Governing equations: Navier-Stokes equations, Two-dimensional shallow-water equations, Saint-Venant equations, compressible water

hammer flow equations (Material were taken from various internet sources)

Arturo S. Leon, Oregon State University (Fall 2011)

© Arturo S. Leon OSU, Fall 2011

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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

Operators

Gradient

Divergence

Laplaciany

ux

uuu

div

2

2

2

22

yu

xuu

y

ux

uu ∂∂,∂

∂

© Arturo S. Leon OSU, Fall 2011

4

j

ij

ii

x

y

x

x

uu

t

ua

z

ww

y

wv

x

wu

t

wa

z

vw

y

vv

x

vu

t

va

z

uw

y

uv

x

uu

t

ua

3

2

1

5

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.

6

)6(1

)5(1

)4(1

zyxz

pgf

zyxy

pgf

zyxx

pgf

zzyzxzzz

zyyyxyyy

zxyxxxxx

7

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

10

9

8

72,2,2

y

w

z

v

z

u

x

w

x

v

y

u

x

w

x

v

x

u

zyyz

zxxz

yxxy

zzyyxx

8

notationEinsteinxx

u

x

pgf

z

w

y

w

x

w

z

pgf

z

v

y

v

x

v

y

pgf

z

u

y

u

x

u

x

pgf

jj

i

iii

zz

yy

xx

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

1

)13(1

)12(1

)11(1

Substituting eqs. 7-10 into eqs. 4-6, we get

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)1A(1 2

ijj

i

ij

ij

i gxx

u

x

p

x

uu

t

u

)2A(0

0

i

i

x

u

z

w

y

v

x

u

The three N-S momentum equations can be written in compact form as

The equation of continuity for an incompressible fluid

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TurbulenceThe 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

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 other end of the spectra we have the smallest eddies which are by viscous forces (stresses) dissipated into internal energy.

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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 NR~2300 and in boundary layers at NR~100000

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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.

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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.

ppp

www

vvv

uuu

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Reynolds Time-averaged Navier-Stokes Equations

These are obtained from the N-S equations and include the flow turbulence effect as well.

directionxinequationMomentum

1

z

uwu

zy

uvu

y

x

uuu

xx

pg

z

uw

y

uv

x

uu

t

ux

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RNS Equations

tensorstressReynold

tensorstressviscouswhere

0

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jiRij

i

j

j

i

vij

i

ij

Rij

j

vij

ij

ij

i

uu

x

u

x

u

x

u

gxxx

p

x

uu

t

u

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Reynold StressesThe 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.

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Closure Problem

3 velocity components, one pressure and 6 Reynold stress terms = 10 unknowns

No. of equations=4 As No. of unknowns >No. 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.

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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.

j

i

i

jtji x

U

x

Uuu

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k-ε Turbulence Model

the Reynold stress tensor is calculated via the Boussinesq approximation

2

kct

Two transport equations are solved which describe the transport of the turbulent kinetic energy, k and its dissipation, ε. The eddy viscosity is calculated as

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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

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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,

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2D Saint Venant Equations

eqmomentumyh

vuvgn

y

zgh

y

hgh

y

hv

x

huv

t

hv

eqmomentumxh

vuugn

x

zgh

x

hgh

y

huv

x

hu

t

hu

eqcontinuityy

hv

x

hu

t

h

b

b

31

222

2

31

222

2

)()()(

)()()(

.0)()(

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1D Saint Venant Equation

0

t

A

t

Q

)( 0 fSSgx

hg

x

UU

t

U

The friction slope Sf is usually obtained from a uniform flow formula such as Manning or chezy.

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Simplified Equations of Saint Venant

wavewavewavesteadywave

kinematicdiffusivequasidyndynamic

SSx

h

x

u

g

u

t

u

g fo

.

1

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Relative Weight of Each Termin SV Equation

)10(

)10(

)10(

)10(1

3

3

5

5

OS

OS

Ox

u

g

u

Ot

u

g

f

o

Order of magnitude of each term In SV equation for a flood on river Rhone