Numerical Hydraulics Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa Lecture 1: The equations

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

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

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

Balance over a control volume unit normal to surface boundary  volume  flux Balance of quantity m: minus sign, as orientation of normal to surface and flux are in opposite direction

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

Continuity equation m = M (Mass),  =  (Density), j = u  (Mass flux) yields the continuity equation for the mass: For incompressible fluids (  = const.) we get: For compressible fluids an equation of state is required:

Other approach: General principle: in 1D x xx Flux in Flux out Storage is change in extensive quantity Gain/loss from volume sources/sinks Conservation law in words: Cross-sectional area A Volume V = A  x x+  x Time interval [t, t+  t]

General principle in 1D Division by  t  xA yields: In the limit  t,  x to 0:

General principle in 3D or

Mass balance: in 1D x xx Storage can be seen as change in intensive quantity Conservation equation for water volume x+  x Time interval [t, t+  t ] Density assumed constant! V=A  x

Mass balance: in 1D continued In the limit

Generalization to 3D or

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

Momentum equation (equation of motion) Example: momentum in x-direction m = Mu x (x-momentum),  =  u x (density), (momentum flux), s x force density (volume- and surface forces) in x-direction inserted into the balance equation yields the x- component of the Navier-Stokes equations: In a rotating coordinate system the Coriolis-force has to be taken into account pressure force gravity force friction force per unit volume

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 Momentum equation (equation of motion) Newton: Ma = F

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 f R depends on the rate of deformation. The relation between the two is given by a material law. Momentum equation (equation of motion)

Friction force

The strain forms a tensor of 2nd rank T he 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

The material law Water is in a very good approximation a Newtonian fluid: strain tensor  tensor of deformation Deformations comprise shear, rotation and compression

Deformation x y rotation x y shearing x y compression

Relative volume change per time Compression

Shearing and rotation xx yy

The shear rate is The angular velocity of rotation is Shearing and rotation

x,y,z represented by x i with i=1,2,3 Anti-symmetric part (angular velocity of rotation) frictionless Symmetric part (shear velocity) contains the friction rotation and shear components General tensor of deformation

Material law according to Newton Most general version 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  is the usual (first) viscosity, is called second viscosity with

Resulting friction term for momentum equation Compression force due to friction Friction force on volume element It can be shown that If one assumes that during pure compression the entropy of a fluid does not increase (no dissipation).

Navier-Stokes equations Under isothermal conditions (T = const.) one has thus together with the continuity equation 4 equations for the 4 unknown functions u x, u y, u z, and p in space and time. They are completed by the equation of state for  (p) as well as initial and boundary conditions.

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

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

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.

Energy equation m = E,  =  (e+u²/2) inner+kinetic energy per unit volume, j =  u=  (e+u²/2)  u, s work done on the control volume by volume and surface forces, dissipation by heat conduction

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

Solute transport equation m = M solute,  = c concentration, (advection and diffusion), s solute sources and sinks Advection-diffusion equation for passive scalar transport in microscopic view.