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# STAdvanced Timber Design CE 504

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This 8 page Class Notes was uploaded by Savanna Cruickshank on Friday October 23, 2015. The Class Notes belongs to CE 504 at University of Idaho taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/227781/ce-504-university-of-idaho in Civil Engineering at University of Idaho.

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Date Created: 10/23/15

CE 504 Computational Hydrology Surface Water Equations Fritz R Fiedler l Derivation of Shallow Water Equations 2 Applicability 3 Wave Celerity Derivation There are several ways the surface water equations can be derived It is rare that surface water is modeled in three dimensions however we can begin with the three dimensional NavierStokes equations and derive the depthaveraged twodimensional shallow water equations These can easily be simplified to one dimension By assuming that certain terms in the equations are negligible the equations can be further simplified The following derivation pertains to overland ow derivation for other surface water applications e g channel ow wetlands shallow ocean circulation is similar Continuity The general three dimensional continuity equation in standard cartesian coordinates is 6 Q 6 w 0 1 6x By 62 where u v and w are uid velocities in the X y and 2 directions respectively Kinematic boundary conditions are applied at the uid surface and porous bed these essentially define a velocity at each boundary For overland ow these conditions are Ewz uz aivz ii r 2 dt 2 6t 2 6x 2 By E w uz aivz 3 dt 6t 6x 6y were r is the rainfall rate f is the infiltration or eXfiltration rate and the variables 21 and 2 refer to the ground and water surface elevations respectively To obtain the two dimensional continuity equation Equation 1 is vertically averaged by integrating each term over the ow depth 22 7 21 and then dividing by this depth The kinematic boundary conditions are used where applicable ie the third term of Equation 1 The termbyterm integrations are 22 1 la uF 1 a uZZJ u 4 z2 z1 Zl x z2 21 6x Z 6x 1 IQdZ1 zz 5 22 21 6y zz z1 6y 2 6y Z1 Fritz R F iedler 18 last updated 252006 22 1 6 The kinematic boundary conditions are required in Equation 6 resulting in 6 6 6 6 6 6 iiuz iw iJJVZ ii i f 7 22 21 6t 6t 2 6x 6x K 2 6y 6y This can be simpli ed by regrouping multiplying by the depth of ow h 22 7 21 and noting that the velocity is uniform with depth ie ul uz u to obtain dZ 62 22 21 W 1 136w 1 22 Zl Zl ha u t h v r f 8 6t 6x 6x 6y 6y or 6lthugtmf 9 6t 6x 6y Momentum The NavierStokes momentum equation in the x direction is a a 6 wa J a pyvzuF 10 6t 6x 6y 62 6x where p is the uid density p is pressure u is the dynamic viscosity and Fx is the gravitational force in the x direction The left side of this equation is the sum of the inertia forces and the right side is the sum of the forces applied to the control volume This equation is also vertically averaged by integrating over and dividing by the depth of ow and applying the kinematic boundary conditions where appropriate First each term of the left side is averaged over depth l Z26u 6u d2 ll 22 21p239Z6t p6t l Zz6u2 6u2 d2 12 22 21 6x p6x l Z26uv 6uv pla dm a 13 2 1 21 y y l pJ6ude l pusz 14 22 21 21 62 22 21 Z The kinematic boundary conditions are required to complete the integration in Equation 14 resulting in 1 u aiuz aZ ZVZ Eli r guz gvz f 15 22 21 6t 2 6x 2 6y 6t 6x 6y Collecting terms multiplying by the depth of ow h and assuming uniform velocity with depth in the x and y directions the vertically averaged left side of Equation 10 becomes Fritz R F iedler 28 last updated 252006 61m 61th2 6huv p1 6t 6x 6y In terms of shear stresses 139 the right side of Equation 10 can be written 6 6 6139 X 6 p T y iFx 17 6x 6x 6y 62 By assuming that the horizontal shear components are small compared to the vertical components ie velocity variation in the horizontal directions is small compared to the vertical velocity variation Equation 17 reduces to uruf 16 6 10 6i F 18 6x 62 The first term in Equation 18 represents the unbalanced pressure force in the x direction and when vertically averaged becomes 1 6h h 19 h pg 6x The third term in Equation 18 is the gravitational force in the x direction 1 62 h 1 20 h pg ax J It is necessary to vertically integrate the second term of Equation 18 which represents the viscous force 1 21 a 1 J T 612 1 62 h The last step results from assuming that the shear stress at the water surface 22 is zero and the shear stress at the ground surface acts in the x direction the 21 superscript implies that it is the shear at the ground surface Combining these terms and multiplying by h as before the vertically averaged right side of Equation 10 is 6h 62 h 1 121 22 pg 6x 6x J x The X and y direction vertically averaged momentum equations for overland ow are then Zz z 1 r 21 22 21 21 6hu 6th2 6huv 6h 62 h 1 Z 23 6t 6x 6y quotr Hf pg 6x 6x TX 2 p 6huv6hv WVfpgh 121 24 K6t 6x 6y 6y 6y y Vertically averaging the zdirection momentum equation assuming that vertical acceleration of uid particles is small compared to the gravitational acceleration and that the shear stress due to the vertical velocity component is small results in the hydrostatic equation Letting p hu discharge per unit area in X direction q hv discharge per unit area in y direction q r 7 f net lateral in ow S f 11 p S rj p St7 6216x and St7y 6216y the equations become x Fritz R F iedler 38 last updated 252006 6h 6p 6q 0 25 6t 6x a ql 6p 6 p2 ghz 6 m p h 0 26 6t 6xh 2 6y h gm Sf hq 6q a q2 gh2 6 M q hS 0 27 6t 6yh 2 6x h guys hq The friction slopes Sfx and Sfy are typically modeled with wellknown empirical equations for laminar e g DarcyWeisbach or turbulent eg Manning s Equation ow regimes In two dimensions and in terms of unit discharges in the X and y directions the DarcyWeisbach formulation for bed shear is 12 p 2 2 sf p 11 q 28 and 12 q 2 2 Sfy 29 where f is the DarcyWeisback friction factor For laminar ows f is a linear function of the Reynolds number K 0 f Rf 30 and in two dimensions Re is computed p2 q2 12 R2 f 31 where K is a parameter related to ground surface roughness and vis the kinematic Viscosity of water Equations 30 and 31 are substituted into Equations 28 and 29 resulting in K 0 W7 S 32 fr 8g h ltqu S 33 fy 8gh3 While the DarcyWeisbach formulation can also be used for turbulent ows with f becoming constant it is more common to use Manning s equation In terms of p and q in two dimensions Manning s equation results in the friction slopes 2 2 205 S znpov 61 11103 n2qp2 q205 S W 35 These equations can be compactly represented in vector form a U 6GU 6HU SU 36 at 6x 03 Fritz R F iedler 48 last updated 252006 where U 14 p qT 37 2 2 A 3 T GU p h 2 h 38 H U 124 QT 39 q h 2 h SU 4 ghS Sf qu ghSy Sfy 17 40 Alternate DerivationForms The Reynolds Transport Theorem RTT is commonly used to derive the Saint Venant equations eg Chow et al 1988 which are essentially the onedimensional form of Equation 36 This approach comprises a balance of uid mass and momentum over a control volume For continuity the unsteady variable density RTT equation for conservation of mass in a control volume is gmpdwgpvmw 41 where the first term represents the change in storage within the control volume and the second term represents net out ow from the control volume The momentum RTT equation is JJJVpdv HVpVodA 2F 42 where the first term is the change in momentum within the control volume the second term is the net out ow of momentum from the control volume the sum of these two terms is equal to the sum of forces applied to the control volume gravity friction unbalanced pressure In the conservation form the onedimensional Saint Venant equations with no lateral in ow are o 43 6t x up 16 Q2 4 S S 0 44 A6 maxii gax g f and in the non conservation form the continuity and momentum equations are u ha u0 45 6t 6x 6x Bu Bu 6h S S 0 46 at Max gax g f In general the conservation form should be used to simulate discontinuous ows eg mixed subcritical and supercritical regimes Proceeding from left to right the momentum equation terms represent local acceleration convective acceleration unbalanced pressure force gravity force and friction force These equations are often Fritz R F iedler 58 last updated 252006 referred to as the dynamic wave equations When the convective acceleration term is neglected the diffusion wave approximation results When both the convective acceleration and unbalanced pressure force terms are neglected the kinematic wave approximation is obtained Steadystate equations are obtained by neglecting the time derivatives Applicability Surface water simulation in hydrology is often broken into two categories overland ow and channel ow Typically overland ow is simulated as a thin sheet using the kinematic wave approximation in one dimension While this approach has been successfully used to match hillslope hydrographs by adjusting the friction parameter and infiltration if applicable it does not truly represent the smallscale dynamics that occur on a given hillslope Diffusion and dynamic wave approaches have also been used to simulate overland ow in two dimensions Channel ow is often modeled using the kinematic wave approximation but the diffusion and dynamic wave approaches are frequently used as well Disturbances do not propagate upstream using kinematic wave theory thus this approach can not be used when backwater effects are significant Only uniform ow can be modeled with kinematic waves There have been numerous studies addressing the applicability of the kinematic wave approximation Woolhiser and Liggett 1967 developed the Kinematic Wave Number to judge if the kinematic wave approximation is valid for 1D overland ow on a plane slope It is given by S L 2 47 hOFO where S is the bed slope L0 is the plane length hg is the normal ow depth and F g ugghg0395 is the Froude number at normal depth computed using Manning s equation an is the mean uniform ow velocity and hg the uniform ow depth Morris and Woolhiser 1980 and Daluz Vieira 1983 determined that the kinematic wave approximation is valid for KF2 Z 5 and F0 lt 2 Hager and Hager 1985 determined that the kinematic wave approximation is valid for a plane or channel reach if imam lt 3 48 and q 1 S 05 lt 007 49 in SI units where n is the Manning s resistance parameter Wave Celerity Lighthill and Whitham 1955 state that kinematic waves result when there is a unique relationship between ow depth and discharge or ow area and discharge For example Manning s equation in SI units can be written quotPl3 35 hi 051 0 Fritz R F iedler 68 last updated 252006 where P is the wetted perimeter The general form of this relationship is A aQ 51 If Equation 51 is differentiated with respect to time and substituted into the continuity equation the result is 6Q p71 aQ 5 5Q ql 52 Kinematic waves are created by changes in discharge The total derivative of discharge 1s 6Q 6Q d dx dt 53 Q x it which is rearranged Q ag 54 6x dx 6t dx By comparing Equations 54 and 52 it is apparent that Q 55 dx qi and 71 56 dt a Q Equation 55 states that discharge increases with lateral in ow which is intuitively obvious Equation 56 de nes the kinematic wave celerity velocity Herein designated ck Equation 51 can be differentiated and rearranged to show that Q dx dA dt Ck 57 The kinematic wave celerity is not equal to the mean flow velocity The mean velocity can be computed for example in a wide rectangular channel as u Qh Also in a wide rectangular channel QQauhuha uc 58 6A ah ah 6 k Substituting Manning s equation 0 uhith3S 5u2u2u 59 k 6h 71 f 3 3 It can be shown that the dynamic wave celerity in a rectangular channel is given by Cd xgh 60 However dynamic waves move in both the upstream and downstream directions u Ca 61 and g u cd 62 References Fritz R F iedler 78 last updated 252006 Chow V T D R Maidment and L W Mays Applied Hydrology McGraw Hill Inc 1988 Daluz Vieira J H Conditions governing the use of approximations for the SaintVenant equations for shallow water ow Journal of Hydrology 60 4358 1983 Hager W H and K Hager Application limits for the kinematic wave approximation Nordic Hydrology 16 203212 1985 Lighthill M J and G B Whitham On kinematic waves 1 Flood movement in long rivers in Proceedings Royal Society London Series A Volume 229 281316 1955 Morris E M and D A Wollhiser Unsteady onedimensional ow over a plane partial equilibrium and recession hydrographs Water Resources Research 162 355360 1980 Wollhiser D A and J A Liggett Unsteady onedimensional ow over a plane the rising hydrograph Water Resources Research 33 753771 1967 Fritz R F iedler 88 last updated 252006

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