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5362 CE Surface Water Modeling Essay 3.2 1. Introduction This essay provides a brief derivation of the St. Venant equations for one-dimensional (1-D) open channel ow. The equations were originally developed in the 1850s, so the concept is not very new. The tools have changed since that time; computational methods have greatly increased the utility of these equations. In general, 1-D unsteady ow would be considered state-of-practice computation; every engineer would be expected to be able to make such calculations (albeit using software). 2-D computation is not routine, but within the realm of consulting practice again using general purpose software. 3-D computation as of this writing (circa 2009) is still in the realm of state-of-art, and would not be within the capability a typical consulting rm. 2. The Computational Cell The fundamental computational element is a computational cell or a reach. Figure 1 is a sketch of a portion of a channel. The left-most section is uphill (and upstream) of the rightmost section. The section geometry is arbitrary, but is drawn to look like a channel cross section. The length of the reach (distance between each section along the ow path) is x. The depth of liquid in the section is z, the width at the free surface is B(z), the functional relationship established by the channel geometry. The ow into the reach on the upstream face is Q Q/x x/2. The ow out of the reach on the downstream face is Q + Q/x x/2. The direction is strictly a sign convention and the development does not require ow in a single direction. The topographic slope is S0 , assumed relatively constant in each reach, but can vary between reaches. 3. Assumptions The development of the unsteady ow equations uses several assumptions: (1) The pressure distribution at any section is hydrostatic this assumption allows computation of pressure force as a function of depth. (2) Wavelengths are long relative to ow depth this is called the shallow wave theory. (3) Channel slopes are small enough so that the topographic slope is roughly equal to the tangent of the angle formed by the channel bottom and the horizontal.
1
CE 5362 Surface Water Modeling
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Figure 1. Reach/Computational Cell (4) The ow is one-dimensional this assumption implies that longitudinal dimension is large relative to cross sectional dimension. Generally river ows will meet this assumption, it fails in estuaries where the spatial dimensions (length and width) are roughly equal. Thus rivers that are hundreds of feet wide imply that reaches are miles long. If this assumption cannot be met, then 2-D methods are more appropriate. (5) Friction is modeled by Chezy or Mannings type empirical models. The particular friction model does not really matter, but historically these equations have used the friction slope concept as computed from one of these empirical models. The tools that are used to build the equations are conservation of mass and linear momentum. 4. Conservation of Mass The conservation of mass in the cell is the statement that mass entering and leaving the cell is balanced by the accumulation or lass of mass within the cell. For pedagogical clarity, this section goes through each part of a mass balance then assembles into a dierence equation of interest.
REVISION NO. 1
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Mass Entering: Mass enters from the left of the cell in our sketch. This direction only establishes a direction convention and negative ux means the arrow points in the direction opposite of that in the sketch. In the notation of the sketch mass entering in a short time interval is: Q x ) t Min = (Q x 2
(1)
where is the uid density. Notice that the mass ux is evaluated at the cell interface and not the centroid, while by convention is assumed to be dened as an average cell property. Mass Leaving: Mass leaves from the right of the cell in our sketch. In the notation of the sketch mass leaving is: Q x Mout = (Q + ) t x 2
(2)
Mass Accumulating: Mass accumulating within the reach is stored in the prism depicted in the sketch by the dashed lines. The product of density and prism volume is the mass added to (or removed from) storage. The rise in water surface in a short time interval is z t. The plan view area of the prism is t B(z) x. The product of these two terms is the mass added to storage, expressed as: z Mstorage = ( t) B(z) x t
(3)
Equating the accumulation to the net inow produces Q x Q x z t) B(z) x = (Q ) t (Q + ) t t x 2 x 2
(4)
(
This is the mass balance equation for the reach. If the ow is isothermal, and essentially incompressible then the density is a constant and can be removed from both sides of the equation. z Q x Q x t) B(z) x = (Q ) t (Q + ) t t x 2 x 2
(5)
(
Rearranging the right hand side produces REVISION NO. 1 Page 3 of 10
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(6)
(
Q x Q x Q z t) B(z) x = t t = x t t x 2 x 2 x
Dividing both sides by x t yields z Q ) B(z) = t x
(7)
(
This equation is the conventional representation of the conservation of mass in 1-D open channel ow. If the equation includes lateral inow the equation is adjusted to include this additional mass term. The usual lateral inow is treated as a discharge per unit length added into the mass balance as expressed in Equation 32. z Q ) B(z) + =q t x
(8)
(
This last equation is one of the two equations that comprise the St. Venant equations. The other equation is developed from the conservation of linear momentum the next section. 5. Conservation of Momentum The conservation of momentum is the statement of the change in momentum in the reach is equal to the net momentum entering the reach plus the sum of the forces on the water in the reach. As in the mass balance, each component will be considered separately for pedagogical clarity. Figure 2 is a sketch of the reach element under consideration, on some non-zero sloped surface. Momentum Entering: Momentum entering on the side left of the sketch is QV = V 2 A
(9)
Momentum Leaving: Momentum leaving on the right side of the sketch is ( QV )x = V 2 A + ( V 2 A)x x x
(10)
QV +
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Figure 2. Equation of motion denition sketch Momentum Accumulating: The momentum accumulating is the rate of change of linear momentum: dL d (mV ) (11) = = ( AV x) = x ( AV ) dt dt t t Forces on the liquid in the reach: Gravity forces: The gravitational force on the element is the product of the mass in the element and the downslope component of acceleration. The mass in the element is Ax The x-component of acceleration is g sin(), which is S0 for small values of . The resulting force of gravity is is the product of these two values: Fg = g AS0 x
(12)
Friction forces: Friction force is the product of the shear stress and the contact area. In the reach the contact area is the product of the reach length and average wetted perimeter. Ff r = Pw x Page 5 of 10
(13) REVISION NO. 1
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where Pw = A/R, R is the hydraulic radius. A good approximation for shear stress in unsteady ow is = gRSf . Sf is the slope of the energy grade line at some instant and is also called the friction slope. This slope can be empirically determined by a variety of models, typically Chezys or Mannings equation is used. In either of these two models, we are using a STEADY FLOW equation of motion to mimic unsteady behavior nothing wrong, and it is common practice, but this decision does limit the frequency response of the model (the ability to change fast hence the shallow wave theory assumption!). The resulting friction model is Ff r = gASf x
(14)
Pressure forces: [Set the equations, backll discussion next version]
(15)
Fp =
A
dF
Figure 3. Pressure integral sketch (16) REVISION NO. 1 dF = (z h)g(h)dh Page 6 of 10
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where (h) is the width of the panel at a given distance above the channel bottom (h) at any section. Fp
(17)
Fp
net
= Fp
up
down
(18)
Fp
net
= Fp (Fp +
Fp Fp x) = x x x
Z
(19)
Fp x = [ x x z = g[ x
Z
g(z h)(h)dh]x
0
Z
(20)
Fp
net
(h)dh +
0 0
(z h)(h)
(h) dh]x x
The rst term integrates to the cross sectional area, the second term is the variation in pressure with position along the channel. The other pressure force to consider is the bank force (the pressure force exerted by the banks on the element). This force is computed using the same type of integral structure except the order is swapped.
Z
(21)
Fp
bank
=[
0
g(z h)
(h) x]dh x
Now we put everything together. d(mV ) dt
(22)
M omentumin M omentumout +
F =
Substitution of the pieces: d(mV ) dt
(23)
M omentumin M omentumout + Fp
net
+ Fbank + Fgravity Ff riction =
Now when the expressions for each expressions for each part
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(24)
( V 2 A)x x Z z Z (h) g dh]x (h)dhx [ g(z h) x 0 x 0 Z (h) x]dh +[ g(z h) x 0 +g AS0 x V 2A V 2A (gRSf x) = x ( g AV ) t
Each row of Equation 24 is in order: (1) Net momentum entering the reach. (2) Pressure force dierential at the end sections. (3) Pressure force on the channel sides. (4) Gravitational force. (5) Frictional force opposing ow. (6) Total acceleration in the reach (change in linear momentum). Canceling terms and dividing by x (isothermal, incompressible ow; reach has nite length) Equation 24 simplies to z (V 2 A) g x x
Z
(25)
(h)dh + g AS0 (gRSf ) =
0
(g AV ) t
The second term integral is the sectional ow area, so it simplies to z (V 2 A) g A + gAS0 gASf = (AV ) x x t
(26)
The term with the square of mean section velocity is expanded by the chain rule, and using continunity becomes (notice the convective acceleration term from the change in area with time) V A V V A (AV ) = A +V =A VA V2 t t t t x x
(27) REVISION NO. 1
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CE 5362 Surface Water Modeling Now expand and construct
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A A V z V V 2V A gA + gA(S0 Sf ) = A VA V2 x x x t x x Cancel common terms and simplify (28) V 2 V z V gA + gA(S0 Sf ) = A x x t
(29)
V A
Equation 33 is the nal form of the momentum equation for practical use. It will be rearranged in the remainder of this essay to t some other purposes, but this is the expression of momentum in the channel reach. Divide by gA and obtain V V z 1 V + (S0 Sf ) = g x x g t
(30) Rearrange
(31)
Sf = S0
1 V z V V x g x g t
Now consider typical ow regimes. (1) The rst two terms (from left to right) are uniform ow, this is an algebraic equation. (2) If the rst four terms are in eect, we have gradually varied ow; an ordinary dierential equation. (3) If all terms are in eect, the have the dynamic ow (shallow wave) conditions; a partial dierential equation. The pair of equations, z Q ) B(z) + =q t x z V V 1 V =0 x g x g t Page 9 of 10
(32)
(
(33)
S0 Sf
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are called the St. Venant equations and comprise a coupled hyperbolic dierential equation system. Solutions ((z, t) and (V, t) functions) are found by a variety of methods including nite dierence, nite element, nite volume, and characteristics methods. In this course we will study a simple nite-dierence scheme to gain some familarity with building solutions, then use prepared tools (SWMM) for more practical problems. 6. Readings Cunge, J.A., Holly, F.M., Verwey, A. 1980. Practical Aspects of Computational River Hydraulics. Pittman Publishing Inc. , Boston, MA. pp. 7-50 in On server at : http://cleveland1.ce.ttu.edu/teaching/ce 5362/ ce 5362 9.9/practical aspects computational river hydraulics.pdf 7. Exercises (1) Verify that the mass balance indeed provides the identities to complete the momentum analysis in Equation 27 8. Author Notes REVISION NO. 1: Initial typeset; need to check mathematics against hand sheets. Need to supply narrative next version.
REVISION NO. 1
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