ENV5056 Numerical Modeling of Flow and Contaminant Transport in Rivers. Equations. Asst. Prof. Dr. Orhan GÜNDÜZ
|
|
- Gwendolyn Charles
- 8 years ago
- Views:
Transcription
1 ENV5056 Numerical Modeling of Flow and Contaminant Transport in Rivers Derivation of Flow Equations Asst. Prof. Dr. Orhan GÜNDÜZ
2 General 3-D equations of incompressible fluid flow Navier-Stokes Equations General 1-D equations of channel flow (can be simplified from N-S eqns or derived separately) Saint-Venant Equations Further simplifications possible based on dimensionality, time dependency and uniformity of flow 2
3 Three-dimensional Hydrodynamic Equations of Flow The three-dimensional hydrodynamic equations of fluid flow are the basic differential equations describing the flow of a Newtonian fluid. In a three-dimensional cartesiancoordinate system, the conservation of mass equation coupled with the Navier- Stokes equations of motion in x, y and z dimensions form the general hydrodynamic equations. They define a wide range of flow phenomena from unsteady, compressible flows to steady, incompressible flows. 3
4 Three-dimensional Hydrodynamic Equations of Flow Developed by Claude-Louis Navierand George Gabriel Stokes, these equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term. Although Navier-Stokes equations only refer to the equations of motion (conservation of momentum), it is commonly accepted to include the equation of conservation of mass. These four equations all together fully describe the fundamental characteristics of fluid motion. 4
5 Some fundamental concepts Velocity vector: v = ui + υ j + wk k j i Unit vectors Dot product of two vectors: [ 1, 2,..., n ] [,,..., ] a = a a a b = b1 b2 bn n a b = a b = a b + a b a b i= 1 i i n n Del operator: = i + j + k x y z 5
6 Some fundamental concepts Gradient of a scalar field, f: f f f f = i + j + k x y z Divergence of a vector field, v: v = ui + υ j + wk div v u υ w = v = + + x y z In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. (Ex: i.i=1x1xcos0=1, i.j=1x1xcos90=0, i.k=1x1xcos90=0) 6
7 Navier-Stokes Equations of Fluid Flow ρ + = t ( ρv) 0 Continuity Equation ρ v + v v = σ + t f Momentum Equation where ρ is fluid density, v is flow velocity vector, σ is stress tensor f is body forces and is del operator. 7
8 Navier-Stokes Equations of Fluid Flow σ τ τ σ τ σ τ τ τ σ xx xy xz = yx yy yz zx zy zz where σ are the normal stresses and τ are shear stresses acting on the fluid 8
9 Navier-Stokes Equations of Fluid Flow Conservation of mass equation: ρ ( ρu) ( ρυ) ( ρw) = t x y z 0 where ρis the density of the fluid, tis time and u, υand ware the components of the velocity vector in x, yand zcoordinates. 9
10 Navier-Stokes Equations of Fluid Flow u u u u σ τ τ t x y z x y z xx yx zx ρ + u + υ + ω = ρ υ υ υ υ τ σ τ u g y t x y z x y z xy yy zy ρ + u + υ + ω = ρ w w w w τ τ σ t x y z x y z xz yz zz ρ + u + υ + ω = ρ g x, g y and g z are gravitational acceleration along x, y and z axes. The terms σ and τare normal and shear stresses acting on the fluid, respectively. The first subscript in the notation indicates the direction of the normal to the plane on which the stress acts, and the second subscript indicates the direction of the stress. g x g z 10
11 Navier-Stokes Equations of Fluid Flow The general equations of motioninclude both velocities and stresses as unknown variables. Using the relationships derived for a compressible Newtonian fluid, one can express the normal and shear stress components in these equations in terms of the velocities: For Newtonian fluids: τ ~ u y Three assumptions needed to define stress terms: 1. The stress tensor is a linear function of the strain rates. 2. The fluid is isotropic. 11
12 Navier-Stokes Equations of Fluid Flow u 2 σ xx = P + 2µ µ v x 3 υ 2 σ yy = P + 2µ µ v y 3 u υ τ xy = τ yx = µ + y x υ w τ yz = τ zy = µ + z y w 2 σ zz = P + 2µ µ v z 3 τ τ µ w u zx = xz = + x z 12
13 Navier-Stokes Equations of Fluid Flow Substituted and simplified: u u u u P u u u 1 u υ w ρ u υ w ρ gx µ µ = t x y z x x y z 3 x x y z υ P 1 u w u υ υ w υ g υ υ υ ρ υ ρ y µ µ υ = t x y z y x y z 3 y x y z w w w w P w w w 1 u υ w ρ u υ w ρ gz µ µ = t x y z z x y z 3 z x y z 13
14 Incompressible Newtonian Fluid Most fluids of practical importance such as water are incompressible, which means that their densities are constant for a wide range of flows. This is a reasonable assumption except for certain extreme situations such as the cases where the fluid is under profound pressures. Since the density of fluid is constant, the continuity equation for incompressible flows can be simplified as: u υ w + + = x y z 0 14
15 Navier-Stokes Equations of Fluid Flow (Incompressible Flow of Newtonian Fluids) u u u u p u u u ρ + u + v + w = + ρ g µ x t x y z x x y z u v v v p v v v ρ + u + v + w = + ρ g y + µ t x y z y x y z u w w w p w w w ρ + u + v + w = + ρ gz + µ t x y z z x y z 15
16 Navier-Stokes Equations of Fluid Flow (Incompressible Flow of Newtonian Fluids) v = 0 Continuity Equation v 2 v v v ρ + = p + ρ g + µ t Momentum Equation Unsteady Acceleration Convective Acceleration Pressure Gradient Other body forces Viscosity 16
17 Saint-Venant Equations of Channel Flow The flow of water through stream channels is a distributed process since the flow rate, velocity and depth vary spatially throughout the channel. Estimates of flow rate or water level at certain locations in the channel system may be obtained using a set of equations that define the conservation of mass and momentum along this channel. This type of a model is based on partial differential equations that allow the flow rate and water level to be computed as a function of space and time. 17
18 Saint-Venant Equations of Channel Flow From a theoretical standpoint, the true flow process in the river system varies in all three spatial coordinate directions (longitudinal, lateral and transverse) as well as time. If a three-dimensional system is used, the resulting equations (modified Navier-Stokes equations for channel flow) would be very complex, and would require considerable amount of field data, which is also spatially variable. In field applications, most of this data could only be described approximately, thus rendering the three dimensional solutions susceptible to data errors. 18
19 Saint-Venant Equations of Channel Flow However, for most practical purposes, the spatial variations in lateral and transverse directions can be neglected and the flow in a river system can be approximated as a one-dimensional process along the longitudinal direction (i.e., in the direction of flow). The Saint Venant equations that were derived in the early 1870s by Barrede Saint-Venant, may be obtained through the application of control volume theory to a differential element of a river reach. The Navier-Stokes equations can be simplified for one-dimensional flow. However, it is more intuitive if these equations are derived from an elemental volume of fluid along a channel. 19
20 Assumptions made in derivation of Saint-Venant Equations of Channel Flow 1. The flow is one-dimensional. The water depth and flow velocity vary only in the direction of flow. Therefore, the flow velocity is constant and the water surface is horizontal across any section perpendicular to the direction of flow. 2. The flow is assumed to vary gradually along the channel so that the hydrostatic pressure distribution prevails and vertical accelerations can be neglected. 3. The channel bottom slope is small. 20
21 Assumptions made in derivation of Saint-Venant Equations of Channel Flow 4. The channel bed is stable such that there is no change in bed elevations in time. 5. The Manning and Chezy equations, which are used in the definition of channel resistance factor in steady, uniform flow conditions, are also used to describe the resistance to flow in unsteady, non-uniform flow applications. 6. The fluid is incompressible and of constant density throughout the flow. 21
22 Cross-sectional view(x-z plane) 22
23 Plan view(x-y plane) Cross-sectional view (Y-Z plane) 23
24 Reynolds Transport Theorem The equations of mass and momentum conservation can be derived starting from the Reynolds Transport Theorem. If B sys and bare defined as extensive and intensive parameters of the system, respectively, the Reynolds Transport Theorem for a fixed non-deforming control volume can be stated as DB sys = ρbd + ρb( V n ) da Dt t cv where ρ and are density and volume of fluid. V is velocity vector. n is the unit vector normal to the control surface and t is time. In this equation, the intensive parameter is defined as extensive property per unit mass. Therefore, if mass is the extensive property, the intensive property becomes unity. cs 24
25 Continuity Equation(Conservation of Mass) The continuity equation can be obtained by simplifying the Reynolds Transport Theorem with the extensive property being the mass of the system. Since the total mass in a system is always constant, the left hand-side of equation becomes zero: 0 = ρd + ρ ( V n) da t cv cs This implies that the time rate of change of mass in the control volume is equal to the difference in cumulative mass inflow and outflow from the control surfaces of the control volume. 25
26 Continuity Equation(Conservation of Mass) Rate of change of mass = Mass inflow Mass outflow This general equation of continuity can be given for the particular case of an open channel with an irregular geometry. As seen in figure, the inflow to the control volume is the sum of the flow Qentering the control volume at the upstream end of the channel and the lateral inflow qentering the control volume as a distributed flow along the side of the channel. In this case, the lateral inflow has the dimensions of flow per unit length of channel. 26
27 Continuity Equation(Conservation of Mass) ρ( Q + qdx) ρ( Q + Q dx) = x ( ρ Adx) t Applying the assumption of constant density and rearranging produces the conservation form of the continuity equation, which is valid for any irregular cross section A Q + q = t x 0 27
28 Momentum Equation It is stated in the Newton's second law for a system that the time rate of change of the linear momentum of the system is equal to the sum of the external forces acting on the system. Using this law in the Reynolds Transport Theorem yields F ρvd ρv ( = + V n) da t cv cs In this case, the intensive property is the velocity of the fluid. This equation reveals that the sum of the forces applied on the system is equal to the time rate of change of momentum stored within the control volume plus the net flow of momentum across the control surfaces 28
29 Momentum Equation For an open channel flow, there are five different forces acting on the control volume: 1. the gravity force along the channel due to weight of water (F g ) 2. the pressure force (F p ) 3. the friction force along the bottom and sides of the channel (F f ) 4. the contraction/expansion force due to abrupt changes in channel cross section (F e ) 5. the wind shear force (F w ) F = F + F + F + F + F g p f e w 29
30 Momentum Equation Gravity Force: The gravity force acting on the control volume shown in figureis a function of the volume of the fluid, which may be given as d = Adx The corresponding weight of the fluid can be expressed as W = ρd g = ρ gadx where gis the gravitational acceleration. The component of weight in the direction of flow becomes the gravity force Fg = ρ gadxsinθ where θis the angle of inclination of the channel bed. With the assumption of a small angle of inclination of the channel bed, the sine of the angle can be approximated as the tangent of the angle: 30
31 Momentum Equation sin( θ ) tan( θ ) which is also equal to the slope of the channel bed, S o. Therefore, for a small angle of inclination of the channel bed, gravity force acting on the control volume can be written as F g = ρ gadxs o 31
32 Momentum Equation Pressure Force: The pressure force is the resultant of the hydrostatic force on the left side of the control volume (F pl ), the hydrostatic force on the right side of the control volume (F rl ) and the pressure force exerted by the banks on the control volume (F pb ) as can be seen from Figure 1.b: F = F F + p pl pr F pb If an element of fluid of thickness dwat an elevation of wfrom the bottom of the channel is immersed at depth y-w, the incremental hydrostatic pressure on this element is computed as ρg(y-w). The corresponding incremental hydrostatic force is then calculated as 32
33 Momentum Equation pl ( ) df = ρ g y w bdw where bis the width of the element across the channel. Integrating this force over the cross section gives the total hydrostatic force on the left end of the control volume pl y 0 ( ) F = ρg y w bdw Using the Taylor's series expansion of the hydrostatic force on the left end, F pl, and ignoring the higher order terms, one might obtain the hydrostaticforceontherightendofthecontrolvolumeas: F pl Fpr = Fpl + dx x 33
34 Momentum Equation The differential of F pl with respect to xin the above equation is computed using the Leibnitz rule for differentiation 0 ( ρ ( ) ) y F g y w b pl dy d0 = dw + ρ g ( y y) b + ρg ( y 0) b x x dx dx where the second and third terms on the right hand-side of the equationare evaluated as zero. The partial derivative term can be expanded using the multiplication rule of differentiation and the differential of F pl with respect to xtakes the following form y y Fpl y b = ρg bdw + ρg ( y w) dw x x x
35 Momentum Equation The first integral in above equation can be simplified as y y y y ρg bdw = ρg bdw = ρg A x x x 0 0 y since A = y 0 bdw The pressure force exerted by the banks on the control volume is related to the rate of change of the width of the channel through the element dxand is given as y b Fpb = ρ g ( y w) dw dx x 0 35
36 Momentum Equation Substituting: F F F = F F + dx + F = dx + F x x pl pl p pl pl pb pb y y y b b Fp = ρg A ρg( y w) dw dx ρ + + g( y w) dw dx x x x 0 0 The resultant pressure force acting on the control volume can be written as: y Fp = ρ g Adx x 36
37 Momentum Equation Friction Force: The friction forces created by shear stress along the bottom and sides of the control volume can be defined in terms of the bed shear stress τ o and can be given as -τ o Pdx, where Pis the wetted perimeter. In accordance with the assumptions stated previously, the bed shear stress can be defined in terms of the steady uniform flow formula: τ = ρgrs = ρg( A/ P) S o f f where Ris the hydraulic radius defined by the ratio of the flow area and the wetted perimeter ands f is the friction slope, which is derived from the Manning's equation and given as 37
38 Momentum Equation 2 n V S f = 2 µ R 2 4 / 3 nis the Manning's roughness coefficient, Vis the flow velocity and µis a constant, which is equal to 1.49 in British units and 1.0 in SI units. Based on the definition of bed shear stress given in aboveequation, the friction force acting on the control volume takes the final form given below F f = ρgas f dx 38
39 Momentum Equation Contraction/Expansion Force: Abrupt contraction or expansion of the channel causes energy loss through turbulence. These losses can be considered similar to the losses in a pipe system. The magnitude of these losses is a function of the change in velocity head, V 2 /2g=(Q/A) 2 /2g, through the length of the channel. The forces associated with these eddy losses can be defined similar to friction force except the term S f is replaced by S e, which is the eddy loss slope representing the loss of energy due to an abrupt contraction or expansion S e = K ec 2g ( Q / A) x 2 where K ec is the non-dimensional expansion or contraction coefficient that is defined as negative for channel expansion and positive for channel contraction. Therefore, contraction/expansion force term becomes: F e = ρgas e dx 39
40 Momentum Equation Wind Shear Force: The wind shear force is caused by the frictional resistance of wind against the free surface of the water. It can be defined as a function of wind shear stress, τ w, and written as τ w Bdx. The wind shear stress is defined as the product of a wind shear factor, W f and fluid density τ w = ρw f W = f C f V r 2 V r where C f is a shear stress coefficient and V r is the velocity of fluid relative to the boundary, which can be written as 40
41 Momentum Equation V Q = A r V w cosω V w is the wind velocity and ωis the angle that wind direction makes with the direction of average fluid velocity, (Q/A). Based on the definition of wind shear stress given in aboveequation,the wind shear force acting on the control volume takes the final form given below F w = ρw f Bdx Finally, the sum of the five forces define the total force on the left-hand side of the momentum equation F = ρgas o dx ρga y x dx ρgas f dx ρgas e dx ρw Bdx f 41
42 Momentum Equation F ρvd ρv ( = + V n) da t cv The two momentum terms on the right-hand side represent the rate of change of storage of momentum in the control volume and the net outflow of momentum across the control surface, respectively The net momentum outflow is the sum of momentum outflow minus the momentum inflow to the control volume. The mass inflow rate to the control volume is the sum of both stream inflow and the lateral inflow and is defined as -ρ(q + qdx). cs 42
43 Momentum Equation The momentum inflow to the control volume is computed by multiplying the two mass inflow rates by their respective velocities and a momentum correction factor, β: V ρ( V n) da = ρ( βvq + βυxqdx) inlet where υ x is the average velocity of lateral inflow in the direction of main channel flow. The momentum coefficient, β, accounts for the non-uniform distribution of velocity at a specific channel cross section. 1 2 β = υ da 2 V A where υis the velocity of the fluid in a small elemental area dain the channel cross section. Generally, the value of the momentum coefficient ranges from 1.01 for straight prismatic channels to 1.33 for river valleys with floodplains 43
44 Momentum Equation The momentum outflow from the control volume is also a function of mass outflow from the control volume, which can be defined as the Taylor series expansion of mass inflow. Hence, the momentum outflow from the control volume is computed as: βvq ( β ) V ρ( V n) da = ρ βvq + dx x outlet Thus, net momentum outflow across the control surface ( βvq) ( βvq) V ρ( V n) da = ρ ( βvq + βυxqdx) + ρ βvq + dx = ρ βυxq dx x x cs 44
45 Momentum Equation The time rate of change of momentum stored in the control volume is written as a function of the volume of the elemental channel length dx. The momentum associated with this elemental volume can be written as ρvadx, or ρqdxand the time rate of change of momentum is given as t Q V ρ d = ρ dx t cv When all terms are combined and substituted back into the momentum equation: F ρvd ρv ( = + V n) da t cv cs ρgas o dx ρga y x dx ρgas f dx ρgas e dx W Bρdx f = ρ βυ xq ( βvq) dx x Q + ρ t dx 45
46 Momentum Equation Thisequationis simplified and rearranged to the following form if all terms are divided by ρdxand Vis replaced by Q/A 0 2 = W B q S S S x y ga x A Q t Q f x e f o υ β β ) / ( 46 x x t The waterdepth termin thisequationcan be replaced by the water surface elevation (stage), h, using the equality z y h + = where zis the channel bottom above a datum such as mean sea level as seen in figure. The derivative of thiswith respect to xis written as x z x y x h + =
47 Momentum Equation However, the term z/xis equal to the negative of the slope of the channel, so theequationcan also be written as S o x y x h = The momentum equation can now be expressed in terms of hby 47 The momentum equation can now be expressed in terms of hby 0 2 = W B q S S x h ga x A Q t Q f x e f υ β β ) / (
48 Saint-Venant Equations The continuity and momentum equations are always addressed together and form the conservation form of the Saint-Venant equations 0 = + q x Q t A = W B q S S x h ga x A Q t Q f x e f υ β β ) / ( These equations are the governing equations of one-dimensional unsteady flow in open channels and were originally developed by the French scientist Barrede Saint-Venantin 1872
49 Saint-Venant Equations Steady State Under steady state assumption, Saint-Venant equations simplify to: dq q dx = 0 β dx 2 d( Q / A) dh + ga + S f + Se βqυ x + Wf B = dx 0 49
Open channel flow Basic principle
Open channel flow Basic principle INTRODUCTION Flow in rivers, irrigation canals, drainage ditches and aqueducts are some examples for open channel flow. These flows occur with a free surface and the pressure
More informationAppendix 4-C. Open Channel Theory
4-C-1 Appendix 4-C Open Channel Theory 4-C-2 Appendix 4.C - Table of Contents 4.C.1 Open Channel Flow Theory 4-C-3 4.C.2 Concepts 4-C-3 4.C.2.1 Specific Energy 4-C-3 4.C.2.2 Velocity Distribution Coefficient
More informationChapter 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.
More information1 The basic equations of fluid dynamics
1 The basic equations of fluid dynamics The main task in fluid dynamics is to find the velocity field describing the flow in a given domain. To do this, one uses the basic equations of fluid flow, which
More informationCHAPTER 9 CHANNELS APPENDIX A. Hydraulic Design Equations for Open Channel Flow
CHAPTER 9 CHANNELS APPENDIX A Hydraulic Design Equations for Open Channel Flow SEPTEMBER 2009 CHAPTER 9 APPENDIX A Hydraulic Design Equations for Open Channel Flow Introduction The Equations presented
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationOpen Channel Flow. M. Siavashi. School of Mechanical Engineering Iran University of Science and Technology
M. Siavashi School of Mechanical Engineering Iran University of Science and Technology W ebpage: webpages.iust.ac.ir/msiavashi Email: msiavashi@iust.ac.ir Landline: +98 21 77240391 Fall 2013 Introduction
More informationdu u U 0 U dy y b 0 b
BASIC CONCEPTS/DEFINITIONS OF FLUID MECHANICS (by Marios M. Fyrillas) 1. Density (πυκνότητα) Symbol: 3 Units of measure: kg / m Equation: m ( m mass, V volume) V. Pressure (πίεση) Alternative definition:
More informationWhen the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.
Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs
More informationFloodplain Hydraulics! Hydrology and Floodplain Analysis Dr. Philip Bedient
Floodplain Hydraulics! Hydrology and Floodplain Analysis Dr. Philip Bedient Open Channel Flow 1. Uniform flow - Manning s Eqn in a prismatic channel - Q, V, y, A, P, B, S and roughness are all constant
More information1. Fluids Mechanics and Fluid Properties. 1.1 Objectives of this section. 1.2 Fluids
1. Fluids Mechanics and Fluid Properties What is fluid mechanics? As its name suggests it is the branch of applied mechanics concerned with the statics and dynamics of fluids - both liquids and gases.
More informationViscous 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...........................
More informationCBE 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,
More informationOPEN-CHANNEL FLOW. Free surface. P atm
OPEN-CHANNEL FLOW Open-channel flow is a flow of liquid (basically water) in a conduit with a free surface. That is a surface on which pressure is equal to local atmospheric pressure. P atm Free surface
More informationBasic 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
More informationFluid Dynamics and the Navier-Stokes Equation
Fluid Dynamics and the Navier-Stokes Equation CMSC498A: Spring 12 Semester By: Steven Dobek 5/17/2012 Introduction I began this project through a desire to simulate smoke and fire through the use of programming
More informationChapter 13 OPEN-CHANNEL FLOW
Fluid Mechanics: Fundamentals and Applications, 2nd Edition Yunus A. Cengel, John M. Cimbala McGraw-Hill, 2010 Lecture slides by Mehmet Kanoglu Copyright The McGraw-Hill Companies, Inc. Permission required
More informationUrban Hydraulics. 2.1 Basic Fluid Mechanics
Urban Hydraulics Learning objectives: After completing this section, the student should understand basic concepts of fluid flow and how to analyze conduit flows and free surface flows. They should be able
More informationGoverning Equations of Fluid Dynamics
Chapter 2 Governing Equations of Fluid Dynamics J.D. Anderson, Jr. 2.1 Introduction The cornerstone of computational fluid dynamics is the fundamental governing equations of fluid dynamics the continuity,
More informationFluids and Solids: Fundamentals
Fluids and Solids: Fundamentals We normally recognize three states of matter: solid; liquid and gas. However, liquid and gas are both fluids: in contrast to solids they lack the ability to resist deformation.
More informationWhat is the most obvious difference between pipe flow and open channel flow????????????? (in terms of flow conditions and energy situation)
OPEN CHANNEL FLOW 1 3 Question What is the most obvious difference between pipe flow and open channel flow????????????? (in terms of flow conditions and energy situation) Typical open channel shapes Figure
More informationCIVE2400 Fluid Mechanics Section 2: Open Channel Hydraulics
CIVE400 Fluid Mechanics Section : Open Channel Hydraulics. Open Channel Hydraulics.... Definition and differences between pipe flow and open channel flow.... Types of flow.... Properties of open channels...
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationNUMERICAL 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
More informationScalars, Vectors and Tensors
Scalars, Vectors and Tensors A scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. Examples are hydrostatic pressure and temperature. A vector
More informationIntroduction to COMSOL. The Navier-Stokes Equations
Flow Between Parallel Plates Modified from the COMSOL ChE Library module rev 10/13/08 Modified by Robert P. Hesketh, Chemical Engineering, Rowan University Fall 2008 Introduction to COMSOL The following
More informationA LAMINAR FLOW ELEMENT WITH A LINEAR PRESSURE DROP VERSUS VOLUMETRIC FLOW. 1998 ASME Fluids Engineering Division Summer Meeting
TELEDYNE HASTINGS TECHNICAL PAPERS INSTRUMENTS A LAMINAR FLOW ELEMENT WITH A LINEAR PRESSURE DROP VERSUS VOLUMETRIC FLOW Proceedings of FEDSM 98: June -5, 998, Washington, DC FEDSM98 49 ABSTRACT The pressure
More information1 Fundamentals of. open-channel flow 1.1 GEOMETRIC ELEMENTS OF OPEN CHANNELS
1 Fundamentals of open-channel flow Open channels are natural or manmade conveyance structures that normally have an open top, and they include rivers, streams and estuaries. n important characteristic
More informationChapter 9. Steady Flow in Open channels
Chapter 9 Steady Flow in Open channels Objectives Be able to define uniform open channel flow Solve uniform open channel flow using the Manning Equation 9.1 Uniform Flow in Open Channel Open-channel flows
More informationCalculating resistance to flow in open channels
Alternative Hydraulics Paper 2, 5 April 2010 Calculating resistance to flow in open channels http://johndfenton.com/alternative-hydraulics.html johndfenton@gmail.com Abstract The Darcy-Weisbach formulation
More informationLecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows
Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3.- 1 Basics: equations of continuum mechanics - balance equations for mass and momentum - balance equations for the energy and the chemical
More informationLecture 5 Hemodynamics. Description of fluid flow. The equation of continuity
1 Lecture 5 Hemodynamics Description of fluid flow Hydrodynamics is the part of physics, which studies the motion of fluids. It is based on the laws of mechanics. Hemodynamics studies the motion of blood
More information2.0 BASIC CONCEPTS OF OPEN CHANNEL FLOW MEASUREMENT
2.0 BASIC CONCEPTS OF OPEN CHANNEL FLOW MEASUREMENT Open channel flow is defined as flow in any channel where the liquid flows with a free surface. Open channel flow is not under pressure; gravity is the
More informationFLUID MECHANICS FOR CIVIL ENGINEERS
FLUID MECHANICS FOR CIVIL ENGINEERS Bruce Hunt Department of Civil Engineering University Of Canterbury Christchurch, New Zealand? Bruce Hunt, 1995 Table of Contents Chapter 1 Introduction... 1.1 Fluid
More informationFor Water to Move a driving force is needed
RECALL FIRST CLASS: Q K Head Difference Area Distance between Heads Q 0.01 cm 0.19 m 6cm 0.75cm 1 liter 86400sec 1.17 liter ~ 1 liter sec 0.63 m 1000cm 3 day day day constant head 0.4 m 0.1 m FINE SAND
More informationChapter 28 Fluid Dynamics
Chapter 28 Fluid Dynamics 28.1 Ideal Fluids... 1 28.2 Velocity Vector Field... 1 28.3 Mass Continuity Equation... 3 28.4 Bernoulli s Principle... 4 28.5 Worked Examples: Bernoulli s Equation... 7 Example
More informationLecture 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
More information2.016 Hydrodynamics Reading #2. 2.016 Hydrodynamics Prof. A.H. Techet
Pressure effects 2.016 Hydrodynamics Prof. A.H. Techet Fluid forces can arise due to flow stresses (pressure and viscous shear), gravity forces, fluid acceleration, or other body forces. For now, let us
More informationChapter 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
More informationDimensional analysis is a method for reducing the number and complexity of experimental variables that affect a given physical phenomena.
Dimensional Analysis and Similarity Dimensional analysis is very useful for planning, presentation, and interpretation of experimental data. As discussed previously, most practical fluid mechanics problems
More informationNUMERICAL ANALYSIS OF OPEN CHANNEL STEADY GRADUALLY VARIED FLOW USING THE SIMPLIFIED SAINT-VENANT EQUATIONS
TASK QUARTERLY 15 No 3 4, 317 328 NUMERICAL ANALYSIS OF OPEN CHANNEL STEADY GRADUALLY VARIED FLOW USING THE SIMPLIFIED SAINT-VENANT EQUATIONS WOJCIECH ARTICHOWICZ Department of Hydraulic Engineering, Faculty
More information4 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
More informationContents. Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 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
More informationFundamentals of Fluid Mechanics
Sixth Edition. Fundamentals of Fluid Mechanics International Student Version BRUCE R. MUNSON DONALD F. YOUNG Department of Aerospace Engineering and Engineering Mechanics THEODORE H. OKIISHI Department
More informationOpen Channel Flow 2F-2. A. Introduction. B. Definitions. Design Manual Chapter 2 - Stormwater 2F - Open Channel Flow
Design Manual Chapter 2 - Stormwater 2F - Open Channel Flow 2F-2 Open Channel Flow A. Introduction The beginning of any channel design or modification is to understand the hydraulics of the stream. The
More informationExperiment (13): Flow channel
Introduction: An open channel is a duct in which the liquid flows with a free surface exposed to atmospheric pressure. Along the length of the duct, the pressure at the surface is therefore constant and
More informationTopic 8: Open Channel Flow
3.1 Course Number: CE 365K Course Title: Hydraulic Engineering Design Course Instructor: R.J. Charbeneau Subject: Open Channel Hydraulics Topics Covered: 8. Open Channel Flow and Manning Equation 9. Energy,
More informationINTRODUCTION TO FLUID MECHANICS
INTRODUCTION TO FLUID MECHANICS SIXTH EDITION ROBERT W. FOX Purdue University ALAN T. MCDONALD Purdue University PHILIP J. PRITCHARD Manhattan College JOHN WILEY & SONS, INC. CONTENTS CHAPTER 1 INTRODUCTION
More informationCHAPTER 4 FLOW IN CHANNELS
CHAPTER 4 FLOW IN CHANNELS INTRODUCTION 1 Flows in conduits or channels are of interest in science, engineering, and everyday life. Flows in closed conduits or channels, like pipes or air ducts, are entirely
More informationAbaqus/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
More informationChapter 5 MASS, BERNOULLI AND ENERGY EQUATIONS
Fluid Mechanics: Fundamentals and Applications, 2nd Edition Yunus A. Cengel, John M. Cimbala McGraw-Hill, 2010 Chapter 5 MASS, BERNOULLI AND ENERGY EQUATIONS Lecture slides by Hasan Hacışevki Copyright
More information11 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
More informationHydraulic Jumps and Non-uniform Open Channel Flow, Course #507. Presented by: PDH Enterprises, LLC PO Box 942 Morrisville, NC 27560 www.pdhsite.
Hydraulic Jumps and Non-uniform Open Channel Flow, Course #507 Presented by: PDH Enterprises, LLC PO Box 942 Morrisville, NC 27560 www.pdhsite.com Many examples of open channel flow can be approximated
More informationCHAPTER 4 OPEN CHANNEL HYDRAULICS
CHAPTER 4 OPEN CHANNEL HYDRAULICS 4. Introduction Open channel flow refers to any flow that occupies a defined channel and has a free surface. Uniform flow has been defined as flow with straight parallel
More informationM6a: Open Channel Flow (Manning s Equation, Partially Flowing Pipes, and Specific Energy)
M6a: Open Channel Flow (, Partially Flowing Pipes, and Specific Energy) Steady Non-Uniform Flow in an Open Channel Robert Pitt University of Alabama and Shirley Clark Penn State - Harrisburg Continuity
More informationDifferential Balance Equations (DBE)
Differential Balance Equations (DBE) Differential Balance Equations Differential balances, although more complex to solve, can yield a tremendous wealth of information about ChE processes. General balance
More informationBasic 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
More informationChapter 10. Flow Rate. Flow Rate. Flow Measurements. The velocity of the flow is described at any
Chapter 10 Flow Measurements Material from Theory and Design for Mechanical Measurements; Figliola, Third Edition Flow Rate Flow rate can be expressed in terms of volume flow rate (volume/time) or mass
More informationChapter 10. Open- Channel Flow
Updated: Sept 3 2013 Created by Dr. İsmail HALTAŞ Created: Sept 3 2013 Chapter 10 Open- Channel Flow based on Fundamentals of Fluid Mechanics 6th EdiAon By Munson 2009* *some of the Figures and Tables
More informationHigh Speed Aerodynamics Prof. K. P. Sinhamahapatra Department of Aerospace Engineering Indian Institute of Technology, Kharagpur
High Speed Aerodynamics Prof. K. P. Sinhamahapatra Department of Aerospace Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 06 One-dimensional Gas Dynamics (Contd.) We
More informationCE 3500 Fluid Mechanics / Fall 2014 / City College of New York
1 Drag Coefficient The force ( F ) of the wind blowing against a building is given by F=C D ρu 2 A/2, where U is the wind speed, ρ is density of the air, A the cross-sectional area of the building, and
More informationLECTURE 9: Open channel flow: Uniform flow, best hydraulic sections, energy principles, Froude number
LECTURE 9: Open channel flow: Uniform flow, best hydraulic sections, energy principles, Froude number Open channel flow must have a free surface. Normally free water surface is subjected to atmospheric
More informationFluid Flow in T-Junction of Pipes
LAPPEENRANTA UNIVERSITY OF TECHNOLOGY Department of Information Technology Laboratory of Applied Mathematics Paritosh R. Vasava Fluid Flow in T-Junction of Pipes The topic of this Master s thesis was approved
More informationSTATE OF FLORIDA DEPARTMENT OF TRANSPORTATION DRAINAGE HANDBOOK OPEN CHANNEL. OFFICE OF DESIGN, DRAINAGE SECTION November 2009 TALLAHASSEE, FLORIDA
STATE OF FLORIDA DEPARTMENT OF TRANSPORTATION DRAINAGE HANDBOOK OPEN CHANNEL OFFICE OF DESIGN, DRAINAGE SECTION TALLAHASSEE, FLORIDA Table of Contents Open Channel Handbook Chapter 1 Introduction... 1
More informationDimensional 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
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More information- momentum conservation equation ρ = ρf. These are equivalent to four scalar equations with four unknowns: - pressure p - velocity components
J. Szantyr Lecture No. 14 The closed system of equations of the fluid mechanics The above presented equations form the closed system of the fluid mechanics equations, which may be employed for description
More informationMathematics and Computation of Sediment Transport in Open Channels
Mathematics and Computation of Sediment Transport in Open Channels Junping Wang Division of Mathematical Sciences National Science Foundation May 26, 2010 Sediment Transport and Life One major problem
More informationXI / PHYSICS FLUIDS IN MOTION 11/PA
Viscosity It is the property of a liquid due to which it flows in the form of layers and each layer opposes the motion of its adjacent layer. Cause of viscosity Consider two neighboring liquid layers A
More informationHydraulics Laboratory Experiment Report
Hydraulics Laboratory Experiment Report Name: Ahmed Essam Mansour Section: "1", Monday 2-5 pm Title: Flow in open channel Date: 13 November-2006 Objectives: Calculate the Chezy and Manning coefficients
More informationCBE 6333, R. Levicky 1 Review of Fluid Mechanics Terminology
CBE 6333, R. Levicky 1 Review of Fluid Mechanics Terminology The Continuum Hypothesis: We will regard macroscopic behavior of fluids as if the fluids are perfectly continuous in structure. In reality,
More informationHeat Transfer Prof. Dr. Ale Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati
Heat Transfer Prof. Dr. Ale Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati Module No. # 04 Convective Heat Transfer Lecture No. # 03 Heat Transfer Correlation
More informationL 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,
More information8.2 Elastic Strain Energy
Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for
More informationMODELING FLUID FLOW IN OPEN CHANNEL WITH CIRCULAR CROSS SECTION DADDY PETER TSOMBE MASTER OF SCIENCE. (Applied Mathematics)
MODELING FLUID FLOW IN OPEN CHANNEL WITH CIRCULAR CROSS SECTION DADDY PETER TSOMBE MASTER OF SCIENCE (Applied Mathematics) JOMO KENYATTA UNIVERSITY OF AGRICULTURE AND TECHNOLOGY 2011 Modeling fluid flow
More informationBackwater Rise and Drag Characteristics of Bridge Piers under Subcritical
European Water 36: 7-35, 11. 11 E.W. Publications Backwater Rise and Drag Characteristics of Bridge Piers under Subcritical Flow Conditions C.R. Suribabu *, R.M. Sabarish, R. Narasimhan and A.R. Chandhru
More informationElectromagnetism Laws and Equations
Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E- and D-fields............................................. Electrostatic Force............................................2
More informationCE 204 FLUID MECHANICS
CE 204 FLUID MECHANICS Onur AKAY Assistant Professor Okan University Department of Civil Engineering Akfırat Campus 34959 Tuzla-Istanbul/TURKEY Phone: +90-216-677-1630 ext.1974 Fax: +90-216-677-1486 E-mail:
More informationNatural Convection. Buoyancy force
Natural Convection In natural convection, the fluid motion occurs by natural means such as buoyancy. Since the fluid velocity associated with natural convection is relatively low, the heat transfer coefficient
More informationLecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014
Lecture 07: Work and Kinetic Energy Physics 2210 Fall Semester 2014 Announcements Schedule next few weeks: 9/08 Unit 3 9/10 Unit 4 9/15 Unit 5 (guest lecturer) 9/17 Unit 6 (guest lecturer) 9/22 Unit 7,
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More informationDistinguished Professor George Washington University. Graw Hill
Mechanics of Fluids Fourth Edition Irving H. Shames Distinguished Professor George Washington University Graw Hill Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok
More informationHeat Transfer From A Heated Vertical Plate
Heat Transfer From A Heated Vertical Plate Mechanical and Environmental Engineering Laboratory Department of Mechanical and Aerospace Engineering University of California at San Diego La Jolla, California
More informationCHAPTER ONE Fluid Fundamentals
CHPTER ONE Fluid Fundamentals 1.1 FLUID PROPERTIES 1.1.1 Mass and Weight Mass, m, is a property that describes the amount of matter in an object or fluid. Typical units are slugs in U.S. customary units,
More informationCHAPTER 5 OPEN-CHANNEL FLOW
CHAPTER 5 OPEN-CHANNEL FLOW 1. INTRODUCTION 1 Open-channel flows are those that are not entirely included within rigid boundaries; a part of the flow is in contract with nothing at all, just empty space
More informationMIKE 21 FLOW MODEL HINTS AND RECOMMENDATIONS IN APPLICATIONS WITH SIGNIFICANT FLOODING AND DRYING
1 MIKE 21 FLOW MODEL HINTS AND RECOMMENDATIONS IN APPLICATIONS WITH SIGNIFICANT FLOODING AND DRYING This note is intended as a general guideline to setting up a standard MIKE 21 model for applications
More information2O-1 Channel Types and Structures
Iowa Stormwater Management Manual O-1 O-1 Channel Types and Structures A. Introduction The flow of water in an open channel is a common event in Iowa, whether in a natural channel or an artificial channel.
More informationFLOW RATE MEASUREMENTS BY FLUMES
Fourteenth International Water Technology Conference, IWTC 14 010, Cairo, Egypt 1 FLOW RATE MEASUREMENTS BY FLUMES W. Dbrowski 1 and U. Polak 1 Department of Environmental Engineering, Cracow University
More informationElasticity Theory Basics
G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationEXAMPLE: Water Flow in a Pipe
EXAMPLE: Water Flow in a Pipe P 1 > P 2 Velocity profile is parabolic (we will learn why it is parabolic later, but since friction comes from walls the shape is intuitive) The pressure drops linearly along
More informationCHAPTER 3 STORM DRAINAGE SYSTEMS
CHAPTER 3 STORM DRAINAGE SYSTEMS 3.7 Storm Drains 3.7.1 Introduction After the tentative locations of inlets, drain pipes, and outfalls with tail-waters have been determined and the inlets sized, the next
More informationVector has a magnitude and a direction. Scalar has a magnitude
Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationCE 6303 MECHANICS OF FLUIDS L T P C QUESTION BANK PART - A
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
More informationMechanical Properties - Stresses & Strains
Mechanical Properties - Stresses & Strains Types of Deformation : Elasic Plastic Anelastic Elastic deformation is defined as instantaneous recoverable deformation Hooke's law : For tensile loading, σ =
More informationThis makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5
1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,
More informationATM 316: Dynamic Meteorology I Final Review, December 2014
ATM 316: Dynamic Meteorology I Final Review, December 2014 Scalars and Vectors Scalar: magnitude, without reference to coordinate system Vector: magnitude + direction, with reference to coordinate system
More informationTransport Phenomena I
Transport Phenomena I Andrew Rosen December 14, 013 Contents 1 Dimensional Analysis and Scale-Up 4 1.1 Procedure............................................... 4 1. Example................................................
More informationReview of Vector Analysis in Cartesian Coordinates
R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.
More informationPractice Problems on Boundary Layers. Answer(s): D = 107 N D = 152 N. C. Wassgren, Purdue University Page 1 of 17 Last Updated: 2010 Nov 22
BL_01 A thin flat plate 55 by 110 cm is immersed in a 6 m/s stream of SAE 10 oil at 20 C. Compute the total skin friction drag if the stream is parallel to (a) the long side and (b) the short side. D =
More information