# Fluid Mechanics. Chapter 9 Surface Resistance. Dr. Amer Khalil Ababneh

Size: px
Start display at page:

Transcription

1 Fluid Mechanics Chapter 9 Surface Resistance Dr. Amer Khalil Ababneh

2 Wind tunnel used for testing flow over models.

3 Introduction Resistances exerted by surfaces are a result of viscous stresses which create resistance (or drag) to motion as a body travels through a fluid. Aeronautical engineers and naval architects are vitally interested in the drag on an airplane or the surface. The phenomena responsible for shear stress are viscosity, and velocity gradients presented in Chapter 2. In addition the concepts of the boundary layer and separation, introduced in Chapter 4, will be further expanded. Will consider surface resistance in two flow situations: 1) uniform flows 2) Non-uniform flows; i.e., boundary-layer flows - Laminar boundary-layer flows - Turbulent boundary-layer flows

4 9.1 Surface Resistance with Uniform Laminar Flow A one-dimensional laminar flow with parallel streamlines occurs for example between two plates: one is stationary and the other moving and also between two stationary parallel plates. These flows are uniform and steady. These flows illustrate the connections between velocity gradient and shear stress. Differential Equation for Uniform Laminar flow Consider the control volume shown in Fig. 9.1, which is aligned with the flow direction s. The streamlines are inclined at an angle θ wi respect to the horizontal plane. The control volume has dimensions Ds Dy unity; that is, the control volume has a unit length into the page. By application of the momentum equation, the sum of the forces acting in the s-direction is equal to the net outflow of momentum from the control volume. The flow is uniform so the outflow of momentum is equal to the inflow and the momentum equation reduces to

5 Figure 9.1 Control volume for analysis of uniform flow with parallel streamlines.

6 There are three types of forces that act on the control volume: - Pressure forces - Shear forces - Weight The pressure forces in s direction, The net force due to shear stress is, The weight along the flow direction is, Summing the above forces and dividing by DsDy lead to,

7 Substituting for shear stress t = m du/dy, leads to (9.3) where μ is constant. This equation is now applied to a flow between two plates; one is moving and the other is stationary.

8 Flow Produced by a Moving Plate (Couette Flow) Consider the flow between the two plates shown in Fig The lower plate is fixed, and the upper plate is moving with a speed U. The plates are separated by a distance L. In this problem there is no pressure gradient in the flow direction (dp/ds = 0), and the streamlines are in the horizontal direction (dz/ds = 0),so Eq. (9.3) reduces to along with the two boundary conditions: Integrating this equation twice gives Applying the boundary conditions results in Figure 9.2 Flow generated by a moving plate (Couette flow).

9 The shear stress is constant and equal to This flow is known as a Couette flow after a French scientist, M. Couette, who did pioneering work on the flow between parallel plates and rotating cylinders. It has application in the design of lubrication systems. EXAMPLE 9.1 SHEAR STRESS I, COUETTE FLOW SAE 30 lubricating oil at T = 38 C flows between two parallel plates, one fixed and the other moving at 1.0 m/s. Plates are spaced 0.3 mm apart. What is the shear stress on the plates? Solution: This is significant.

10 Non-Uniform Flows: Boundary-Layer Flows 9.2 Qualitative Description of the Boundary Layer The boundary layer is the region adjacent to a surface over which the velocity changes from the free- stream value (with respect to the object) to zero at the surface. This region, which is generally very thin, occurs because of the viscosity of the fluid. The velocity gradient at the surface is responsible for the viscous shear stress and surface resistance. The boundary-layer development for flow past a thin plate oriented parallel to the flow direction shown in Fig. 9.4a. The thickness of the boundary layer, δ, is defined as the distance from the surface to the point where the velocity is 99% of the free-stream velocity. The actual thickness of a boundary layer may be 2% 3% of the plate length, so the boundary-layer thickness shown in Fig. 9.4a is exaggerated at least by a factor of five to show details of the flow field. Fluid passes over the top and underneath the plate, so two boundary layers are depicted (one above and one below the plate). For convenience, the surface is assumed to be stationary, and the free-stream fluid is moving at a velocity U o.

11 Figure 9.4 Development of boundary layer and shear stress along a thin, flat plate. (a) Flow pattern above and below the plate. (b) Shear-stress distribution on either side of plate.

12 The development and growth of the boundary layer occurs because of the no-slip condition at the surface; that is, the fluid velocity at the surface must be zero. As the fluid particles next to the plate pass close to the leading edge of the plate, a retarding force (from the shear stress) begins to act on the particles to slow them down. As these particles progress farther downstream, they continue to be subjected to shear stress from the plate, so they continue to decelerate. In addition, these particles (because of their lower velocity) retard other particles adjacent to them but farther out from the plate. Thus the boundary layer becomes thicker, or grows, in the downstream direction. The broken line in Fig. 9.4a identifies the outer limit of the boundary layer. As the boundary layer becomes thicker, the velocity gradient at the wall becomes smaller and the local shear stress is reduced. The initial section of the boundary layer is the laminar boundary layer. In this region the flow is smooth and steady. Thickening of the laminar boundary layer continues smoothly in the downstream direction until a point is reached where the boundary layer becomes unstable. Beyond this point, the critical point, small disturbances in the flow will grow and spread, leading to turbulence. The boundary becomes fully turbulent at the transition point. The region between the critical point and the transition point is called the transition region.

13 The turbulent boundary layer is characterized by intense crossstream mixing as turbulent eddies transport high-velocity fluid from the boundary layer edge to the region close to the wall. This cross-stream mixing gives rise to a high effective viscosity, which can be three orders of magnitude higher than the actual viscosity of the fluid itself. The effective viscosity, due to turbulent mixing is not a property of the fluid but rather a property of the flow, namely, the mixing process. Because of this intense mixing, the velocity profile is much fuller than the laminar-flow velocity profile as shown in Fig. 9.4a. This situation leads to an increased velocity gradient at the surface and a larger shear stress. The shear-stress distribution along the plate is shown in Fig. 9.4b. It is easy to visualize that the shear stress must be relatively large near the leading edge of the plate where the velocity gradient is steep, and that it becomes progressively smaller as the boundary layer thickens in the downstream direction. At the point where the boundary layer becomes turbulent, the shear stress at the boundary increases because the velocity profile changes producing a steeper gradient at the surface.

14 9.3 Laminar Boundary Layer - Boundary-Layer Equations In 1904 Prandtl, 1 first stated the essence of the boundary-layer hypothesis, which is that viscous effects are concentrated in a thin layer of fluid (the boundary layer) next to solid boundaries. In 1908, Blasius, one of Prandtl's students, obtained a solution for the flow in a laminar boundary layer on a flat plate with a constant free-stream velocity. One of Blasius's key assumptions was that the shape of the nondimensional velocity distribution did not vary from section to section along the plate. That is, he assumed that a plot of the relative velocity, u/u 0, versus the relative distance from the boundary, y/δ, would be the same at each section. With this assumption and with Prandtl's equations of motion for boundary layers, Blasius obtained a numerical solution for the relative velocity distribution, shown in Fig In this plot, x is the distance from the leading edge of the plate, and Re x is the Reynolds number based on the free-stream velocity and the length along the plate (Re x = U 0 x/ν). In Fig. 9.5 the outer limit of the boundary layer (u/u 0 = 0.99) occurs at approximately,

15 Figure 9.5 Velocity distribution in laminar boundary layer.

16 Since y = δ at this point, the following relationship is derived for the boundary-layer thickness in laminar flow on a flat plate The Blasius solution also showed that which can be used to find the shear stress at the surface. The velocity gradient at the boundary becomes (at a section; x =constant)

17 Shear Stress The shear stress at the boundary is obtained from Surface Resistance Because the shear stress at the boundary, t o, varies along the plate, it is necessary to integrate this stress over the entire surface to obtain the total surface resistance, F s. For one side of the plate, (9.13)

18 EXAMPLE 9.3 LAMINAR BOUNDARY-LAYER THICKNESS AND SHEAR STRESS Crude oil at 70 F(n = 10-4 ft2/s, S = 0.86) with a free-stream velocity of 1 ft/s flows past a thin, flat plate that is 4 ft wide and 6 ft long in a direction parallel to the flow. The flow is laminar. Determine and plot the boundary-layer thickness and the shear stress distribution along the plate.

19 The results for Example 9.3 are plotted in the accompanying figure and listed in Table 9.1. Solution Reynolds-number variation with distance Boundary-layer thickness Shear-stress distribution

20 Table 9.1 RESULT δ AND τo FOR DIFFERENT VALUES OF x x = 0.1 ft x = 1.0 ft x = 2 ft x = 4 ft x = 6 ft x1/ τ0, psf δ, ft δ, in

21 To find the resistive force, substitute in Eq. (9.13) for t 0 and integrate, giving (9.14)

22 Shear-Stress Coefficients It is convenient to express the shear stress at the boundary, t 0, and the total shearing force F s in terms of π-groups involving the kinetic pressure of the free stream,. The local shear-stress coefficient, c f, is defined as Substituting the value for t 0 in the above gives c f as a function of Reynolds number based on the distance from the leading edge. The total shearing force, given by Eq. (9.13), can also be expressed as a π-group

23 where A is the plate area. This π-group is called the average shear-stress coefficient. Substituting Eq. (9.14) into the definition of C f, gives EXAMPLE 9.4 RESISTANCE CALCULATION FOR LAMINAR BOU#DARY LAYER ON A FLAT PLATE Crude oil at 70 F (ν = 10-4 ft2 /s, S = 0.86.) with a free-stream velocity of 1 ft/s flows past a thin, flat plate that is 4 ft wide and 6 ft long in a direction parallel to the flow. The flow is laminar. Determine the resistance on one side of the plate.

24 Solution Reynolds number. Value for C f Total shear force.

25 9.4 Boundary Layer Transition Transition is the zone where the laminar boundary layer changes into a turbulent boundary layer as shown in Fig. 9.4a. As the laminar boundary layer continues to grow, the viscous stresses are less capable of damping disturbances in the flow. A point is then reached where disturbances occurring in the flow are amplified, leading to turbulence. The critical point occurs at a Reynolds number of about 10 5 (Re cr 10 5 ) based on the distance from the leading edge. Vortices created near the wall grow and mutually interact, ultimately leading to a fully turbulent boundary layer at the transition point, which nominally occurs at a Reynolds number of (Re tr ). For purposes of simplicity in this text, it will be assumed that the boundary layer changes from laminar to turbulent flow at a Reynolds number 500,000. Transition to a turbulent boundary layer can be influenced by several other flow conditions, such as free-stream turbulence, pressure gradient, wall roughness, wall heating, and wall cooling. With appropriate roughness elements at the leading edge, the boundary layer can become turbulent at the very beginning of the plate. In this case it is said that the boundary layer is tripped at the leading edge.

26 9.5 Turbulent Boundary Layer In the majority of practical problems the boundary layer is turbulent which is primarily responsible for surface shear force, or surface resistance. Velocity Distribution The velocity distribution in the turbulent boundary layer is more complicated than the laminar boundary layer. The turbulent boundary has three zones of flow that require different equations for the velocity distribution in each zone, as opposed to the single relationship of the laminar boundary layer. Figure 9.6 shows a portion of a turbulent boundary layer in which the three different zones of flow are identified. The zone adjacent to the wall is the viscous sublayer; the zone immediately above the viscous sublayer is the logarithmic region; and, finally, beyond that region is the velocity defect region. Each of these velocity zones will be discussed separately. Figure 9.6 Sketch of zones in turbulent boundary layer. Viscous Sublayer The zone

27 The three zones in a turbulent boundary layer. Figure 9.6 Sketch of zones in turbulent boundary layer.

28 Boundary-Layer Thickness and Shear-Stress Correlations Unlike the laminar boundary layer, there is no analytically derived equation for the thickness of the turbulent boundary layer. It can be shown that the thickness of the turbulent boundary layer is where x is the distance from the leading edge of the plate and Rex is U 0 x/ν. Many empirical expressions have been proposed for the local shear-stress distribution for the turbulent boundary layer on a flat plate. One of the simplest correlations is

29 The corresponding average shear-stress coefficient is The variation of C f with Reynolds number is shown by the solid line in Fig This curve corresponds to a boundary layer that begins as a laminar boundary layer and then changes to a turbulent boundary layer after the transition Reynolds number. This is the normal condition for a flat-plate boundary layer. Table 9.3 summarizes the equations for boundary-layer-thickness, and for local shear-stress and average shear-stress coefficients for the boundary layer on a flat plate. The boundary layer, however, usually consists of laminar and turbulent. To account for this combined effect, it can be shown that the average shear-stress coefficient for this case is,

30 Figure 9.12 Average shear-stress coefficients.

31 Table 9.3 SUMMARY OF EQUATIONS FOR BOU"DARY LAYER ON A FLAT PLATE Laminar Flow Rex, Turbulent Flow Rex, ReL < ReL Boundary-Layer Thickness, δ Local Shear-Stress Coefficient, c f Average Shear-Stress Coefficient, C f

32 If the boundary layer is tripped by some roughness or leadingedge disturbance (such as a wire across the leading edge), the boundary layer is turbulent from the leading edge. This is shown by the dashed line in Fig For this condition the boundary layer thickness, local shear-stress coefficient, and average shear-stress coefficient are fit reasonably well by the following equations, which are valid up to a Reynolds number of 10 7.

33 EXAMPLE 9.6 LAMINAR/TURBULENT BOUNDARY LAYER ON FLAT PLATE Assume that air 20 C and normal atmospheric pressure flows over a smooth, flat plate with a velocity of 30 m/s. The initial boundary layer is laminar and then becomes turbulent at a transitional Reynolds number of The plate is 3 m long and 1 m wide. What will be the average resistance coefficient C f for the plate? Also, what is the total shearing resistance of one side of the plate, and what will be the resistance due to the turbulent part and the laminar part of the boundary layer? Air properties at 20 C:

34 Solution Reynolds number based on plate length Average shear-stress coefficient Total shear force Transition point

35 Laminar average shear-stress coefficient Laminar shear force Turbulent shear force

36 EXAMPLE 9.7 RESISTANCE FORCE WITH TRIPPED BOUNDARY LAYER Air at 20 C flows past a smooth, thin plate with a free-stream velocity of 20 m/s. Plate is 3 m wide and 6 m long in the direction of flow and boundary layer is tripped at the leading edge. Solution Reynolds number Reynolds number is less than Average shear-stress coefficient

37 Resistance force

### DAY 19: Boundary Layer

DAY 19: Boundary Layer flat plate : let us neglect the shape of the leading edge for now flat plate boundary layer: in blue we highlight the region of the flow where velocity is influenced by the presence

### M E 320 Professor John M. Cimbala Lecture 38

M E 320 Professor John M. Cimbala Lecture 38 Today, we will: Discuss displacement thickness in a laminar boundary layer Discuss the turbulent boundary layer on a flat plate, and compare with laminar flow

### Turbulence Laboratory

Objective: CE 319F Elementary Mechanics of Fluids Department of Civil, Architectural and Environmental Engineering The University of Texas at Austin Turbulence Laboratory The objective of this laboratory

### Principles of Convection

Principles of Convection Point Conduction & convection are similar both require the presence of a material medium. But convection requires the presence of fluid motion. Heat transfer through the: Solid

UNIT IV BOUNDARY LAYER AND FLOW THROUGH PIPES Definition of boundary layer Thickness and classification Displacement and momentum thickness Development of laminar and turbulent flows in circular pipes

### UNIT II CONVECTION HEAT TRANSFER

UNIT II CONVECTION HEAT TRANSFER Convection is the mode of heat transfer between a surface and a fluid moving over it. The energy transfer in convection is predominately due to the bulk motion of the fluid

### Basic Fluid Mechanics

Basic Fluid Mechanics Chapter 6A: Internal Incompressible Viscous Flow 4/16/2018 C6A: Internal Incompressible Viscous Flow 1 6.1 Introduction For the present chapter we will limit our study to incompressible

### BOUNDARY LAYER FLOWS HINCHEY

BOUNDARY LAYER FLOWS HINCHEY BOUNDARY LAYER PHENOMENA When a body moves through a viscous fluid, the fluid at its surface moves with it. It does not slip over the surface. When a body moves at high speed,

### Chapter 3 Lecture 8. Drag polar 3. Topics. Chapter-3

Chapter 3 ecture 8 Drag polar 3 Topics 3.2.7 Boundary layer separation, adverse pressure gradient and favourable pressure gradient 3.2.8 Boundary layer transition 3.2.9 Turbulent boundary layer over a

### External Flow and Boundary Layer Concepts

1 2 Lecture (8) on Fayoum University External Flow and Boundary Layer Concepts By Dr. Emad M. Saad Mechanical Engineering Dept. Faculty of Engineering Fayoum University Faculty of Engineering Mechanical

### Problem 4.3. Problem 4.4

Problem 4.3 Problem 4.4 Problem 4.5 Problem 4.6 Problem 4.7 This is forced convection flow over a streamlined body. Viscous (velocity) boundary layer approximations can be made if the Reynolds number Re

### ENGR Heat Transfer II

ENGR 7901 - Heat Transfer II External Flows 1 Introduction In this chapter we will consider several fundamental flows, namely: the flat plate, the cylinder, the sphere, several other body shapes, and banks

### Department of Mechanical Engineering

Department of Mechanical Engineering AMEE401 / AUTO400 Aerodynamics Instructor: Marios M. Fyrillas Email: eng.fm@fit.ac.cy HOMEWORK ASSIGNMENT #2 QUESTION 1 Clearly there are two mechanisms responsible

### 7.6 Example von Kármán s Laminar Boundary Layer Problem

CEE 3310 External Flows (Boundary Layers & Drag, Nov. 11, 2016 157 7.5 Review Non-Circular Pipes Laminar: f = 64/Re DH ± 40% Turbulent: f(re DH, ɛ/d H ) Moody chart for f ± 15% Bernoulli-Based Flow Metering

### Turbulence - Theory and Modelling GROUP-STUDIES:

Lund Institute of Technology Department of Energy Sciences Division of Fluid Mechanics Robert Szasz, tel 046-0480 Johan Revstedt, tel 046-43 0 Turbulence - Theory and Modelling GROUP-STUDIES: Turbulence

### Unit operations of chemical engineering

1 Unit operations of chemical engineering Fourth year Chemical Engineering Department College of Engineering AL-Qadesyia University Lecturer: 2 3 Syllabus 1) Boundary layer theory 2) Transfer of heat,

### Turbulence Modeling I!

Outline! Turbulence Modeling I! Grétar Tryggvason! Spring 2010! Why turbulence modeling! Reynolds Averaged Numerical Simulations! Zero and One equation models! Two equations models! Model predictions!

### FLUID MECHANICS. Chapter 9 Flow over Immersed Bodies

FLUID MECHANICS Chapter 9 Flow over Immersed Bodies CHAP 9. FLOW OVER IMMERSED BODIES CONTENTS 9.1 General External Flow Characteristics 9.3 Drag 9.4 Lift 9.1 General External Flow Characteristics 9.1.1

### Convection. forced convection when the flow is caused by external means, such as by a fan, a pump, or atmospheric winds.

Convection The convection heat transfer mode is comprised of two mechanisms. In addition to energy transfer due to random molecular motion (diffusion), energy is also transferred by the bulk, or macroscopic,

### Introduction to Turbulence AEEM Why study turbulent flows?

Introduction to Turbulence AEEM 7063-003 Dr. Peter J. Disimile UC-FEST Department of Aerospace Engineering Peter.disimile@uc.edu Intro to Turbulence: C1A Why 1 Most flows encountered in engineering and

### Given the water behaves as shown above, which direction will the cylinder rotate?

water stream fixed but free to rotate Given the water behaves as shown above, which direction will the cylinder rotate? ) Clockwise 2) Counter-clockwise 3) Not enough information F y U 0 U F x V=0 V=0

### PHYSICAL MECHANISM OF CONVECTION

Tue 8:54:24 AM Slide Nr. 0 of 33 Slides PHYSICAL MECHANISM OF CONVECTION Heat transfer through a fluid is by convection in the presence of bulk fluid motion and by conduction in the absence of it. Chapter

### Fluid: Air and water are fluids that exert forces on the human body.

Fluid: Air and water are fluids that exert forces on the human body. term fluid is often used interchangeably with the term liquid, from a mechanical perspective, Fluid: substance that flows when subjected

### Physical Properties of Fluids

Physical Properties of Fluids Viscosity: Resistance to relative motion between adjacent layers of fluid. Dynamic Viscosity:generally represented as µ. A flat plate moved slowly with a velocity V parallel

### Transport processes. 7. Semester Chemical Engineering Civil Engineering

Transport processes 7. Semester Chemical Engineering Civil Engineering 1. Elementary Fluid Dynamics 2. Fluid Kinematics 3. Finite Control Volume Analysis 4. Differential Analysis of Fluid Flow 5. Viscous

### Empirical Co - Relations approach for solving problems of convection 10:06:43

Empirical Co - Relations approach for solving problems of convection 10:06:43 10:06:44 Empirical Corelations for Free Convection Use T f or T b for getting various properties like Re = VL c / ν β = thermal

### Chapter 5 Phenomena of laminar-turbulent boundary layer transition (including free shear layers)

Chapter 5 Phenomena of laminar-turbulent boundary layer transition (including free shear layers) T-S Leu May. 3, 2018 Chapter 5: Phenomena of laminar-turbulent boundary layer transition (including free

### Fluid Mechanics Prof. T.I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture - 17 Laminar and Turbulent flows

Fluid Mechanics Prof. T.I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture - 17 Laminar and Turbulent flows Welcome back to the video course on fluid mechanics. In

### In Chapter 6 we considered the general and theoretical aspects of forced

cen58933_ch07.qxd 9/4/2002 12:12 PM Page 367 EXTERNAL FORCED CONVECTION CHAPTER 7 In Chapter 6 we considered the general and theoretical aspects of forced convection, with emphasis on differential formulation

### Visualization of flow pattern over or around immersed objects in open channel flow.

EXPERIMENT SEVEN: FLOW VISUALIZATION AND ANALYSIS I OBJECTIVE OF THE EXPERIMENT: Visualization of flow pattern over or around immersed objects in open channel flow. II THEORY AND EQUATION: Open channel:

### Figure 3: Problem 7. (a) 0.9 m (b) 1.8 m (c) 2.7 m (d) 3.6 m

1. For the manometer shown in figure 1, if the absolute pressure at point A is 1.013 10 5 Pa, the absolute pressure at point B is (ρ water =10 3 kg/m 3, ρ Hg =13.56 10 3 kg/m 3, ρ oil = 800kg/m 3 ): (a)

### Masters in Mechanical Engineering. Problems of incompressible viscous flow. 2µ dx y(y h)+ U h y 0 < y < h,

Masters in Mechanical Engineering Problems of incompressible viscous flow 1. Consider the laminar Couette flow between two infinite flat plates (lower plate (y = 0) with no velocity and top plate (y =

### Day 24: Flow around objects

Day 24: Flow around objects case 1) fluid flowing around a fixed object (e.g. bridge pier) case 2) object travelling within a fluid (cars, ships planes) two forces are exerted between the fluid and the

### Boundary-Layer Theory

Hermann Schlichting Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22

### Introduction to Marine Hydrodynamics

1896 1920 1987 2006 Introduction to Marine Hydrodynamics (NA235) Department of Naval Architecture and Ocean Engineering School of Naval Architecture, Ocean & Civil Engineering First Assignment The first

### 4.2 Concepts of the Boundary Layer Theory

Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell 4.2 Concepts of the Boundary Layer Theory It is difficult to solve the complete viscous flow fluid around a body unless the geometry is very

### Turbulence is a ubiquitous phenomenon in environmental fluid mechanics that dramatically affects flow structure and mixing.

Turbulence is a ubiquitous phenomenon in environmental fluid mechanics that dramatically affects flow structure and mixing. Thus, it is very important to form both a conceptual understanding and a quantitative

### Contents. Microfluidics - Jens Ducrée Physics: Laminar and Turbulent Flow 1

Contents 1. Introduction 2. Fluids 3. Physics of Microfluidic Systems 4. Microfabrication Technologies 5. Flow Control 6. Micropumps 7. Sensors 8. Ink-Jet Technology 9. Liquid Handling 10.Microarrays 11.Microreactors

### 2.3 The Turbulent Flat Plate Boundary Layer

Canonical Turbulent Flows 19 2.3 The Turbulent Flat Plate Boundary Layer The turbulent flat plate boundary layer (BL) is a particular case of the general class of flows known as boundary layer flows. The

### O. A Survey of Critical Experiments

O. A Survey of Critical Experiments 1 (A) Visualizations of Turbulent Flow Figure 1: Van Dyke, Album of Fluid Motion #152. Generation of turbulence by a grid. Smoke wires show a uniform laminar stream

Table of Contents Foreword.... xiii Preface... xv Chapter 1. Fundamental Equations, Dimensionless Numbers... 1 1.1. Fundamental equations... 1 1.1.1. Local equations... 1 1.1.2. Integral conservation equations...

### An-Najah National University Civil Engineering Department. Fluid Mechanics. Chapter 1. General Introduction

1 An-Najah National University Civil Engineering Department Fluid Mechanics Chapter 1 General Introduction 2 What is Fluid Mechanics? Mechanics deals with the behavior of both stationary and moving bodies

### PART 1B EXPERIMENTAL ENGINEERING. SUBJECT: FLUID MECHANICS & HEAT TRANSFER LOCATION: HYDRAULICS LAB (Gnd Floor Inglis Bldg) BOUNDARY LAYERS AND DRAG

1 PART 1B EXPERIMENTAL ENGINEERING SUBJECT: FLUID MECHANICS & HEAT TRANSFER LOCATION: HYDRAULICS LAB (Gnd Floor Inglis Bldg) EXPERIMENT T3 (LONG) BOUNDARY LAYERS AND DRAG OBJECTIVES a) To measure the velocity

### 7. Basics of Turbulent Flow Figure 1.

1 7. Basics of Turbulent Flow Whether a flow is laminar or turbulent depends of the relative importance of fluid friction (viscosity) and flow inertia. The ratio of inertial to viscous forces is the Reynolds

### Convection Heat Transfer. Introduction

Convection Heat Transfer Reading Problems 12-1 12-8 12-40, 12-49, 12-68, 12-70, 12-87, 12-98 13-1 13-6 13-39, 13-47, 13-59 14-1 14-4 14-18, 14-24, 14-45, 14-82 Introduction Newton s Law of Cooling Controlling

### Laminar Flow. Chapter ZERO PRESSURE GRADIENT

Chapter 2 Laminar Flow 2.1 ZERO PRESSRE GRADIENT Problem 2.1.1 Consider a uniform flow of velocity over a flat plate of length L of a fluid of kinematic viscosity ν. Assume that the fluid is incompressible

### Iran University of Science & Technology School of Mechanical Engineering Advance Fluid Mechanics

1. Consider a sphere of radius R immersed in a uniform stream U0, as shown in 3 R Fig.1. The fluid velocity along streamline AB is given by V ui U i x 1. 0 3 Find (a) the position of maximum fluid acceleration

### CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW

CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW 4.1 Introduction Boundary layer concept (Prandtl 1904): Eliminate selected terms in the governing equations Two key questions (1) What are the

### What we know about Fluid Mechanics. What we know about Fluid Mechanics

What we know about Fluid Mechanics 1. Survey says. 3. Image from: www.axs.com 4. 5. 6. 1 What we know about Fluid Mechanics 1. MEB (single input, single output, steady, incompressible, no rxn, no phase

### Chapter 3 NATURAL CONVECTION

Fundamentals of Thermal-Fluid Sciences, 3rd Edition Yunus A. Cengel, Robert H. Turner, John M. Cimbala McGraw-Hill, 2008 Chapter 3 NATURAL CONVECTION Mehmet Kanoglu Copyright The McGraw-Hill Companies,

### Chemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017

Chemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017 Objective: Text: To introduce the basic concepts of fluid mechanics and heat transfer necessary for solution of engineering

### Chapter 8: Flow in Pipes

Objectives 1. Have a deeper understanding of laminar and turbulent flow in pipes and the analysis of fully developed flow 2. Calculate the major and minor losses associated with pipe flow in piping networks

### MYcsvtu Notes HEAT TRANSFER BY CONVECTION

www.mycsvtunotes.in HEAT TRANSFER BY CONVECTION CONDUCTION Mechanism of heat transfer through a solid or fluid in the absence any fluid motion. CONVECTION Mechanism of heat transfer through a fluid in

### UNIT 4 FORCES ON IMMERSED BODIES. Lecture-01

1 UNIT 4 FORCES ON IMMERSED BODIES Lecture-01 Forces on immersed bodies When a body is immersed in a real fluid, which is flowing at a uniform velocity U, the fluid will exert a force on the body. The

### 6.2 Governing Equations for Natural Convection

6. Governing Equations for Natural Convection 6..1 Generalized Governing Equations The governing equations for natural convection are special cases of the generalized governing equations that were discussed

### So far, we have considered conduction, which is the mechanism of heat

cen58933_ch06.qxd 9/4/2002 12:05 PM Page 333 FUNDAMENTALS OF CONVECTION CHAPTER 6 So far, we have considered conduction, which is the mechanism of heat transfer through a solid or a quiescent fluid. We

### 1. Introduction, tensors, kinematics

1. Introduction, tensors, kinematics Content: Introduction to fluids, Cartesian tensors, vector algebra using tensor notation, operators in tensor form, Eulerian and Lagrangian description of scalar and

### HEAT TRANSFER BY CONVECTION. Dr. Şaziye Balku 1

HEAT TRANSFER BY CONVECTION Dr. Şaziye Balku 1 CONDUCTION Mechanism of heat transfer through a solid or fluid in the absence any fluid motion. CONVECTION Mechanism of heat transfer through a fluid in the

### External Forced Convection. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

External Forced Convection Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Drag and Heat Transfer in External flow Fluid flow over solid bodies is responsible

### Chapter 6 An introduction of turbulent boundary layer

Chapter 6 An introduction of turbulent boundary layer T-S Leu May. 23, 2018 Chapter 6: An introduction of turbulent boundary layer Reading assignments: 1. White, F. M., Viscous fluid flow. McGraw-Hill,

### Momentum (Newton s 2nd Law of Motion)

Dr. Nikos J. Mourtos AE 160 / ME 111 Momentum (Newton s nd Law of Motion) Case 3 Airfoil Drag A very important application of Momentum in aerodynamics and hydrodynamics is the calculation of the drag of

### BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW

Proceedings of,, BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW Yunho Jang Department of Mechanical and Industrial Engineering University of Massachusetts Amherst, MA 01002 Email:

### External Flows. Dye streak. turbulent. laminar transition

Eternal Flos An internal flo is surrounded by solid boundaries that can restrict the development of its boundary layer, for eample, a pipe flo. An eternal flo, on the other hand, are flos over bodies immersed

### τ du In his lecture we shall look at how the forces due to momentum changes on the fluid and viscous forces compare and what changes take place.

4. Real fluids The flow of real fluids exhibits viscous effect, that is they tend to stick to solid surfaces and have stresses within their body. You might remember from earlier in the course Newtons law

### Chapter 8: Flow in Pipes

8-1 Introduction 8-2 Laminar and Turbulent Flows 8-3 The Entrance Region 8-4 Laminar Flow in Pipes 8-5 Turbulent Flow in Pipes 8-6 Fully Developed Pipe Flow 8-7 Minor Losses 8-8 Piping Networks and Pump

### CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer

CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer You are assigned to design a fallingcylinder viscometer to measure the viscosity of Newtonian liquids. A schematic

### Turbulent Boundary Layers & Turbulence Models. Lecture 09

Turbulent Boundary Layers & Turbulence Models Lecture 09 The turbulent boundary layer In turbulent flow, the boundary layer is defined as the thin region on the surface of a body in which viscous effects

### Outlines. simple relations of fluid dynamics Boundary layer analysis. Important for basic understanding of convection heat transfer

Forced Convection Outlines To examine the methods of calculating convection heat transfer (particularly, the ways of predicting the value of convection heat transfer coefficient, h) Convection heat transfer

### CONVECTIVE HEAT TRANSFER

CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 3 LAMINAR BOUNDARY LAYER FLOW LAMINAR BOUNDARY LAYER FLOW Boundary

### Watershed Sciences 6900 FLUVIAL HYDRAULICS & ECOHYDRAULICS

Watershed Sciences 6900 FLUVIAL HYDRAULICS & ECOHYDRAULICS WEEK Four Lecture 6 VELOCITY DISTRIBUTION Joe Wheaton FOR TODAY, YOU SHOULD HAVE READ 1 LET S GET ON WITH IT TODAY S PLAN VELOCITY DISTRIBUTIONS

### Stream line, turbulent flow and Viscosity of liquids - Poiseuille s Method

Stream line, turbulent flow and Viscosity of liquids - Poiseuille s Method Dr D. Arun Kumar Assistant Professor Department of Physical Sciences Bannari Amman Institute of Technology Sathyamangalam General

### Chapter 10. Solids and Fluids

Chapter 10 Solids and Fluids Surface Tension Net force on molecule A is zero Pulled equally in all directions Net force on B is not zero No molecules above to act on it Pulled toward the center of the

### CHAPTER 6 Fluids Engineering. SKMM1922 Introduction of Mechanical Engineering

CHAPTER 6 Fluids Engineering SKMM1922 Introduction of Mechanical Engineering Chapter Objectives Recognize the application of fluids engineering to such diverse fields as microfluidics, aerodynamics, sports

### ROAD MAP... D-1: Aerodynamics of 3-D Wings D-2: Boundary Layer and Viscous Effects D-3: XFLR (Aerodynamics Analysis Tool)

AE301 Aerodynamics I UNIT D: Applied Aerodynamics ROAD MAP... D-1: Aerodynamics o 3-D Wings D-2: Boundary Layer and Viscous Eects D-3: XFLR (Aerodynamics Analysis Tool) AE301 Aerodynamics I : List o Subjects

### Tutorial 10. Boundary layer theory

Tutorial 10 Boundary layer theory 1. If the velocity distribution law in a laminar boundary layer over a flat plate is assumes to be of the form, determine the velocity distribution law. At y = 0, u= 0

### FLUID FORCES ON NON-STREAMLINE BODIES BACKGROUND NOTES AND DESCRIPTION OF THE FLOW PHENOMENA

FLUID FORCES ON NON-STREAMLINE BODIES BACKGROUND NOTES AND DESCRIPTION OF THE FLOW PHENOMENA 1. INTRODUCTION The purpose of this Item is to provide background information which will assist the user of

### Momentum Equation-Derivation-1 Consider the 2D continuity equation and integrate over the height of the boundary layer

Module 2 : Convection Lecture 25 : Integral Boundary Layer Equations Objectives In this class: Integral equations for the momentum and thermal boundary layers are obtained Solution of these equations for

### Topics in Fluid Dynamics: Classical physics and recent mathematics

Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:

### MOMENTUM TRANSPORT Velocity Distributions in Turbulent Flow

TRANSPORT PHENOMENA MOMENTUM TRANSPORT Velocity Distributions in Turbulent Flow Introduction to Turbulent Flow 1. Comparisons of laminar and turbulent flows 2. Time-smoothed equations of change for incompressible

### Fundamental Concepts of Convection : Flow and Thermal Considerations. Chapter Six and Appendix D Sections 6.1 through 6.8 and D.1 through D.

Fundamental Concepts of Convection : Flow and Thermal Considerations Chapter Six and Appendix D Sections 6.1 through 6.8 and D.1 through D.3 6.1 Boundary Layers: Physical Features Velocity Boundary Layer

### Fluid Mechanics II Viscosity and shear stresses

Fluid Mechanics II Viscosity and shear stresses Shear stresses in a Newtonian fluid A fluid at rest can not resist shearing forces. Under the action of such forces it deforms continuously, however small

### 6. Laminar and turbulent boundary layers

6. Laminar and turbulent boundary layers John Richard Thome 8 avril 2008 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 1 / 34 6.1 Some introductory ideas Figure 6.1 A boundary

### ME 431A/538A/538B Homework 22 October 2018 Advanced Fluid Mechanics

ME 431A/538A/538B Homework 22 October 2018 Advanced Fluid Mechanics For Friday, October 26 th Start reading the handout entitled Notes on finite-volume methods. Review Chapter 7 on Dimensional Analysis

### Lecture-4. Flow Past Immersed Bodies

Lecture-4 Flow Past Immersed Bodies Learning objectives After completing this lecture, you should be able to: Identify and discuss the features of external flow Explain the fundamental characteristics

### PEMP ACD2505. M.S. Ramaiah School of Advanced Studies, Bengaluru

Two-Dimensional Potential Flow Session delivered by: Prof. M. D. Deshpande 1 Session Objectives -- At the end of this session the delegate would have understood PEMP The potential theory and its application

### FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER

FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY OF AGRICULTURE DEPARTMENT OF AGRICULTURAL MACHINERY AND TECHNOLOGIES ENGINEERING 1 5. FLOW IN PIPES 5.1.3. Pressure and Shear Stress

### Boundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity

Boundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity Tobias Knopp D 23. November 28 Reynolds averaged Navier-Stokes equations Consider the RANS equations with

### Lecture 6 Friction. Friction Phenomena Types of Friction

Lecture 6 Friction Tangential forces generated between contacting surfaces are called friction forces and occur to some degree in the interaction between all real surfaces. whenever a tendency exists for

### ME 144: Heat Transfer Introduction to Convection. J. M. Meyers

ME 144: Heat Transfer Introduction to Convection Introductory Remarks Convection heat transfer differs from diffusion heat transfer in that a bulk fluid motion is present which augments the overall heat

### CEE 3310 External Flows (Boundary Layers & Drag), /2 f = 0.664

CEE 3310 External Flows (Boundary Layers & Drag), 2010 161 7.12 Review Boundary layer equations u u x + v u y = u ν 2 y 2 Blasius Solution δ x = 5.0 and c Rex 1/2 f = 0.664 Re 1/2 x within 10% of Von Kármán

CE 6303 MECHANICS OF FLUIDS L T P C QUESTION BANK 3 0 0 3 UNIT I FLUID PROPERTIES AND FLUID STATICS PART - A 1. Define fluid and fluid mechanics. 2. Define real and ideal fluids. 3. Define mass density

### Universal Viscosity Curve Theory

TM Universal Viscosity Curve Theory Turbine Flow Meters and Flow Viscosity Introduction Like any transducer, a turbine flow meter is sensitive to physical parameters other than the one which is of interest.

### Friction Factors and Drag Coefficients

Levicky 1 Friction Factors and Drag Coefficients Several equations that we have seen have included terms to represent dissipation of energy due to the viscous nature of fluid flow. For example, in the

### SENTHIL SELIYAN ELANGO ID: UB3016SC17508 AIU HYDRAULICS (FLUID DYNAMICS)

SENTHIL SELIYAN ELANGO ID: UB3016SC17508 AIU HYDRAULICS (FLUID DYNAMICS) ATLANTIC INTERNATIONAL UNIVERSITY INTRODUCTION Real fluids The flow of real fluids exhibits viscous effect, which are they tend

### CONVECTIVE HEAT TRANSFER

CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 4 HEAT TRANSFER IN CHANNEL FLOW BASIC CONCEPTS BASIC CONCEPTS Laminar

### Chapter 6. Losses due to Fluid Friction

Chapter 6 Losses due to Fluid Friction 1 Objectives To measure the pressure drop in the straight section of smooth, rough, and packed pipes as a function of flow rate. To correlate this in terms of the

### CEE 3310 External Flows (Boundary Layers & Drag, Nov. 14, Re 0.5 x x 1/2. Re 1/2

CEE 3310 External Flows (Boundary Layers & Drag, Nov. 14, 2016 159 7.10 Review Momentum integral equation τ w = ρu 2 dθ dx Von Kármán assumed and found δ x = 5.5 Rex 0.5 u(x, y) U = 2y δ y2 δ 2 δ = 5.5

### Introduction to Heat and Mass Transfer. Week 12

Introduction to Heat and Mass Transfer Week 12 Next Topic Convective Heat Transfer» Heat and Mass Transfer Analogy» Evaporative Cooling» Types of Flows Heat and Mass Transfer Analogy Equations governing