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Numerical Hydraulics

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Numerical Hydraulics

Lecture 1: The equations

Wolfgang Kinzelbach with

Marc Wolf and

Cornel Beffa

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

- 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

- 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

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

- 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

- 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:

Time interval [t, t+Dt]

Storage is change in

extensive quantity

Cross-sectional area A

Volume V = ADx

Fluxin

Fluxout

Gain/loss from volume

sources/sinks

Dx

x+Dx

x

Conservation law in words:

Division by DtDxA yields:

In the limit Dt, Dx to 0:

General principle in 3D

or

Density assumed constant!

Storage can be seen as change

in intensive quantity

Time interval [t, t+Dt]

V=ADx

Dx

x+Dx

x

Conservation equation for water volume

In the limit

or

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

- 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

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

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:

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.

- 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

- Water is in a very good approximation a Newtonian fluid:
strain tensor a tensor of deformation

- Deformations comprise shear, rotation and compression

y

y

y

x

x

x

rotation

shearing

compression

Relative volume

change per time

Dy

Dx

The shear rate is

The angular velocity of rotation is

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 i=1,2,3

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

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).

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.

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

- 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

- 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.

- m = E, f = r(e+u²/2) inner+kinetic energy per unit volume, j = fu=r(e+u²/2)u,
s work done on the control volume by volume and surface forces, dissipation by heat conduction

- 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

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

Advection-diffusion equation for passive scalar

transport in microscopic view.