Essentials of Fluid Mechanics for CFD Engineers

What are Essentials Topics of Fluid Mechanics for CFD Modeling


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 (Zeroth-order 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) : 3N
  • Total component for scalar: 30 = 1

2.2 Vector (First-order tensor)

  • A first-order 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.


  • 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 co-ordinate

  • Cylindrical co-ordinate

  • Spherical co-ordinate

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 Non-Solenoid 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

Continuity Equation | Fluid mechanics, Continuity, Mass flow rate

  • 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:

  • Anti-Cyclonic 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’.
  • Large-eddy has Reynolds number in terms of length scale hence it is almost inviscid and Large-eddy 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 thermo-dynamic Effects: Buoyancy forces or surface
    • Physio-Chemical or concentration-based forces: Environmental or Bio-fluid 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 Magneto-Hydrodynamics Flows

  • Magneto-fluid dynamics: Fluid Dynamics of electrically-conducting 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 Magneto-hydrodynamics 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 Non-linear viscosity

  • Most of fluid does not show a linear variation of shear stress with the rate of strain
  • For such non-linear viscous fluid is called Non-Newtonian fluid

Apparent viscosity - Wikipedia

  • Viscosity changes due to change in force, pressure gradient, temperature or concentration of liquid or solid particles

12.4: Viscosity and Laminar Flow; Poiseuille's Law - Physics LibreTexts

  • The following figure shows the effect of non-linear viscosity on velocity profiles

Viscosity of Newtonian and Non-Newtonian Fluids

  • Example of Non-Newtonian 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 non-slip 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  non-linearity inertia term of Navier-Stokes(NS) equations that causes disturbances in fluid flows


For more detailed understanding of turbulent flow, Read  Turbulent Flow Physics and Methods of Analysis 

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