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Governing equations of Fluid Flow

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Title: Governing equations of Fluid Flow


1
FUNDAMENTAL EQUATIONS, CONCEPTS AND
IMPLEMENTATION OF NUMERICAL SIMULATION IN FREE
SURFACE FLOW
2
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.

3
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4
  • 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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6
  • The stresses are related to fluid element
    displacements by invoking the Stokes viscosity
    law for an incompressible fluid.

7
Substituting eqs. 7-10 into eqs. 4-6, we get
8
The three N-S momentum equations can be written
in compact form as

The equation of continuity for an incompressible
fluid
9
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

10
  • 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

11
  • 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.

12
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.

13
  • Reynolds Time-averaged Navier-Stokes Equations
  • These are obtained from the N-S equations and
    include the flow turbulence effect as well.

14
RNS Equations
15
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.

16
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.

17
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.

18
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

19
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

20
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,

21
2D Saint Venant Equations
22
1D Saint Venant Equation
The friction slope Sf is usually obtained from a
uniform flow formula such as Manning or chezy.
23
Simplified Equations of Saint Venant
24
Relative Weight of Each Termin SV Equation
Order of magnitude of each term In SV equation
for a flood on river Rhone
25
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.

26
Classification of Grids
  • Shape
  • Orthogonality
  • Structure
  • Blocks
  • Position of variables
  • Grid movements

27
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

28
Initial Condition
  • Values of flow depth, velocity, pressure etc must
    be assigned at the start of the calculation run.
  • Hot start

29
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.

30
Methods of Solution
  • Finite Difference Method
  • Finite element method
  • Finite volume method
  • Strategies
  • Implicit
  • Explicit
  • CFL condition
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