Governing equations of Fluid Flow

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FUNDAMENTAL EQUATIONS, CONCEPTS AND
IMPLEMENTATION OF NUMERICAL SIMULATION IN FREE
SURFACE FLOW
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Governing Equations of Fluid Flow

• Navier-Stokes Equations
• A system of 4 nonlinear PDE of mixed hyperbolic
  parabolic type describing the fluid hydrodynamics
  in 3D.
• Three equations of conservation of momentum in
  cartesian coordinate system plus equation of
  continuity embodying the principal of
  conservation of mass.
• Expression of Fma for a fluid in a differential
  volume.

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• The acceleration vector contains local
  acceleration and covective terms
• The force vector is broken into a surface force
  and a body force per unit volume.
• The body force vector is due only to gravity
  while the pressure forces and the viscous shear
  stresses make up the surface forces.

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• The stresses are related to fluid element
  displacements by invoking the Stokes viscosity
  law for an incompressible fluid.

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Substituting eqs. 7-10 into eqs. 4-6, we get
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The three N-S momentum equations can be written
in compact form as

The equation of continuity for an incompressible
fluid
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Turbulence

• The free surface flows occurring in nature is
  almost always turbulent. Turbulence is
  characterized by random fluctuating motion of
  the fluid masses in three dimensions. A few
  characteristic of the turbulence are
• 1. Irregularity
• Turbulent flow is irregular, random and chaotic.
  The flow consists of a spectrum of different
  scales (eddy sizes) where largest eddies are of
  the order of the flow geometry (i.e. flow depth,
  jet width, etc). At the

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• other end of the spectra we have the smallest
  eddies which are by viscous forces (stresses)
  dissipated into internal energy.
• 2. Diffusuvity The turbulence increases the
  exchange of momentum in flow thereby increasing
  the resistance (wall friction) in internal flows
  such as in channels and pipes.
• 3. Large Reynolds Number Turbulent flow occurs at
  high Reynolds number. For example, the transition
  to turbulent flow in pipes occurs at NR2300 and
  in boundary layers at NR100000

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• 4.Three-dimensional Turbulent flow is always
  three-dimensional. However, when the equations
  are time averaged we can treat the flow as
  two-dimensional.
• 5. Dissipation Turbulent flow is dissipative,
  which means that kinetic energy in the small
  (dissipative) eddies are transformed into
  internal energy. The small eddies receive the
  kinetic energy from slightly larger eddies. The
  slightly larger eddies receive their energy from
  even larger eddies and so on. The largest eddies
  extract their energy from the mean flow. This
  process of transferred energy from the largest
  turbulent scales (eddies) to the smallest is
  called cascade process.

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Turbulence

• . The random , chaotic nature of turbulence is
• treated by dividing the instantaneous
• values of velocity components and
• pressure into a mean value and a
• fluctuating value, i.e.
• Why decompose variables ?
• Firstly, we are usually interested in the mean
  values rather than the time histories. Secondly,
  when we want to solve the Navier-Stokes equation
  numerically it would require a very fine grid to
  resolve all turbulent scales and it would also
  require a fine resolution in time since turbulent
  flow is always unsteady.

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• Reynolds Time-averaged Navier-Stokes Equations
• These are obtained from the N-S equations and
  include the flow turbulence effect as well.

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RNS Equations
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Reynold Stresses

• The continuity equation remains unchanged except
  that instantaneous velocity components are
  replaced by the time-averaged ones. The three
  momentum equations on the LHS are changed only to
  the extent that the inertial and convective
  acceleration terms are now expressed in terms of
  time averaged velocity components. The most
  significant change is that on the LHS we now have
  the Reynold stresses. These are time-averaged
  products of fluctuating velocity components and
  are responsible for considerable momentum
  exchange in turbulent flow.

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Closure Problem

• 3 velocity components, one pressure and 6 Reynold
  stress terms 10 unknowns
• No. of equations4
• As No. of unknowns gtNo. of equations, the problem
  is indeterminate. One need to close the problem
  to obtain a solution.
• The turbulence modeling tries to represent the
  Reynold stresses in terms of the time-averaged
  velocity components.

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Turbulence Models

• Boussinesq Model
• An algebraic equation is used to compute a
  turbulent viscosity, often called eddy viscosity.
  The Reynolds stress tensor is expressed in terms
  of the time-averaged velocity gradients and the
  turbulent viscosity.

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k-e Turbulence Model
Two transport equations are solved which
describe the transport of the turbulent kinetic
energy, k and its dissipation, e. The eddy
viscosity is calculated as

• the Reynold stress tensor is calculated via the
  Boussinesq approximation

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RNS Equations and River Flow Simulation

• RNS equations are seldom used for the river flow
  simulation. Reasons being
• High Cost
• Long Calculation time
• Flow structure
• Method of choice for flows in rivers, streams and
  overland flow is 2D and 1D Saint Venant equations
  or Shallow water equations

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2D Saint Venant Equation

• Obtained from RNS equations by depth-averaging.
• Suitable for flow over a dyke, through the
  breach, over the floodplain.
• Assumptions hydrostatic pressure distribution,
  small channel slope,

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2D Saint Venant Equations
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1D Saint Venant Equation
The friction slope Sf is usually obtained from a
uniform flow formula such as Manning or chezy.
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Simplified Equations of Saint Venant
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Relative Weight of Each Termin SV Equation
Order of magnitude of each term In SV equation
for a flood on river Rhone
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Calculation Grid

• Breaking up of the flow domain into small cells
  is central to CFD. Grid or mesh refers to the
  totality of such cells.
• In Open channel flow simulation the vertices of a
  cell define a unique point
• (x,y,,z)
• The governing equations are discretized into
  algebraic equations and solved over the volume of
  a cell.

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Classification of Grids

• Shape
• Orthogonality
• Structure
• Blocks
• Position of variables
• Grid movements

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Boundary Conditions

• Inflow b. c
• If Frlt1, specify discharge or velocity.
• If Frgt1, specify discharge or velocity and depth
• Outflow b.c
• Zero depth gradient or Newmann b.c
• Specify depth
• Specify Fr1

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Initial Condition

• Values of flow depth, velocity, pressure etc must
  be assigned at the start of the calculation run.
• Hot start

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Wall Boundary Condition

• No slip condition require very fine meshing
  adjacent to the wall requiring lot of CPU time.
  Flow close to the wall is not resolved but wall
  laws derived from the universal velocity
  distribution are used.

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Methods of Solution

• Finite Difference Method
• Finite element method
• Finite volume method
• Strategies
• Implicit
• Explicit
• CFL condition