The Shallow Water Equations Derivation Procedure

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The Shallow Water Equations

Clint Dawson and Christopher M. Mirabito

Institute for Computational Engineering and SciencesUniversity of Texas at Austin

[email protected]

September 29, 2008
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IntroductionDerivation of the SWE

The Shallow Water Equations (SWE)

What are they?

The SWE are a system of hyperbolic/parabolic PDEs governing fluidflow in the oceans (sometimes), coastal regions (usually), estuaries(almost always), rivers and channels (almost always).

The general characteristic of shallow water flows is that the verticaldimension is much smaller than the typical horizontal scale. In thiscase we can average over the depth to get rid of the verticaldimension.

The SWE can be used to predict tides, storm surge levels and

coastline changes from hurricanes, ocean currents, and to studydredging feasibility.

SWE also arise in atmospheric flows and debris flows.

C. Mirabito The Shallow Water Equations
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The SWE (Cont.)

How do they arise?

The SWE are derived from the Navier-Stokes equations, whichdescribe the motion of fluids.

The Navier-Stokes equations are themselves derived from theequations for conservation of mass and linear momentum.

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Derivation of the Navier-Stokes EquationsBoundary Conditions

SWE Derivation Procedure

There are 4 basic steps:

1 Derive the Navier-Stokes equations from the conservation laws.

2 Ensemble average the Navier-Stokes equations to account for the

turbulent nature of ocean flow. See [1, 3, 4] for details.3 Specify boundary conditions for the Navier-Stokes equations for a

water column.

4 Use the BCs to integrate the Navier-Stokes equations over depth.

In our derivation, we follow the presentation given in [1] closely, but wealso use ideas in [2].


I d i D i i f h N i S k E i
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Conservation of Mass

Consider mass balance over a control volume . Then

d

dt

dV

Time rate of changeof total mass in =

(v) n dA

Net mass flux acrossboundary of ,

where

is the fluid density (kg/m3),

v = u

vw is the fluid velocity (m/s), and

n is the outward unit normal vector on .


Introd ction Deri ation of the Na ier Stokes Eq ations
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Conservation of Mass: Differential Form

Applying Gausss Theorem gives

d

dt

dV =

(v) dV.

Assuming that is smooth, we can apply the Leibniz integral rule:

t+ (v)

dV = 0.

Since is arbitrary,

t+ (v) = 0


Introduction Derivation of the Navier Stokes Equations
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Conservation of Linear Momentum

Next, consider linear momentum balance over a control volume . Then

d

dt

v dV

Time rate ofchange of totalmomentum in

=

(v)v n dA

Net momentum fluxacross boundary of +

b dV

Body forcesacting on +

Tn dA

External contactforces actingon

,

where

b is the body force density per unit mass acting on the fluid (N/kg),and

T is the Cauchy stress tensor (N/m2). See [5, 6] for more detailsand an existence proof.


Introduction Derivation of the Navier-Stokes Equations
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Derivation of the Navier Stokes EquationsBoundary Conditions

Conservation of Linear Momentum: Differential Form

Applying Gausss Theorem again (and rearranging) gives

d

dt

v dV +

(vv) dV

b dV

T dV = 0.

Assuming v is smooth, we apply the Leibniz integral rule again:

t(v) + (vv) b T

dV = 0.

Since is arbitrary,

t(v) + (vv) b T = 0


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Derivation of the Navier Stokes EquationsBoundary Conditions

Conservation Laws: Differential Form

Combining the differential forms of the equations for conservation ofmass and linear momentum, we have:

t + (v) = 0

t(v) + (vv) = b + T

To obtain the Navier-Stokes equations from these, we need to make someassumptions about our fluid (sea water), about the density , and aboutthe body forces b and stress tensor T.


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Derivation of the SWEq

Boundary Conditions

Sea water: Properties and Assumptions

It is incompressible. This means that does not depend on p. Itdoes not necessarily mean that is constant! In ocean modeling, depends on the salinity and temperature of the sea water.

Salinity and temperature are assumed to be constant throughout our

domain, so we can just take as a constant. So we can simplify theequations:

v = 0,

tv + (vv) = b + T.

Sea water is a Newtonian fluid. This affects the form ofT.


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Derivation of the SWE Boundary Conditions

Body Forces and Stresses in the Momentum Equation

We know that gravity is one body force, so

b = g + bothers,

where

g is the acceleration due to gravity (m/s2), and

bothers are other body forces (e.g. the Coriolis force in rotatingreference frames) (N/kg). We will neglect for now.

For a Newtonian fluid,

T = pI + Twhere p is the pressure (Pa) and T is a matrix of stress terms.


Introductionf S

Derivation of the Navier-Stokes EquationsC
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The Navier-Stokes Equations

So our final form of the Navier-Stokes equations in 3D are:

v = 0,

tv + (vv) = p+ g + T,


IntroductionD i ti f th SWE

Derivation of the Navier-Stokes EquationsB d C diti
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The Navier-Stokes Equations

Written out:

u

x+

v

y+

w

z= 0

(1)

(u)

t +

(u2)

x +

(uv)

y +

(uw)

z =

(xx p)

x +

xy

y +

xz

z(2)

(v)

t+

(uv)

x+

(v2)

y+

(vw)

z=

xy

x+

(yy p)

y+

yz

z(3)

(w)

t+

(uw)

x+

(vw)

y+

(w2)

z= g +

xz

x+

yz

y+

(zz p)

z(4)



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A Typical Water Column

= (t, x, y) is the elevation (m) of the free surface relative to thegeoid.

b = b(x, y) is the bathymetry (m), measured positive downwardfrom the geoid.

H = H(t, x, y) is the total depth (m) of the water column. Notethat H = + b.



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A Typical Bathymetric Profile

Bathymetry of the Atlantic Trench. Image courtesy USGS.



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Boundary Conditions

We have the following BCs:1 At the bottom (z = b)

No slip: u= v = 0No normal flow:

ub

x+ v

b

y+ w = 0 (5)

Bottom shear stress:

bx = xxb

x+ xy

b

y+ xz (6)

where bx is specified bottom friction (similarly for y direction).2 At the free surface (z = )

No relative normal flow:

t

+ ux

+ vy

w = 0 (7)

p= 0 (done in [2])Surface shear stress:

sx = xx

x xy

y+ xz (8)

where the surface stress (e.g. wind) sx is specified (similary for yC. Mirabito The Shallow Water Equations




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z-momentum Equation

Before we integrate over depth, we can examine the momentum equationfor vertical velocity. By a scaling argument, all of the terms except thepressure derivative and the gravity term are small.Then the z-momentum equation collapses to

p

z= g

implying thatp = g( z).

This is the hydrostatic pressure distribution. Then

p

x= g

x(9)

with similar form for py

.



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y

The 2D SWE: Continuity Equation

We now integrate the continuity equation v = 0 from z = b toz = . Since both b and depend on t, x, and y, we apply the Leibnizintegral rule:

0 =

b

v dz

=

b

u

x+

v

y

dz + w|z= w|z=b

=

x

b

u dz +

y

b

v dz u|z=

x+ u|z=b

b

x

v|z=

y+ v|z=b

b

y

+ w|z= w|z=b



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The Continuity Equation (Cont.)

Defining depth-averaged velocities as

u =1

H

b

u dz, v =1

H

b

v dz,

we can use our BCs to get rid of the boundary terms. So thedepth-averaged continuity equation is

H

t+

x(Hu) +

y(Hv) = 0 (10)



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LHS of the x- and y-Momentum Equations

If we integrate the left-hand side of the x-momentum equation overdepth, we get:

b

[

t

u+

x

u2 +

y

(uv) +

z

(uw)] dz

=

t(Hu) +

x(Hu2) +

y(Huv) +

Diff. adv.

terms

(11)

The differential advection terms account for the fact that the average ofthe product of two functions is not the product of the averages.We get a similar result for the left-hand side of the y-momentumequation.



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RHS ofx- and y-Momentum Equations

Integrating over depth gives us

gHx + sx bx + x bxx + y bxygH

y+ sy by +

x

b

xy +y

b

yy(12)





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At long last. . .

Combining the depth-integrated continuity equation with the LHS andRHS of the depth-integrated x- and y-momentum equations, the 2D(nonlinear) SWE in conservative form are:

H

t+

x(Hu) +

y(Hv) = 0

t(Hu) +

x

Hu2

+

y(Huv) = gH

x+

1

[sx bx + Fx]

t(Hv) +

x(Huv) +

y Hv2

= gH

y+

1

[sy by + Fy]

The surface stress, bottom friction, and Fx and Fy must still determinedon a case-by-case basis.





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References

C. B. Vreugdenhil: Numerical Methods for Shallow Water Flow,Boston: Kluwer Academic Publishers (1994)

E. J. Kubatko: Development, Implementation, and Verification ofhp-Discontinuous Galerkin Models for Shallow Water Hydrodynamics

and Transport, Ph.D. Dissertation (2005)

S. B. Pope: Turbulent Flows, Cambridge University Press (2000)

J. O. Hinze: Turbulence, 2nd ed., New York: McGraw-Hill (1975)

J. T. Oden: A Short Course on Nonlinear Continuum Mechanics,Course Notes (2006)

R. L. Panton: Incompressible Flow, Hoboken, NJ: Wiley (2005)

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The Shallow Water Equations Derivation Procedure