This document provides an overview of fluid mechanics and wave hydrodynamics. It defines key terms like fluids, waves, currents and tides. It describes the fundamental equations of fluid mechanics including continuity, Euler, Navier-Stokes and Bernoulli's equations. It also covers topics like classification of flows, wave generation, propagation and transformation processes like refraction, reflection and diffraction. The document is intended as an introduction to fluid mechanics and wave concepts for students of coastal and harbour engineering.
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
Abstract: Vorticity is a curl of fluid velocity and the Absolute ξ_a = ξ +f where ξ is the relative
vorticity while f is the coriolis parameter. North word moving air acquires and increasing value for f
and hence decreasing value for ξ. The concept of circulation is related that of vorticity, and it has a
number of important applications horizontal closed curves in either hemisphere will have positive
circulation vorticity is negative while the curvature is negative the vorticity is positive.
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
Abstract: Vorticity is a curl of fluid velocity and the Absolute ξ_a = ξ +f where ξ is the relative
vorticity while f is the coriolis parameter. North word moving air acquires and increasing value for f
and hence decreasing value for ξ. The concept of circulation is related that of vorticity, and it has a
number of important applications horizontal closed curves in either hemisphere will have positive
circulation vorticity is negative while the curvature is negative the vorticity is positive.
Fluid flow-Mention fluid properties such as viscosity, compressibility and surface tension of fluids.
Hydrostatics (Fluidststics) influencing fluid flow.
Fluid dynamics‐ Bernoulli’s theorem, flow of fluids in pipes, laminar and turbulent flow.
A fluid is a substance that continually deforms (flows) under an applied shear stress.
Fluids are a subset of the phases of matter and include liquids, gases.
Fluid flow may be defined as the flow of substances that do not permanently resist distortion
The subject of fluid flow can be divided into fluid static's and fluid dynamics
FLUID STATICS
Consider a column of liquid with two openings Which are provided at the wall of the vessel at different height
The rate of flow through these openings are different due to the pressure exerted at the different heights are different
Consider a stationary column the pressure P is acting on the surface of the fluid, column is maintained at constant pressure by applying pressure
The force acting below and above the point 1 are evaluated
Substituting the force with pressure x area of cross section in the above equation
B.TECH. DEGREE COURSE
SCHEME AND SYLLABUS
(2002-03 admission onwards)
MAHATMA GANDHI UNIVERSITY,mg university, KTU
KOTTAYAM
KERALA
Module 1
Introduction - Proprties of fluids - pressure, force, density, specific weight, compressibility, capillarity, surface tension, dynamic and kinematic viscosity-Pascal’s law-Newtonian and non-Newtonian fluids-fluid statics-measurement of pressure-variation of pressure-manometry-hydrostatic pressure on plane and curved surfaces-centre of pressure-buoyancy-floation-stability of submerged and floating bodies-metacentric height-period of oscillation.
Module 2
Kinematics of fluid motion-Eulerian and Lagrangian approach-classification and representation of fluid flow- path line, stream line and streak line. Basic hydrodynamics-equation for acceleration-continuity equation-rotational and irrotational flow-velocity potential and stream function-circulation and vorticity-vortex flow-energy variation across stream lines-basic field flow such as uniform flow, spiral flow, source, sink, doublet, vortex pair, flow past a cylinder with a circulation, Magnus effect-Joukowski theorem-coefficient of lift.
Module 3
Euler’s momentum equation-Bernoulli’s equation and its limitations-momentum and energy correction factors-pressure variation across uniform conduit and uniform bend-pressure distribution in irrotational flow and in curved boundaries-flow through orifices and mouthpieces, notches and weirs-time of emptying a tank-application of Bernoulli’s theorem-orifice meter, ventury meter, pitot tube, rotameter.
Module 4
Navier-Stoke’s equation-body force-Hagen-Poiseullie equation-boundary layer flow theory-velocity variation- methods of controlling-applications-diffuser-boundary layer separation –wakes, drag force, coefficient of drag, skin friction, pressure, profile and total drag-stream lined body, bluff body-drag force on a rectangular plate-drag coefficient for flow around a cylinder-lift and drag force on an aerofoil-applications of aerofoil- characteristics-work done-aerofoil flow recorder-polar diagram-simple problems.
Module 5
Flow of a real fluid-effect of viscosity on fluid flow-laminar and turbulent flow-boundary layer thickness-displacement, momentum and energy thickness-flow through pipes-laminar and turbulent flow in pipes-critical Reynolds number-Darcy-Weisback equation-hydraulic radius-Moody;s chart-pipes in series and parallel-siphon losses in pipes-power transmission through pipes-water hammer-equivalent pipe-open channel flow-Chezy’s equation-most economical cross section-hydraulic jump.
Understand the physical mechanism of convection and its classification.
Visualize the development of velocity and thermal boundary layers during flow over surfaces.
Gain a working knowledge of the dimensionless Reynolds, Prandtl, and Nusselt numbers.
Distinguish between laminar and turbulent flows, and gain an understanding of the mechanisms of momentum and heat transfer in turbulent flow.
Derive the differential equations that govern convection on the basis of mass, momentum, and energy balances, and solve these equations for some simple cases such as laminar flow over a flat plate.
Non dimensionalize the convection equations and obtain the functional forms of friction and heat transfer coefficients.
Use analogies between momentum and heat transfer, and determine heat transfer coefficient from knowledge of friction coefficient.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
3. INTRODUCTION
The behaviour of 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
8. FORCES ACTING ON FLUIDS 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 most of 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 rise and 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 flow of 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 THE FORCES 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 is an 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 OCEAN WAVES
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 OCEAN WAVES
CLASSIFICATION OF OCEAN WAVES
As per water depth As per originAs per apparent shape
17
18. FUNDAMENTALS OF FLUID FLOW
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
It is 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 and we 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 incompressible fluids,
𝜕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 OF MOTION
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’s equation 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 acceleration has 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
𝜕2 𝑣
𝜕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’s equation 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 equation is 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 FLOW PROBLEMS
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
31. PARAMETERS TO DEFINE A 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 WAVE THEORY
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 LAPLACE EQUATION
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
As wave 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
Total reflection 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
A phenomena 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
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 is given 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
Solves unsteady 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
Uses Navier-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 averages NS 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 along with this the incompressible continuity equation
are called shallow water equations.
44
45. CONCLUSION
45
Delft3d Wave solves 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.