Germany. Abstract All processes, in nature as well as in technical systems, can be described by fundamental equations—the conservation equations. These equations can be derived using conservation princi- ples and have to be solved for the situation under consideration. This can be done without explicitly investigating the dimensions of the quantities involved. However, an important consideration in all equations used in fluid mechanics and thermodynamics is dimensional homogeneity. One can use the idea of dimensional consistency in order to group variables together into dimensionless parameters which are less numerous than the original variables. This method is known as dimen- sional analysis. This paper starts with a discussion on dimensions and about the pi theorem of Buckingham. This theorem relates the number of quantities with dimensions to the number of dimensionless groups needed to describe a situation. After establishing this basic relationship between quantities with dimensions and dimensionless groups, the conservation equations for processes in fluid mechanics (Cauchy and Navier–Stokes equations, continuity equation, energy equation) are explained. By non-dimensionalizing these equations, certain dimensionless groups appear (e.g. Reynolds number, Froude number, Grashof number, Weber number, Prandtl number). The physical significance and importance of these groups are explained and the simplifications of the underlying equations for large or small dimensionless parameters are described. Finally, some examples for selected processes in nature and engineering are given to illustrate the method. 1 Introduction If we compare a small leaf with a large one, or a child with its parents, we have the feeling that a ‘similarity’ of some sort exists.
    [Show full text]
  • Aerodynamics Material - Taylor & Francis
    CopyrightAerodynamics material - Taylor & Francis ______________________________________________________________________ 257 Aerodynamics Symbol List Symbol Definition Units a speed of sound ⁄ a speed of sound at sea level ⁄ A area aspect ratio ‐‐‐‐‐‐‐‐ b wing span c chord length c Copyrightmean aerodynamic material chord- Taylor & Francis specific heat at constant pressure of air · root chord tip chord specific heat at constant volume of air · / quarter chord total drag coefficient ‐‐‐‐‐‐‐‐ , induced drag coefficient ‐‐‐‐‐‐‐‐ , parasite drag coefficient ‐‐‐‐‐‐‐‐ , wave drag coefficient ‐‐‐‐‐‐‐‐ local skin friction coefficient ‐‐‐‐‐‐‐‐ lift coefficient ‐‐‐‐‐‐‐‐ , compressible lift coefficient ‐‐‐‐‐‐‐‐ compressible moment ‐‐‐‐‐‐‐‐ , coefficient , pitching moment coefficient ‐‐‐‐‐‐‐‐ , rolling moment coefficient ‐‐‐‐‐‐‐‐ , yawing moment coefficient ‐‐‐‐‐‐‐‐ ______________________________________________________________________ 258 Aerodynamics Aerodynamics Symbol List (cont.) Symbol Definition Units pressure coefficient ‐‐‐‐‐‐‐‐ compressible pressure ‐‐‐‐‐‐‐‐ , coefficient , critical pressure coefficient ‐‐‐‐‐‐‐‐ , supersonic pressure coefficient ‐‐‐‐‐‐‐‐ D total drag induced drag Copyright material - Taylor & Francis parasite drag e span efficiency factor ‐‐‐‐‐‐‐‐ L lift pitching moment · rolling moment · yawing moment · M mach number ‐‐‐‐‐‐‐‐ critical mach number ‐‐‐‐‐‐‐‐ free stream mach number ‐‐‐‐‐‐‐‐ P static pressure ⁄ total pressure ⁄ free stream pressure ⁄ q dynamic pressure ⁄ R
    [Show full text]
  • A Semi-Hydrostatic Theory of Gravity-Dominated Compressible Flow
    Generated using version 3.2 of the official AMS LATEX template 1 A semi-hydrostatic theory of gravity-dominated compressible flow 2 Thomas Dubos ∗ IPSL-Laboratoire de Météorologie Dynamique, Ecole Polytechnique, Palaiseau, France Fabrice voitus CNRM-Groupe d’étude de l’Atmosphere Metéorologique, Météo-France, Toulouse, France ∗Corresponding author address: Thomas Dubos, LMD, École Polytechnique, 91128 Palaiseau, France. E-mail: [email protected] 1 3 Abstract 4 From Hamilton’s least action principle, compressible equations of motion with density diag- 5 nosed from potential temperature through hydrostatic balance are derived. Slaving density 6 to potential temperature suppresses the degrees of freedom supporting the propagation of 7 acoustic waves and results in a sound-proof system. The linear normal modes and dispersion 8 relationship for an isothermal state of rest on f- and β- planes are accurate from hydrostatic 9 to non-hydrostatic scales, except for deep internal gravity waves. Especially the Lamb wave 10 and long Rossby waves are not distorted, unlike with anelastic or pseudo-incompressible 11 systems. 12 Compared to similar equations derived by Arakawa and Konor (2009), the semi-hydrostatic 13 system derived here possesses an additional term in the horizontal momentum budget. This 14 term is an apparent force resulting from the vertical coordinate not being the actual height 15 of an air parcel, but its hydrostatic height, i.e. the hypothetical height it would have after 16 the atmospheric column it belongs to has reached hydrostatic balance through adiabatic 17 vertical displacements of air parcels. The Lagrange multiplier λ introduced in Hamilton’s 18 principle to slave density to potential temperature is identified as the non-hydrostatic ver- 19 tical displacement, i.e.
    [Show full text]
  • Chapter 5 Dimensional Analysis and Similarity
    Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences.
    [Show full text]
  • Introduction to Compressible Computational Fluid Dynamics James S
    Introduction to Compressible Computational Fluid Dynamics James S. Sochacki Department of Mathematics James Madison University [email protected] Abstract This document is intended as an introduction to computational fluid dynamics at the upper undergraduate level. It is assumed that the student has had courses through three dimensional calculus and some computer programming experience with numer- ical algorithms. A course in differential equations is recommended. This document is intended to be used by undergraduate instructors and students to gain an under- standing of computational fluid dynamics. The document can be used in a classroom or research environment at the undergraduate level. The idea of this work is to have the students use the modules to discover properties of the equations and then relate this to the physics of fluid dynamics. Many issues, such as rarefactions and shocks are left out of the discussion because the intent is to have the students discover these concepts and then study them with the instructor. The document is used in part of the undergraduate MATH 365 - Computation Fluid Dynamics course at James Madi- son University (JMU) and is part of the joint NSF Grant between JMU and North Carolina Central University (NCCU): A Collaborative Computational Sciences Pro- gram. This document introduces the full three-dimensional Navier Stokes equations. As- sumptions to these equations are made to derive equations that are accessible to un- dergraduates with the above prerequisites. These equations are approximated using finite difference methods. The development of the equations and finite difference methods are contained in this document. Software modules and their corresponding documentation in Fortran 90, Maple and Matlab can be downloaded from the web- site: http://www.math.jmu.edu/~jim/compressible.html.
    [Show full text]
  • Compressible Flow
    94 c 2004 Faith A. Morrison, all rights reserved. Compressible Fluids Faith A. Morrison Associate Professor of Chemical Engineering Michigan Technological University November 4, 2004 Chemical engineering is mostly concerned with incompressible flows in pipes, reactors, mixers, and other process equipment. Gases may be modeled as incompressible fluids in both microscopic and macroscopic calculations as long as the pressure changes are less than about 20% of the mean pressure (Geankoplis, Denn). The friction-factor/Reynolds-number correlation for incompressible fluids is found to apply to compressible fluids in this regime of pressure variation (Perry and Chilton, Denn). Compressible flow is important in selected application, however, including high-speed flow of gasses in pipes, through nozzles, in tur- bines, and especially in relief valves. We include here a brief discussion of issues related to the flow of compressible fluids; for more information the reader is encouraged to consult the literature. A compressible fluid is one in which the fluid density changes when it is subjected to high pressure-gradients. For gasses, changes in density are accompanied by changes in temperature, and this complicates considerably the analysis of compressible flow. The key difference between compressible and incompressible flow is the way that forces are transmitted through the fluid. Consider the flow of water in a straw. When a thirsty child applies suction to one end of a straw submerged in water, the water moves - both the water close to her mouth moves and the water at the far end moves towards the lower- pressure area created in the mouth. Likewise, in a long, completely filled piping system, if a pump is turned on at one end, the water will immediately begin to flow out of the other end of the pipe.
    [Show full text]
  • An Asymptotic-Preserving All-Speed Scheme for the Euler and Navier-Stokes Equations
    An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations Floraine Cordier1;2;3, Pierre Degond2;3, Anela Kumbaro1 1 CEA-Saclay DEN, DM2S, SFME, LETR F-91191 Gif-sur-Yvette, France. fl[email protected], [email protected] 2 Universit´ede Toulouse; UPS, INSA, UT1, UTM ; Institut de Math´ematiquesde Toulouse ; F-31062 Toulouse, France. 3 CNRS; Institut de Math´ematiquesde Toulouse UMR 5219 ; F-31062 Toulouse, France. [email protected] Abstract We present an Asymptotic-Preserving 'all-speed' scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretiza- tion, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incom- pressible regime. The divergence-free condition on the velocity in the incompress- ible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and Two-dimensional numerical results provide a validation of the Asymptotic-Preserving 'all-speed' properties. Key words: Low Mach number limit, Asymptotic-Preserving, all-speed, compressible arXiv:1108.2876v1 [math-ph] 14 Aug 2011 flows, incompressible flows, Navier-Stokes equations, Euler equations.
    [Show full text]
  • Role of the Sonic Scale in the Growth of Magnetic Field in Compressible
    Role of the sonic scale in the growth of magnetic field in compressible turbulence Itzhak Fouxon∗ and Michael Mondy Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel We study the growth of small fluctuations of magnetic field in supersonic turbulence, the small- scale dynamo. The growth is due to the fastest turbulent eddies above the resistive scale. We observe that for supersonic turbulence these eddies are effectively incompressible which creates a robust structure of the growth. The eddies are localised below the sonic scale ls defined as the scale where the typical velocity of the turbulent eddies equals the speed of sound. Thus the flow below ls is effectively incompressible and the field growth proceeds as in incompressible flow. At large Mach numbers ls is much smaller than the integral scale of turbulence so the fastest growing mode of the magnetic field belongs to small-scale turbulence. We derive this mode and the associated growth rate numerically in a white noise in time model of turbulence. The relevance of this model relies on considering evolution time larger than the correlation time of turbulence. PACS numbers: Introduction. The problem of the growth of small fluc- istic time-scale of turbulence at lres, a fact well-known in tuations of magnetic field in turbulent flows of a con- the incompressible case [2{4, 9] that carries over to com- ducting fluid is one of the most fundamental problems pressible turbulence. The local Mach number is usually of magnetohydrodynamics (MHD) [1, 2]. Here the back small at lres so that the flow compressibility, possibly reaction of the magnetic field on the flow is neglected due large at large scales, is irrelevant and the growth pro- to the field’s smallness.
    [Show full text]
  • 1 David Apsley 3. APPROXIMATIONS and SIMPLIFIED EQUATIONS
    3. APPROXIMATIONS AND SIMPLIFIED EQUATIONS SPRING 2021 3.1 Steady-state vs time-dependent flow 3.2 Two-dimensional vs three-dimensional flow 3.3 Incompressible vs compressible flow 3.4 Inviscid vs viscous flow 3.5 Hydrostatic vs non-hydrostatic flow 3.6 Boussinesq approximation for density 3.7 Depth-averaged (shallow-water) equations 3.8 Reynolds-averaged equations (turbulent flow) Examples Fluid dynamics is governed by equations for mass, momentum and energy. The momentum equation for a viscous fluid is called the Navier-Stokes equation; it is based upon: • continuum mechanics; • the momentum principle; • shear stress proportional to velocity gradient. A fluid for which the last is true is called a Newtonian fluid. This is the case for most fluids in engineering. However, there are important non-Newtonian fluids; e.g. mud, cement, blood, paint, polymer solutions. CFD is very useful for these, as their governing equations are usually impossible to solve analytically. The full equations are time-dependent, 3-dimensional, viscous, compressible, non-linear and highly coupled. However, in most cases it is possible to simplify analysis by adopting a reduced equation set. Some common approximations are listed below. Reduction of dimension: • steady-state; • two-dimensional. Neglect of some fluid property: • incompressible; • inviscid. Simplified forces: • hydrostatic; • Boussinesq approximation for density. Approximations based upon averaging: • depth-averaging (shallow-water equations); • Reynolds-averaging (turbulent flows). The consequences of these approximations are examined in the following sections. CFD 3 – 1 David Apsley 3.1 Steady-State vs Time-Dependent Flow Many flows are naturally time-dependent. Examples include waves, tides, turbines, reciprocating pumps and internal combustion engines.
    [Show full text]
  • Turbulence Modeling of Compressible Flows with Large Density Variation
    Turbulence modeling of compressible flows with large density variation by Igor A. Grigoriev April 2016 Technical Report Royal Institute of Technology Department of Mechanics SE-100 44 Stockholm, Sweden Akademisk avhandling som med tillst˚and av Kungliga Tekniska H¨ogskolan i Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen fredagen den 1 April 2016 kl 10.15 i D3, Lindstedtsv¨agen 5, Kungliga Tekniska H¨ogskolan, Stockholm. ©I.A. Grigoriev 2016 Universitetsservice US-AB, Stockholm 2016 Turbulence modeling of compressible flows with large density variation Igor A. Grigoriev Linn´eFLOW Centre, KTH Mechanics, Royal Institute of Technology SE-100 44 Stockholm, Sweden Abstract In this study we highlight the influence of mean dilatation and mean density gradient on the Reynolds stress modeling of compressible, heat-releasing and supercritical turbulent flows. Firstly, the modeling of the rapid pressure-strain correlation has been extended to self-consistently account for the influence of mean dilatation. Secondly, an algebraic model for the turbulent density flux has been developed and coupled to the tensor equation for Reynolds stress anisotropy via a 'local mean acceleration', a generalization of the buoyancy force. We applied the resulting differential Reynolds stress model (DRSM) and the corresponding explicit algebraic Reynolds stress model (EARSM) to homo- geneously sheared and compressed or expanded two-dimensional mean flows. Both formulations have shown that our model preserves the realizability of the turbulence, meaning that the Reynolds stresses do not attain unphysical val- ues, unlike earlier approaches. Comparison with rapid distortion theory (RDT) demonstrated that the DRSM captures the essentials of the transient behaviour of the diagonal anisotropies and gives good predictions of the turbulence kinetic energy.
    [Show full text]
  • Dimensional Analysis and Modeling
    cen72367_ch07.qxd 10/29/04 2:27 PM Page 269 CHAPTER DIMENSIONAL ANALYSIS 7 AND MODELING n this chapter, we first review the concepts of dimensions and units. We then review the fundamental principle of dimensional homogeneity, and OBJECTIVES Ishow how it is applied to equations in order to nondimensionalize them When you finish reading this chapter, you and to identify dimensionless groups. We discuss the concept of similarity should be able to between a model and a prototype. We also describe a powerful tool for engi- I Develop a better understanding neers and scientists called dimensional analysis, in which the combination of dimensions, units, and of dimensional variables, nondimensional variables, and dimensional con- dimensional homogeneity of equations stants into nondimensional parameters reduces the number of necessary I Understand the numerous independent parameters in a problem. We present a step-by-step method for benefits of dimensional analysis obtaining these nondimensional parameters, called the method of repeating I Know how to use the method of variables, which is based solely on the dimensions of the variables and con- repeating variables to identify stants. Finally, we apply this technique to several practical problems to illus- nondimensional parameters trate both its utility and its limitations. I Understand the concept of dynamic similarity and how to apply it to experimental modeling 269 cen72367_ch07.qxd 10/29/04 2:27 PM Page 270 270 FLUID MECHANICS Length 7–1 I DIMENSIONS AND UNITS 3.2 cm A dimension is a measure of a physical quantity (without numerical val- ues), while a unit is a way to assign a number to that dimension.
    [Show full text]