Engineering Fluid Mechanics Lecture Notes 2024-2025 PDF

ENGINEERING FLUID MECHANICS MECH326 / MECH627 Henry Ng – [email protected] MODULE STAFF Lecturer Dr. Henry Ng Harrison Hughes/Walker, Room UG52 [email protected] TIMETABLE MECH 326 / MECH 627 ENGINEERING FLUID MECHANICS LECTURES: Monday 15:00 – 17:0...

ENGINEERING FLUID MECHANICS MECH326 / MECH627 Henry Ng – [email protected] MODULE STAFF Lecturer Dr. Henry Ng Harrison Hughes/Walker, Room UG52 [email protected] TIMETABLE MECH 326 / MECH 627 ENGINEERING FLUID MECHANICS LECTURES: Monday 15:00 – 17:00 (Hele-Shaw LT) Tuesday 15:00 – 17:00 (Hele-Shaw LT) Department-wide – WK7: no classes WK8 – No Tuesday Lecture: Class test for MECH326 COURSE MATERIAL Engineering Fluid Mechanics = 15 credit module CANVAS – First Point of Contact Lecture slides (self contained set of notes) Videos Tutorial sessions COMMUNICATION & EMAIL REPLY POLICY Email reviewed once per day Subject line: Must start with MECH326 or MECH627 Replies usually within 2 business days or less Buffer allows identification of class-wide issues (if any) Maximises efficiency & consistency Please only follow up if more then 48 hours elapsed from initial message Exception (Very few) Important, individual and personal issues – Reply ASA(Reasonably)P – Student Experience Team ASSESSMENT MECH326: Three-hour exam in January – 80% MCQ Class Test – 20% Week 8 – Check calendar to confirm MECH627: Three-hour exam in January – 80% 2 assignments – 20% total (10% each) Week 6 – Check calendar to confirm Week 12 – Check calendar to confirm ASSIGNMENTS: MECH627 ONLY! 2 Problem sheets with 3/4 questions – 10 marks each (total: 20% of the final mark) Purposes: (1) encourage study / revision throughout the semester; (2) self-assessment (feedback!) Submissions ONLY through CANVAS Group working OK but submissions MUST be individual Submission deadlines are strict. DO NOT ask for extensions, etc. – Submit Mitigating Circumstance if necessary – Student experience team Worked solutions available ~1 week after submission deadline  MODULE OVERVIEW Hydrodynamic similarity (dimensional analysis) Pipe flow Navier-Stokes equations Boundary layer theory Potential flow Compressible flow RECOMMENDED TEXTBOOKS White, F. M., Fluid Mechanics, McGraw-Hill, 6th edition, 2008 Fox, R. W., and McDonald, A. T., Introduction to Fluid Mechanics, John Wiley & Sons, Inc., 4th edition, 1994 White, F. M., Viscous Fluid Flow, McGraw-Hill, 3rd edition, 2006 Anderson, J. D., Fundamentals of Aerodynamics, McGraw-Hill, 2nd edition, 1991 Anderson, J. D., Modern Compressible Flow – With Historical Perspective, McGraw-Hill, 3rd edition, 2004 Massey, B. S., Mechanics of Fluids, Van Nostrand Reinhold, 6th edition, 1989 RECOMMENDED TEXTBOOKS (CONTINUED) Douglas, J. F. and Matthews, R. D., Solving Problems in: Fluid Mechanics. Volume 1, Longman, 3rd edition, 1996 Streeter, V. L., Wylie, E. B., and Bedford, K. W., Fluid Mechanics, McGraw- Hill, 9th edition, 1998 Granger, R. A., Fluid Mechanics, Dover Publications, Inc., 1995 Munson, B. R., Young, D. F., and Okiishi, T. H., Fundamentals of Fluid Mechanics, John Wiley & Sons, Inc., 2nd edition, 1994 MODULE AIMS/MOTIVATION Understanding fundamental concepts in fluid mechanics which are of practical importance in engineering In preliminary design Engineering analysis of the aerodynamic characteristics of a vehicle Engineering analysis of flow through turbine, engine, pipe systems or past a wind turbine Need for simple, fast analysis methods Assumptions and simplifications are introduced to reduce complexity of problem MODULE AIMS/MOTIVATION Detailed flow/design analysis Computational Fluid Dynamics (CFD) often used in research institutes and the industry Nowadays based on complex mathematical models containing ‘complete’ model of the fluid flow Complex numerical techniques are required User of CFD needs understanding of underlying models and the fluid dynamics principles/concepts MODULE CONTENT This module will include: ‘Exact’/analytical answers for ‘simple’ problems Approximate/analytical answers for a range of fluid dynamics problems of engineering relevance This understanding will also as background knowledge for CFD simulations typically conducted by engineers MODULE CONTENT: Introduction MODULE CONTENT: Viscosity The role of viscosity in fluids Viscosity of gases and liquids – temperature dependence The occurrence of no-slip boundary conditions on the surface of solid objects MODULE CONTENT: Similarity of flows The concept of similarity Non-dimensionalisation Similarity parameters Reynolds number Mach number Others Foundation for sub-scale testing Physical test – wind tunnels, water tunnels, flow rigs. Numerical simulations – CFD  MODULE CONTENT: Similarity of flows Concept of similarity is of great practical relevance It forms the basis of sub-scale wind tunnel/water tunnel testing It allows a priori predictions of the flow characteristics based on the value of non-dimensional numbers  Flow similarity: sub-scale wind tunnel testing Full-scale model: Wind tunnel conditions should replicate ‘real world’ Sub-scale model: What do the wind speed and density have to be to replicate ‘real world’ conditions? Flow similarity: sub-scale wind tunnel testing For a sub-scale model what wind tunnel conditions give a ‘similar’ flow to the full-scale ‘real’ flow How do the aerodynamic loads for a (sub-scale) wind tunnel model compare to the full-scale vehicle?  MODULE CONTENT: Flow through pipes Laminar flow – exact solutions Turbulent flow – solutions based on empirical laws – approximate solutions Effect of bends/cross-section changes  MODULE CONTENT: Flow through pipes Laminar/turbulent flow through pipes Effect sudden cross-section changes – ‘engineering’ empirical expressions are commonly used to account for these effects  MODULE CONTENT: Boundary layers The Boundary Layer concept Laminar and turbulent boundary layers Transition from laminar to turbulent flow Wall friction in flows with zero pressure gradient Boundary layers are of great practical interest in many engineering applications Characteristics of boundary layer vitally important in the drag of cars, aircraft, etc. Boundary layer separation can create dramatic changes in the flow (and platform performance) Viscous flows: boundary layers Viscous flows: boundary layers Viscous flows: boundary layers Viscous flows: boundary layers Viscous flows: boundary layers  MODULE CONTENT: Potential flow Potential flow theory Approximation of flows with ‘analytical’ techniques Compute streamlines to see what the flow `looks like’ Vorticity and circulation in flows Linear governing equation – superposition principle Approximation of flow around bluff objects Approximation of lift generated by airfoil section Example: Bernoulli’s equation Considering a contracting tube: 𝑅1 = 0.1𝑚 and 𝑅2 = 0.1𝑚. Assume flow is 1D, inviscid, incompressible and irrotational 𝑅1 = 0.3𝑚 𝑅2 = 0.10𝑚 𝑈1 = 5𝑚 ⋅ 𝑠 −1 𝑈2 =? 𝑃1 = 107 𝑃𝑎 𝑃2 =? 𝜌 = 1000𝑘𝑔 ⋅ 𝑚−3 𝜌 = 1000𝑘𝑔 ⋅ 𝑚−3 Example: Bernoulli’s equation - Solution Use Bernoulli’s equation to compute pressure at the outlet; 1 2 1 2 𝑃1 + 𝜌𝑈1 = 𝑃2 + 𝜌𝑈2 2 2 Valid on if flow is inviscid, incompressible and incompressible. Rearrange above for 𝑃2 ; 1 2 1 2 1 𝑃2 = 𝑃1 + 𝑈1 − 𝜌𝑈2 = 𝑃1 + 𝜌(𝑈12 − 𝑈22 ) 2 2 2 𝑃2 = 107 + 500 ⋅ (52 − 452 ) 𝑃2 = 9 ⋅ 106 𝑃𝑎 Potential flow – an `approximate’ flow model Potential flow theory can provide approximations for the velocity field for a flow around an object Bernoulli’s equation can then be used to evaluate the pressure in the fluid based on the velocity changes MODULE CONTENT: Compressible flow Compressible flow through ducts Subsonic and supersonic flow Ducts with constant cross section Effect of heat addition – Rayleigh flow Use of Rayleigh flow tables Critical length of a channel MODULE CONTENT: Compressible flow ‘External’ supersonic flows Oblique shock waves Prandtl-Meyer expansion fan Use of oblique-shock graph Computation of flow through oblique shock Exact solutions for Prandtl-Meyer expansion Aerodynamics of wedge-shaped airfoil section  Compressible flow: Shocks and Fans For supersonic flow – flow can approximated by oblique shock waves and expansion waves MODULE CONTENT: Viscosity Viscosity: dynamic viscosity, shear stress 𝑦 = 0, 𝑢 = 0 𝑦 = 𝛿, 𝑢 = 𝑉 𝑑𝑢 𝑉 = 𝑑𝑦 𝛿 𝑑𝑢 𝜏=𝜇 𝑑𝑦 Viscosity is resistance to velocity shear Viscosity: dimensional analysis The velocity gradient can be approximated by; 𝑑𝑢 𝑉 = 𝑑𝑦 𝛿 And shear stress is; 𝑑𝑢 𝜏=𝜇 𝑑𝑦 So we can write; 𝜏 𝑑𝑢 𝑉 = = 𝜇 𝑑𝑦 𝛿 Or in terms of viscosity; 𝜏𝛿 𝜇= 𝑉 Viscosity: dimensional analysis Which has dimensions of; 𝜏𝛿 𝑀 𝑇 𝑀 𝜇 = = 2⋅𝐿⋅ = 𝑉 𝐿𝑇 𝐿 𝐿𝑇 And units of; 𝜏𝛿 𝑀 𝑇 𝜇 = = 2⋅𝐿⋅ 𝑉 𝐿𝑇 𝐿 𝜏𝛿 𝑠 𝜇 = = 𝑃𝑎 ⋅ 𝑚 ⋅ = 𝑃𝑎 ⋅ 𝑠 𝑉 𝑚 𝜏𝛿 𝑀 𝑘𝑔 𝜇 = = = 𝑉 𝐿𝑇 𝑚 ⋅ 𝑠 Viscosity: Newtonian Flows For most flows, the viscosity is of the fluid is only a function of the temperature (and pressure) The shear stress then scales linearly with the velocity gradient in the flow Therefore viscosity is constant irrespective of shear rate 𝑑𝑢 𝜏 = 𝜇 𝑇, 𝑃 𝑑𝑦 When this is the case the fluid is Newtonian Viscosity: The kinematic viscosity We have just defined the ‘dynamic’ viscosity 𝜇. Sometimes the relative magnitude of the viscous shear stress with respect the fluid inertia is more informative 𝜏 𝜈= 𝑑𝑢 𝜌 𝑑𝑦 This quantity is the ‘kinematic’ viscosity and is; 𝜇 𝜈= 𝜌 Viscosity of a gas: Temperature dependence The viscosity of air can often be approximated by the following correlation; 3 𝐾𝑇 2 𝜇= 𝑇+𝐶 This is “Sutherland’s law” for air which has constants; 1 −2 𝐾= 1.448 ⋅ 10−6 𝑃𝑎 ⋅ 𝑠 ⋅ 𝐾 𝐶 = 110.4𝐾 Because the constant are positive, viscosity of air always increase with temperature 𝑑𝜇 >0 𝑑𝑇 Viscosity of a gas: Temperature dependence The viscosity of air at S.T.P 𝜇 298𝐾 = 1.82 ⋅ 10−5 𝑃𝑎 ⋅ 𝑠 𝜇 𝜈 298𝐾 = = 1.49 ⋅ 10−5 𝑚2 ⋅ 𝑠 −1 𝜌 Viscosity of a liquid: Temperature dependence The viscosity of liquids can often be approximated with the following relationship; 2 𝜇 T0 𝑇0 ln =𝑎+𝑏 +𝑐 𝜇0 𝑇 𝑇 For water, the constants are; 𝑎 = −2.1, 𝑏 = −4.45, 𝑐 = 6.55, 𝑇0 = 273.15𝐾, 𝜇0 = 1.79 ⋅ 10−3 𝑃𝑎 ⋅ 2 Viscosity of a liquid: Temperature dependence So, for water at room temperature i.e. 25∘ 𝐶; 2 𝜇 273.15 273.15 ln −3 = −2.1 − 4.45 + 6.55 1.79 ⋅ 10 298 298 𝜇 = 9.11 ⋅ 10−4 𝑃𝑎 ⋅ 𝑠 And 100∘ 𝐶; 2 𝜇 273.15 273.15 ln −3 = −2.1 − 4.45 + 6.55 1.79 ⋅ 10 373.15 373.15 𝜇 = 2.80 ⋅ 10−4 𝑃𝑎 ⋅ 𝑠 For liquids, viscosity 𝑑𝜇 decreases with temperature 0 𝑑𝑇 For liquids: viscosity decreases with temperature 𝑑𝜇