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COMPUTATIONAL MARINE HYDRODYNAMICS (NA40020)

Ritwik Ghoshal

Spring 2025

Contact:

ritwik@naval.iitkgp.ac.in

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Fluid-flow equations

  • Fluid dynamics is governed by conservation equations for:
    • mass;
    • momentum;
    • energy;
    • Equation of state/ Constitutive relations
  • Although there are many different physical quantities, most satisfy a single generic equation: the scalar-transport or advection-diffusion equation.

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Mass Conservation (Continuity)

  • Mass is neither created nor destroyed.
    • Rate of change of mass in cell = net inward mass flux

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Mass Conservation (Continuity)

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Mass Conservation

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Mass Conservation

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Momentum conservation

Rate of flow per unit area

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Momentum conservation

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Momentum conservation

Euler-equation

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Concept of stress

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Momentum Equation

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Momentum Equation

  • X direction

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  • y-direction

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  • z-direction

(A)

(B)

(C)

Body force

Surface force

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Energy Conservation

  • First Law of thermodynamics

Rate of change of energy (Energy in the cell + net outward energy flux) =

rate of work done on the system + net rate of heat added

E = Total Energy per unit mass

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Energy Conservation

(Per unit volume)

=

=

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Energy Conservation

  • Rate of work done on the system by surface forces acting on x direction

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Energy Conservation

  • Rate of work done on the system by the surface forces acting on x direction

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  • Rate of work done on the system by the surface forces acting on y direction

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  • Rate of work done on the system by the surface forces acting on z direction

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Energy Conservation

  • Total rate of work done by surface forces per unit volume

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Energy Conservation

  • Net rate of heat transfer to the particle due to heat flow in the x-direction

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  • In y direction

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  • In z-direction

Heat flux, q, is a flow of energy per unit of area per unit of time.

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Energy Conservation

  • Total rate of heat added to the system per unit volume

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  • Fourier Law of heat conduction

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  • Rate of heat added due to heat conduction

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Thermal conductivity

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Energy Conservation

  • Energy = internal energy (i) + kinetic energy( )+ potential

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  • Multiply Eq (A) x u , Eq (B) x v and Eq (C) x w

(D)

Source of energy

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Energy Conservation

(E)

Subtracting, (D) – (E)

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Navier-Stokes equation

  • Newtonian fluid:
    • Homogeneous and Isotropic
    • Rate of linear deformation

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    • Rate volumetric deformation

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Newtonian fluid

  • Viscous stresses are proportional to the rates of deformation

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  • For gases
  • For incompressible fluid

Stokes hypothesis

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Navier-Stokes equation derivation

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  • Substituting

New source term

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Navier-Stokes equation

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Internal energy equation

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Governing equations

Conservative form/ Eulerian form

Non-conservative form/ Lagrangian form