Proveden

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Computer Fluid Dynamics E181107
2181106
CFD4
Balancing, transport equations
Remark foils with black background could be
skipped, they are aimed to the more advanced
courses
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2013
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Balancing
CFD4

• CFD is based upon conservation laws
• conservation of mass
• conservation of momentum m.du/dtF (second
  Newtons law)
• conservation of energy dqdupdv (first law of
  thermodynamics)
• System is considered as continuum and described
  by macroscopic variables

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Transported property ?
CFD4
This table presents nomenclature of transported
properties for specific cases of mass, momentum,
energy and component transport. Similarity of
constitutive equations (Newton,Fourier,Fick) is
basis for unified formulation of transport
equations.
? Property related to unit mass P?? related to unit volume (?? is balanced in the fluid element) P diffusive molecular flux of property ? through unit surface. Units of P multiplied by m/s Constitutive laws and transport coefficients c having the same unit m2/s P - c ?P
Mass 1 ? 0
Momentum Viscous stresses Newtons law (kinematic viscosity)
Total energy Enthalpy E h ?E ?h?cpT Heat flux Fouriers law (temperature diffusivity)
Mass fraction of a component in mixture ?A ?A??A diffusion flux of component A Ficks law (diffusion coefficient)
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Integral balancing - Gauss
CFD4
Control volume balance expressed by Gauss theorem
accumulation flux through boundary
d?
?
Divergence of ? projection
of ? to outer normal

• Variable ? can be
• Vector (vector of velocity, momentum, heat flux).
  Surface integral represents flux of vector in the
  direction of outer normal.
• Tensor (tensor of stresses). In this case the
  Gauss theorem represents the balance between
  inner stresses and outer forces acting upon the
  surface, in view of the fact that
  is the vector of forces acting on the oriented
  surface d?.

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Fluid ELEMENT fixed in space
CFD4

• Motion of fluid is described either by
• Lagrangian coordinate system (tracking individual
  particles along streamlines)
• Eulerian coordinate system (fixed in space, flow
  is characterized by velocity field)

Balances in Eulerian description are based upon
identification of fluxes through sides of a box
(FLUID ELEMENT) fixed in space. Sides if the box
in the 3D case are usually marked by letters W/E,
S/N, and B/T.
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Mass balancing (fluid element)
CFD4
Accumulation of mass
Mass flowrate through sides W and E
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Mass balancing
CFD4
Continuity equation written in index notation
Continuity equation written in symbolic form (the
so called conservative form)
Symbolic notation is independent of coordinate
system. For example in the cylindrical coordinate
system (r,?,z) this equation looks different
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Fluid PARTICLE / ELEMENT
CFD4
Time derivatives -at a fixed place -at a moving
coordinate system
In other words Different time derivatives
distinguish between time changes seen by an
observer that is steady (??/?t), an observer
moving at a prescribed velocity (d?/dt), observer
translated with the fluid particle (D?/Dt -
material derivative) or moving and rotating with
the fluid particle (??/?t - Jaumann derivatives).
Modigliani
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Fluid PARTICLE / ELEMENT
CFD4
Fluid element a control volume fixed in space
(filled by fluid). Balancing using fluid elements
results to the conservative formulation,
preferred in the CFD of compressible fluids
Fluid particle group of molecules at a point,
characterized by property ? (related to unit
mass). Balancing using fluid particles results to
the nonconservation form. Rate of change of
property ?(t,x,y,z) during the fluid particle
motion
Material derivative
Projection of gradient to the flow direction
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Balancing ?? in fixed Fluid Element
CFD4
Accumulation ? in FE Outflow of ? from FE
by convection
intensity of inner sources or diffusional fluxes
across the fluid element boundary
This follows from the mass balance
These terms are cancelled
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Balancing ?? in fixed Fluid Element
CFD4
Conservation form (?? balance)
Nonconservation form
Rate of ? increase of fluid particle
Flowrate of ? out of Fluid element
Accumulation of ? inside the fluid element
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Integral balance ?? in Fluid Element
CFD4
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Moving Fluid element
CFD4
moving control volume
Fluid element VdV at time tdt
velocity of particle (flow)
velocity of FE
Integral balance of property ?
Fluid element V at time t
Amount of ? in new FE at tdt
Convection inflow at relative velocity
Diffusional inflow of ?
Terms describing motion of FE are canceled
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Moving Fluid element
CFD4
You can imagine that the FE moves with fluid
particles, with the same velocity, that it
expands or contracts according to changing
density (therefore FE represents a moving cloud
of fluid particle), however the same resulting
integral balance is obtained as for the case of
the fixed FE in space
Internal volumetric sources of ? (e.g. gravity,
reaction heat, microwave)
Diffusive flux of ? superposed to the fluid
velocity u
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Moving Fluid element (Reynolds theorem)
CFD4
You can imagine that the control volume moves
with fluid particles, with the same velocity,
that it expands or contracts according to the
changing density (therefore it represents a
moving cloud of fluid particles), however The
same resulting integral balance is obtained in a
moving element as for the case of the fixed FE in
space
Diffusive flux of ? superposed to the fluid
velocity u
Internal volumetric sources of ? (e.g. gravity,
reaction heat, microwave)
Reynolds transport theorem
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Integral/differential form
CFD4
All integrals can be converted to volume integral
s (Gauss theorem again)
Integral form
Differential form