What are Essentials Topics of Fluid Mechanics for CFD Modeling
by
Dr. Sharad N. Pachpute
1. Introduction of Fluid Flows
 Fluid mechanics deals with flow physics for liquids, gases, or any flowing medium by considering external forces and their effect on the deformation of fluid flows
 It is an essential subject for many engineering and applied sciences
 Mechanical, chemical, civil engineering and environmental science
 Geophysics, Meteorology, Oceanography
 Astrophysics, and Biology
 Due to importance of fluid in real life, nature and industrial applications, this subject has been important many decades for scientists, engineers and astrophysicist
1.1 Major Applications of fluid Flows
 Any application relevant to fluid flow
 Automobile, chemical and processing industries
 Energy sector
 Transportation Sector
 Environmental Fluid Flows
 Weather or meteorology
 Biological Applications
1.2 Outline of Fluid Mechanics
 Fluid mechanics is divided into statics and dynamics
 In fluid static, the result force and moments are zero
 The fluid dynamics which deals with flows in motions, resultant forces or moments can not be zero
 Fluid flows deal with gas and liquids
2. The Concept of Field
 Any physical quantity which generates or carry the force in the fluid is called a field
 Some basic properties of fluid (like density, viscosity, surface tension etc.) with surroundings conditions (walls, moving surfaces etc.) are responsible for the occurrence of flow field.
 The fields can be scalar and vector or both in nature
 Scalar fields are quite easier to understand with than vector fields
 Most of resources, energy and work are in the form scalars. These scalars can be used to create vector fields like velocity, force and stress
 First step is to analyze fluid flows using bas properties of scalar and vector fields
 During flowing process, the properties of working medium changes with a time dependent. Such flows are unsteady
 In fluid flows, physical quantities are function of time (t) and space (x)
 Vector : velocity, stresses acting on fluid
 Scalar: Thermodynamic properties of fluid such pressure, temperature, density , enthalpy etc.
Basic Physical Quantities in Fluid Mechanics
2.1 Scalar (Zerothorder tensor)
 Scalar has a definite magnitude but not a definite direction
 Example: Density, Pressure, Temperature, Concentration
 If N is the order of tensor Total component in 3 directions (X, Y and Z directions) : 3^{N}
 Total component for scalar: 3^{0 }= 1
2.2 Vector (Firstorder tensor)
 A firstorder tensor consists of definite magnitude, with three components and a definite different three directions
 Example: velocity, force, heat flux etc.
 Heat flux vector
 The spread of Fire in a Room is an example of a heat flux vector, a firing rate is not same in all direction
 ▼ is Differential Operator, then the heat flux vector
Here, del (▼)is a differential operator
2.3 Second order Tensor
 A second order tensor has nine definite components and nine definite directions
 Example: Velocity gradient, Stress, Strain, thermal conductivity
 Velocity Gradient:
A velocity gradient consists of deformation and rotation tensor.
 Stress Tensor:
 Thermal conductivity:
 For isotropic materials has only one component and it is a scalar quantity. The thermal conductivity is same in all direction.
 The thermal conductivity is different in all direction which is called an anisotropic material. It has 9 different components.
3.Hydrostatics
 Field variables of pressure and gravity and their effect was presented by the Pascal
 To start with simple Fluid Mechanics traditionally with hydrostatics, only pressure and external force (e.g. gravity) are essential
Where p is the pressure fields, Φ is the external force field potential and ρ is density.
 The static pressure is independent on shapes of water storage
 Due to high pressure at the bottom, the velocity of jet is higher at the bottom and lower for the top jet
4. Physical Properties of Tensor
4.1 Scalar is a mother of Vector Field
 Start from the path integral Work which is equal to the dot product of force and displacement vectors
 Vector is a gradient of Scalar
 The gradient in scalar creates a vector field
System conversions for Gradient for different coordinate system
 Cartesian coordinate
 Cylindrical coordinate
 Spherical coordinate
4.2 Gauss’s Divergence Theorem
 Divergence (Gauss’s) theorem
 This theorem is useful for the expansion or contraction (divergence or convergence) of material inside a volume which is equal to what entering and leaving across the boundary of material
 The divergence theorem is mainly used
 to convert a surface integral into a volume integral.
 to convert a volume integral to a surface integral.
4.3 The Divergence Effect in A Fluid flow
 A velocity field is the major vector field essential to describe a flowing fluid
 The divergence is used to measure the expansion and contraction in fluid flows
 A divergence of vector field can have a positive or negative value
 Diverging Flows: a vector field whose divergence is positive
 Converging Flows: a vector field whose divergence is negative
 Solenoidal Flows: a vector field whose divergence is zero
 Application of convergence and divergence for atmospheric flow over a ground
4.4 Types of NonSolenoid Flows
 Incompressible flow is vector dominated as variation in scalar (density) is negligible
 Example: flow over a car
 For compressible Flow, both vector and scalar are essential for flow analysis
 Example: flow over a fighter jet or space shuttle
5. Mass Conservation or Continuity Equation
5.1 Continuity equation in differential form
5.1 One Dimensional Form Continuity equation
 For compressible flow: Mass flow rate at any cross section = ρAV
 For in compressible flow: flow rate at any cross section , Area*average Velocity =constant
6. Curl of a vector field
 Curl of a vector field is a measure of flow Circulation which is defined as
 Circulation is the amount of force that pushes along a closed boundary or path
 Curl measure the amount of circulation per unit area, circulation density, or rate of rotation (twisting action at a single point)
 Curl has a magnitude equal to the maximum “circulation” at each point
 Its direction is oriented perpendicularly to this plane of circulation for each point
6.1 The Curl of Velocity Field
 A vorticity vector is defined as the curl of a velocity vector
 Example: Atmospheric flow is highly rotational
6.2 Rotational vs Irrotational flow
 Flows with vorticity are called rotational flows
 Flows without vorticity (zero value of curl of velocity) are said to be irrotational flow
 Cyclonic Flows:
 AntiCyclonic Flows:
Vorticity Transport Equation
 Vortex Stretching is in turbulent flow presented by the Vorticity Transport Equation
 In this equation, there are two terms o the RHS: viscous torque and vortex stretching term
 Vortex lines possess vorticity which is defined as the curl of the local velocity vector and its magnitude is indicative of the angular rotation of fluid elements about the local ‘axis’.
 Largeeddy has Reynolds number in terms of length scale hence it is almost inviscid and Largeeddy angular momentum is conserved as it does not have any viscous torque acting on it
 The Energy Cascade is observed. Large eddies have most of the turbulence kinetic energy (TKE) which is transferred to smaller eddy. Hence, the angular velocity for the large eddy is decreasing.
 To maintain their angular momentum conservation, the moment of inertia (MOI) of the large eddies increases and large eddies are stretched.
 Vortex lines become distorted from the induced velocities of the bigger eddies.
 This phenomenon can be explained using the vorticity transport equation
7. Forces Acting on Fluid Flows
7.1 Types of forces acting on Fluid Flows
1) Body (Volumetric) Force:

 Gravity Force
 Electrical Force:
 Dielectric force
 Coulomb force
 Electrostriction force
 Magnetic: Lorentz force
2) Normal Surface Force: Pressure, surface tension
3) Combined Normal and Tangential surface Forces

 Force due to thermodynamic Effects: Buoyancy forces or surface
 PhysioChemical or concentrationbased forces: Environmental or Biofluid mechanics
7.2 Stress Acting on Fluid Flows is mother of Surface Force
 Force acting on surface of a fluid element is the divergence of the Stress tensor field
 The stress tensor is usually divided into its normal and shear stress parts
8. Momentum Acting on Fluid Flows
 The momentum equation is integrated over a system to get drag, lift forces forces due to pressure and viscosity
 Forces due to viscous forces, gravitational forces act on fluid
 Depending on application, Other forces can affect fluid flows such as electrodynamic, electrostatic, and magnetic forces
 Newton’s second law of motion
8.1 Flows Driven by Gravitational Force
 Water fall under gravity is an example of flow driven by a gravitational force
 The momentum equation is given below
 The gravitational force acting on a fluid system is a volume integral of the product of mass and acceleration due to gravity
8.2 Flows Driven by Electromagnetic Force
 In such cases the forces are identified as electrodynamic and electrostatic
 Supplying high voltage between the Collecting Electrode and Discharge Electrode generates a Corona Discharge that produces minus ions
8.3 MagnetoHydrodynamics Flows
 Magnetofluid dynamics: Fluid Dynamics of electricallyconducting fluids and their interactions with magnetic fields.
 There is an interaction of three fields: Magnetic field, Electric field and Plasma flow
 Magnetic field flow (B) can produce Electric field (J).
 The interaction of magnetic and electrical fields can produce Lorentz Force to accelerate the flow
 The principle of Magnetohydrodynamics dynamics is shown below
 Using electric and magnetic field, the flow can be accelerated with an increase in flow
9. Differential Form of Momentum Conservation Equations for Fluid Flows
 Newton’s Second Law to a system of fluid flow
 Newton’s Second law to a control volume of fluid element
 Stress acting on fluid is a function for strain rate and rotation tensor
10. Viscosity of Fluid
10.1 Newtonian (linear) Viscous Fluid
 Newton Law of Viscosity: for straight, parallel and uniform flow, the shear stress , between layers is proportional to the velocity gradient in the direction perpendicular to the layers
 Shear stress = Viscosity * Velocity Gradient
 Newton’s Law of viscosity provides more economical solution to complex flows
 Example: Air, water
10.2 Nonlinear viscosity
 Most of fluid does not show a linear variation of shear stress with the rate of strain
 For such nonlinear viscous fluid is called NonNewtonian fluid
 Viscosity changes due to change in force, pressure gradient, temperature or concentration of liquid or solid particles
 The following figure shows the effect of nonlinear viscosity on velocity profiles
 Example of NonNewtonian Fluids: Honey, tooth paste, Ketch up, slurry flow, salt solution, corn starch, blood, melted butter, shampoo, starch suspension
11. Euler’s Equation
 For the memoir of 1755, Euler obtained the momentum equations of fluid flow
 He equated the product of mass and acceleration aof fluid element to the resultant of the pressures and external body forces acting the fluid element
12. Bernoulli’s Equation
 Bernoulli integrated the Euler equation for a steady, incompressible and inviscid flow
 Bernoulli equation shows that the total energy of fluid for a particular section is constant
13. Boundary Layer
 When fluid flows over a surface, a very thin region is formed adjacent to the nonslip wall. This results in the formation of gradient in velocity normal to the wall
 When fluid flows over a sudden down portion. The fluid separates due to positive pressure gradient
Flow Over A curved Surface
 Two flow regions are formed over a curved surface
 In favorable pressure gradient, the fluid flow accelerates
 In adverse pressure gradient, the fluid flow separates due to wake region
14. Laminar vs Turbulent Flow
 Relative magnitude of inertial and viscous terms is called as Reynolds number
 Increasing velocity of fluid flow (Reynolds number), the nonlinearity inertia term of NavierStokes(NS) equations that causes disturbances in fluid flows
For more detailed understanding of turbulent flow, Read Turbulent Flow Physics and Methods of Analysis
The basics have been explained nicely.
Thanks Dr.Alankrita