Numerical Hydraulics

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Numerical Hydraulics
Lecture 1 The equations

• Wolfgang Kinzelbach with
• Marc Wolf and
• Cornel Beffa

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Contents of course

• The equations
• Compressible flow in pipes
• Numerical treatment of the pressure surge
• Flow in open channels
• Numerical solution of the St. Venant equations
• Waves

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Basic equations of hydromechanics

• The basic equations are transport equations for
• Mass, momentum, energy
• General treatment
• Transported extensive quantity m
• Corresponding intensive quantity f (m/Volume)
• Flux j of quantity m
• Volume-sources/sinks s of quantity m

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Extensive/intensive quantities

• Extensive quantities are additive
• e.g. volume, mass, energy
• Intensive quantities are specific quantities,
  they are not additive
• e.g. temperature, density
• Integration of an intensive quantity over a
  volume yields the extensive quantity

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Balance over a control volume
unit normal to surface
boundary G
flux
volume W
Balance of quantity m
minus sign, as orientation of normal to surface
and flux are in opposite direction
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Differential form

• Using the Gauss integral theorem
• we obtain
• The basic equations of hydromechanics follow
  from this equation for special choices of m, f, s
  and j

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Continuity equation

• m M (Mass), f r (Density), j ur (Mass flux)
  yields the continuity equation for the mass
• For incompressible fluids (r const.) we get
• For compressible fluids an equation of state is
  required

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Other approach General principle in 1D
Time interval t, tDt
Storage is change in extensive quantity
Cross-sectional area A Volume V ADx
Fluxin
Fluxout
Gain/loss from volume sources/sinks
Dx
xDx
x
Conservation law in words
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General principle in 1D
Division by DtDxA yields
In the limit Dt, Dx to 0
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General principle in 3D
or
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Mass balance in 1D
Density assumed constant!
Storage can be seen as change in intensive
quantity
Time interval t, tDt
VADx
Dx
xDx
x
Conservation equation for water volume
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Mass balance in 1D continued
In the limit
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Generalization to 3D
or
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Essential derivative
The total or essential derivative of a
time-varying field quantity is defined by
The total derivative is the derivative along the
trajectory given by the velocity vector
field Using the total derivative the continuity
equation can be written in a different way
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Momentum equation (equation of motion)

• Example momentum in x-direction
• m Mux (x-momentum), f rux (density),
  (momentum flux), sx force density (volume- and
  surface forces) in x-direction inserted into the
  balance equation yields the x-component of the
  Navier-Stokes equations

pressure force gravity force friction
force per unit volume
In a rotating coordinate system the
Coriolis-force has to be taken into account
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Momentum equation (equation of motion)

• Using the essential derivative and the continuity
  equation we obtain
• The x-component of the pressure force
• per unit volume is
• The x-component of gravity
• per unit volume is
• The friction force per unit volume will be
  derived later

Newton Ma F
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Momentum equation (equation of motion)

• In analogy to the x-component the equations for
  the y- and z-component can be derived. Together
  they yield a vector equation

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Momentum equation (equation of motion)

• Writing out the essential derivative we get
• The friction term fR depends on the rate of
  deformation. The relation between the two is
  given by a material law.

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Friction force
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Friction force

• The strain forms a tensor of 2nd rank The normal
  strain only concerns the deviations from the mean
  pressure p due to friction deviatoric stress
  tensor. The tensor is symmetric.
• The friction force per unit volume is

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The material law

• Water is in a very good approximation a Newtonian
  fluid
• strain tensor a tensor of deformation
• Deformations comprise shear, rotation and
  compression

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Deformation
shearing
compression
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Compression
Relative volume change per time
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Shearing and rotation
Dy
Dx
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Shearing and rotation
The shear rate is
The angular velocity of rotation is
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General tensor of deformation
rotation and shear components
Symmetric part (shear velocity) contains the
friction
Anti-symmetric part (angular velocity of
rotation) frictionless
x,y,z represented by xi with i1,2,3
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Material law according to NewtonMost general
version
with
Three assumptions Stress tensor is a linear
function of the strain rates The fluid is
isotropic For a fluid at rest must be
zero so that hydrostatic pressure results
h is the usual (first) viscosity, l is called
second viscosity
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Resulting friction term for momentum equation
Friction force on volume element
Compression force due to friction
It can be shown that
If one assumes that during pure compression the
entropy of a fluid does not increase (no
dissipation).
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Navier-Stokes equations
Under isothermal conditions (T const.) one has
thus together with the continuity equation 4
equations for the 4 unknown functions ux, uy, uz,
and p in space and time. They are completed by
the equation of state for r(p) as well as
initial and boundary conditions.
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Vorticity

• The vorticity is defined as the rotation of the
  velocity field

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Vorticity equation

• Applying the operator to the Navier-Stokes
  equation and using various vector algebraic
  identities one obtains in the case of the
  incompressible fluid
• The Navier-Stokes equation is therefore also a
  transport equation (advection-diffusion equation)
  for vorticity.
• Other approach transport equation for angular
  momentum

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Vorticity equation

• Pressure and gravity do not influence the
  vorticity as they act through the center of mass
  of the mass particles.
• Under varying density a source term for vorticity
  has to be added which acts if the gravitational
  acceleration is not perpendicular to the surfaces
  of equal pressure (isobars).
• In a rotating reference system another source
  term for the vorticity has to be added.

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Energy equation

• m E, f r(eu²/2) innerkinetic energy per
  unit volume, j f?ur(eu²/2)?u,
• s work done on the control volume by volume
  and surface forces, dissipation by heat conduction

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Energy equation

• The new variable e requires a new material
  equation. It follows from the equation of state
• e e(T,p)
• In the energy equation, additional terms can
  appear, representing adsorption of heat radiation

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Solute transport equation

• m Msolute, f c concentration,
• (advection and diffusion), s solute sources
  and sinks

Advection-diffusion equation for passive scalar
transport in microscopic view.