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Hydrodynamics
- 1. Mridula G M MtechCoastal &Harbour Engineering KUFOS WAVE HYDRODYNAMICS 1
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- 3. INTRODUCTION The behaviourof waves in the ocean governs the driving forces responsible for the different kinds of phenomena in the marine environment Three states of matter that exists in nature -- solid, liquid and gas Liquid and gas are referred to as fluids. Main distinction between liquid and gas lies in their rate of change of density If the change of the density of a fluid is negligible, it is then defined as incompressible. 3
- 4. TYPES OF FLUID 1.Ideal fluid : a fluid with no viscosity, no surface tension and is incompressible. 2. Real fluid : A fluid that has viscosity, surface tension and is compressible. 3. Compressible fluid : will reduce its volume in the presence of external pressure. 4. Incompressible fluid : is a fluid that does not change the volume of the fluid due to external pressure. 4
- 5. HISTORY OF FM Real fluid does not yield good estimates on the forces on structures due to fluid flow It was realised by Navier (1822) Thus the viscous flow theory was introduced including a viscous term to the momentum conservation equation. Similarly after few years Stokes (1845) also developed viscous flow theory This momentum conservation equation describing viscous flow is termed as Navier -Stokes Equation 5
- 6. TYPES OF FLOW 1.Steady and unsteady flow ( change of fluid characteristics with respect to time) 2. Uniform and non uniform flow (change of fluid characteristics with respect to space) 3. Laminar and turbulent flow (movement of particles in layers and zig-zag motion) 4. Rotational and irrotational flow (rotation of fluid particles about their mass centers) 6
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- 8. FORCES ACTING ONFLUIDS IN MOTION 1. Gravity force, Fg 2. Pressure force, Fp 3. Viscous force, Fv 4. Turbulent force, Ft 5. Surface tension force, Fs 6. Compressibility force or elastic force, Fe By Newton’s law of motion for fluids, ie , rate of change of momentum is equal to force causing the motion, we have the equation of motion as: Ma = Fg + Fp + Fv + Ft + Fs + Fe Where M is the mass and a is the acceleration of the fluid8
- 9. In mostof the fluid problems Fe and Fs may be neglected, hence Ma = Fg + Fp + Fv + Ft Then above equation is known as Reynold’s Equation of Motion For laminar flows, Ft is negligible, hence Ma = Fg + Fp + Fv Then the above equation is known as Navier Stokes Equation In case of ideal fluids, Fv is zero, hence Ma = Fg + Fp Then the above equation is known as the Euler’s Equation of Motion 9
- 10. INTRODUCTION TO HYDRODYNAMICS Oceans cover 71% of Earth’s surface and contain 97% of Earth’s water Largest ocean is the Pacific ocean and covers about 30% of Earth’s surface In order to explore and exploit the resources, a knowledge on the ocean environment is essential To have a knowledge on the physics of waves ,tides and currents, the subject of wave hydrodynamics is important. 10
- 11. TIDES The riseand fall of water surface due to the combined effect of the gravitational forces exerted by the Sun ,Moon and the rotation of the Earth From Newton’s law of universal gravitation Therefore, greater the mass of objects and the closer they are to each other, the greater the gravitational attraction between them. Because of this, Sun’s tide generating force is about half that of the Moon 1 2 2 m m F d 11
- 12. TIDAL RANGE:the vertical distance between high tide and low tide. Range is upto 15m. CLASSIFICATION 1. Diurnal: have one HT and one LT daily 2. Semidiurnal: have two HT and two LT daily 3. Mixed: there will two HT and two LT daily but of unequal shape 12
- 13. CURRENTS The flowof mass of water due to the existence of a gradient, ie, variation of any of the following • Temperature • Pressure • Salinity • Waves • Density Have magnitude and direction We need the information on current because: o Current exerts forces on structures o Presence of current in an environment dominated by waves, the characteristics of wave will be altered 13
- 14. ACCORDING TO THEFORCES BY WHICH THEY ARE CREATED Wind force Tides Waves Density differences Permanent Periodical Accidental Rotating Reversing Hydraulic Shoreward Longshore Seaward Surface Sub surface Deep CLASSIFICATION OF CURRENTS 14
- 15. WAVES Wave isan oscillation accompanied by the transfer of energy Wind gives energy for the growth of ocean waves The motion of the surface of waves are considered to be oscillatory Water droplet move in a vertical circle as the wave passes. The droplet moves forward with the wave's crest and backward with the trough Waves oscillatory motion 15
- 16. GENERATION OF OCEANWAVES Winds pumps in energy for the growth of the ocean waves Wind energy is partly transformed into wave energy by surface(normal and tangential) shear. As wind continuously blows over the surface, more energy is transferred and wave energy increases, ie, wave height increases Thus generation is depended on 3 factors: • Fetch (area where wind blows) • Velocity • duration 16
- 17. CLASSIFICATION OF OCEANWAVES CLASSIFICATION OF OCEAN WAVES As per water depth As per originAs per apparent shape 17
- 18. FUNDAMENTALS OF FLUIDFLOW 1. Conservation of mass ( Continuity Equation) 2. Euler’s Equation 3. Navier Stokes equation (Conservation of Momentum) 4. Bernoulis Equation ( Conservation of Energy) 18
- 19. CONTINUITY EQUATION Itis an equation that represents the transport of some quantity Mass, momentum, energy and other natural quantities are conserved under their respective appropriate conditions and a variety of physical phenomena may be described using continuity equation 19
- 20. Simplify andwe get: 𝜕𝜌 𝜕𝑡 + 𝜕(𝜌𝑢) 𝜕𝑥 + 𝜕(𝜌𝑣) 𝜕𝑦 = 0 For three dimensions, 𝜕𝜌 𝜕𝑡 + 𝜕(𝜌𝑢) 𝜕𝑥 + 𝜕(𝜌𝑣) 𝜕𝑦 + 𝜕(𝜌𝑤) 𝜕𝑧 = 0 The continuity equation is applicable for steady , unsteady flows and uniform, non-uniform flows and compressible and incompressible fluids. For steady flows, 𝜕ρ 𝜕𝑡 = 0 𝜕(𝜌𝑢) 𝜕𝑥 + 𝜕(𝜌𝑣) 𝜕𝑦 + 𝜕(𝜌𝑤) 𝜕𝑧 = 0 Above equation becomes, 20
- 21. For incompressiblefluids, 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 + 𝜕𝑤 𝜕𝑧 = 0 This equation is the continuity equation, ie, “net mass of fluid flowing across boundaries into an element over a short period must be equal to the amount by which the mass of element increases during the same period”. The mass density of the fluid does not change with x, y, z and t, hence the above equation simplifies to, 21
- 22. EULER’S EQUATION OFMOTION Only pressure forces and the fluid weight or in general, the body force are assumed to be acting on the mass of the fluid motion. Ma = Fg + Fp Mass of fluid in the medium is considered as (ρ.Δx. Δy. Δz) Component of body force in x direction =X(ρ.Δx. Δy. Δz) Net pressure force Fpx acting on the fluid mass : Pressure force per unit volume px p F . x. y. z x px p F x 22
- 23. For Euler’sequation of motion in X direction On solving in X, Y and Z direction we get, X direction = Y direction = Z direction = These equations are called Euler’s equation of motion. ax, ay, az are termed as total accelerations in respective directions Max = Fgx + Fpx x 1 p X a x y 1 p Y a y z 1 p Z a z 23
- 24. Total accelerationhas two components with respect to space and time Euler equations are applicable to compressible and incompressible, non-viscous in steady or unsteady state of flow. x u u u u a u v w t x y z y v v v v a u v w t x y z z w w w w a u v w t x y z local acceleration or temporal acceleration convective acceleration 24
- 25. NAVIER -STOKES EQUATION It is a generalisation of Euler’s equation of motion (inviscid) Ma = Fg + Fp + Fv It is the most important equation in fluid mechanics Rate of change of momentum in an element = sum of net momentum flux in the element and external forces When shear forces are included along with Euler’s equation, an extra force term is introduced 𝐷𝑢 𝐷𝑡 = 𝜕𝑢 𝜕𝑡 + 𝑢 𝜕𝑢 𝜕𝑥 + 𝑣 𝜕𝑢 𝜕𝑦 + 𝑤 𝜕𝑢 𝜕𝑧 = − 1 𝜌 𝜕𝑃 𝜕𝑥 + 𝛻2 𝑢 + 𝑓𝑥 25
- 26. 26 𝜇 𝜕2 𝑣 𝜕𝑥2 + 𝜇 𝜕2𝑣 𝜕𝑦2 + 𝜇 𝜕2 𝑣 𝜕𝑧2 − 𝛻𝑝 = 𝜌 𝜕𝑣 𝜕𝑡 + 𝜌𝑣. 𝛻𝑣 In vector notation ,this can be written as 𝜇 𝛻. 𝛻 𝑣 − 𝛻𝑝 = 𝜌 𝜕𝑣 𝜕𝑡 + 𝜌𝑣. 𝛻𝑣 This is the navier stokes equation 𝜕𝑣 𝜕𝑡 + 𝑣. 𝛻𝑣 = − 1 𝜌 𝛻𝑝 + 𝜇 𝜌 𝛻2 𝑣
- 27. BERNOULI’S EQUATION Bernouli’sequation is related with pressure, velocity and elevation changes of a fluid in motion. “In an ideal fluid, when the flow is steady and continuous, the sum of pressure energy, kinetic energy and potential energy is a constant”. Where, v- fluid flow speed g- acceleration due to gravity z- elevation of the point above a reference plane p- pressure ρ- density of the fluid 27
- 28. Bernouli’s equationis called as the law of conservation of energy. APPLICATIONS It is mainly applied in incompressible fluid problems and also in areas involving energy. The application of Bernouli’s equation is used in measuring devices such as Venturimeter Pitot tube Orificemeter 28
- 29. CLASSIFICATION OF FLOWPROBLEMS Based on flow characteristics and degree of complexity: 1. Laminar and turbulent 2. Time dependent (steady/ transient) 3. Nature of flow field (parabolic/ elliptic) 4. Dimensionality of flow field ( 1D, 2D, 3D) 5. Newtonian and non newtonian 6. Single phase or multiphase flow 29
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- 31. PARAMETERS TO DEFINEA WAVE The main parameters are: 1. Wave height: vertical distance between crest and trough 2. Wave period: time taken to travel one wavelength 3. Wavelength: distance between two successive crest or trough 31
- 32. WAVE THEORY LINEAR WAVETHEORY Developed by Airy in 1845 gave a mathematical description for the progressive waves applicable for a wide range of depth to wavelength ratio. It is assumed that water surface elevation is very small when compared with wave length and water depth and thus it is also known as ‘small amplitude wave theory’. The condition is to obtain the solution to Laplace equation The boundary conditions specified are: o Bottom boundary condition : vertical velocity at bed is zero o Kinematic free surface boundary condition: surface elevation= vertical water particle velocity o Dynamic free surface boundary condition: specifies pressure distribution over free 2 2 2 2 0 x z 32
- 33. SOLUTION TO LAPLACEEQUATION Velocity potential function is given by ∅ = 𝑎𝑔𝑐𝑜𝑠ℎ 𝑘(ℎ + 𝑧) 𝜎 cosh 𝑘ℎ {Sin kx − σ𝑡 } Wave surface elevation η = a cos(𝑘𝑥 − 𝜎𝑡) Wave celerity Dispersion relation 𝜎2 = 𝑔𝑘 tanh(𝑘ℎ) x 2 L L C t k T 2 T 33
- 34. WAVE TRANSFORMATION Aswave enters from deep to coastal waters wavelength, celerity and approach angle reduces. Wave height decreases or increases depending upon the configuration of the coast Wave transformation processes are refraction, diffraction and reflection WAVE REFRACTION A change in alignment of the wave crest line as wave advances from deep to shallow waters is called wave refraction. 34
- 35. WAVE REFLECTION Totalreflection of wave from a barrier is called wave reflection WAVE DIFFRACTION Fanning of the wave crest in the leeward side of a barrier is known as wave diffraction. 35
- 36. WAVE BREAKING Aphenomena in which the surface of the waves folds or rolls over and intersects itself. They are of four types 1. Plunging breaker : wave crest advances faster than celerity 2. Spilling breaker : crest separates and starts rolling down to the front face of the wave 3. Surging breaker : along steeper coast, the wave rolls up and down along the steeper face 4. Collapsing breaker : breaking other than above fall under this category. plunging breaker spilling breaker 36
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- 38. DELFT3D - WAVE Computes the evolution of random short crested waves in coastal regions with deep, intermediate and shallow waters and surrounding currents. Waves are described with action balance equations All information about sea surface is contained in wave energy density spectrum E(σ,Ɵ) distributing wave energy over frequencies(σ) and propagation direction (Ɵ). The spectrum used is action density spectrum, since, in the presence of currents, action density is conserved, ie. N(σ,Ɵ). It is assumed that current is uniform with respect to vertical coordinate The variables are relative frequency σ and wave direction Ɵ 𝐴𝑐𝑡𝑖𝑜𝑛 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 = 𝐸𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦38
- 39. It isgiven by 𝜕𝑁 𝜕𝑡 + 𝜕𝑐 𝑥 𝑁 𝜕𝑥 + 𝜕𝑐 𝑦 𝑁 𝜕𝑦 + 𝜕𝑐 𝜎 𝑁 𝜕𝜎 + 𝜕𝑐 𝜃 𝑁 𝜕𝜃 = 𝑆 𝜎 where 𝑐 𝑥, 𝑐 𝑦 are propagation velocity in X and Y direction 𝜕𝑁 𝜕𝑡 - local rate of change of action density with time 𝜕𝑐 𝑥 𝑁 𝜕𝑥 , 𝜕𝑐 𝑦 𝑁 𝜕𝑦 - propagation of action in geographical space 𝜕𝑐 𝜎 𝑁 𝜕𝜎 - shifting of relative frequency with respect to depth and currents 𝜕𝑐 𝜃 𝑁 𝜕𝜃 - depth induced and current induced refraction S – represents generation, dissipation and non linear wave- wave interactions wind by whitecapping bottom friction depth induced breaking 39
- 40. DELFT3D - FLOW Models 2D and 3D unsteady flow and transports resulted from tidal or meteorological forces. Used to predict flow in shallow seas, coastal waters, estuaries, lagoons, rivers and lakes. For interaction between waves and currents , coupled with DELFT3D- WAVE. If fluid is vertically homogeneous 2D approach is made. Eg storm surge, tsunami, seiches 3D modelling is used where flow field is not vertically homogeneous. Eg dispersion of cooling water in lakes, salt intrusion in estuary, thermal stratification40
- 41. PHYSICAL PROCESS Solvesunsteady shallow water equation in 2D and 3D obtained by solving Navier-Stokes 3D equation for incompressible flows The system of equations include continuity equation, horizontal equation of motion and transport equation Includes mathematical formulations to account for certain phenomena Free surface gradients Coriolis force Turbulence Transport of salt and heat Tidal force at open boundaries Radiation stresses Flow through hydraulic structures ..etc 41
- 42. FLOW MODEL UsesNavier-Stokes equation since it is used to model water flows Ma = Fg + Fp + Fv It is given by This equation is valid if density is constant or Boussinesq approximation is applied. ASSUMPTION 1: BOUSSINESQ APPROXIMATION “If density variations are small then it may be assumed to be a constant in all terms except gravitational term”42
- 43. Reynold’s averagesNS equation is obtained by ASSUMPTION 2: SHALLOW WATER 1. Horizontal length scale is much larger than the vertical length scale 2. Vertical velocity is small in comparison with horizontal velocity Thus the momentum equation in vertical direction reduces to hydrostatic pressure distribution Integrating and neglecting the atmospheric and horizontal pressure gradients we obtain 43
- 44. And alongwith this the incompressible continuity equation are called shallow water equations. 44
- 45. CONCLUSION 45 Delft3d Wavesolves action balance equation and Flow solves Navier Stokes equation for incompressible fluids. From wave model we obtain significant wave height, mean wavelength, wave steepness etc. From Flow model we obtain depth averaged velocity, bed shear stress, horizontal viscosity etc. These models are run for determining fluid velocity and pressure in a given geometry.
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