ADV COMPUTNL FLOWS & TRANSPORT
ADV COMPUTNL FLOWS & TRANSPORT CAM 397
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Date Created: 09/06/15
The Shallow Water Equations Clint Dawson and Christopher M Mirabito lnstltute for Computational Engineering and Sclences niversity of Texas at Aust clint ices utegtltas edu September 29 2008 7qu ln Emmy n The Shallow Water Equations SWE What are they a The SWE are a system of hyperbolicparabolic PDEs governing fluid flow in the oceans sometimes coastal regions usually estuaries almost always rivers and channels almost always 0 The general characteristic of shallow water flows is that the vertical dimension is much smaller than the typical horizontal scale In this case we can average over the depth to get rid of the vertical dimension 0 The SWE can be used to predict tides storm surge levels and coastline changes from hurricanes ocean currents and to study dredging feasibility o SWE also arise in atmospheric flows and debris flows The SWE Co How do they arise o The SWE are derived from the Navier Stokes equations which describe the motion of fluids 3 The NavierStokes equations are themselves derived from the equations for conservation of mass and linear momentum lm benign Hi SWE Derivation Procedure There are 4 basic steps 0 Derive the NavierStokes equations from the conservation laws 9 Ensemble average the NavierStokes equations to account for the turbulent nature of ocean flow See 1 3 4 for details 9 Specify boundary conditions for the Navier Stokes equations for a water column 0 Use the BCs to integrate the Navier Stokes equations over depth In our derivation we follow the presentation given in 1 closely but we also use ideas in 2 lm mermaan Hi Conservation of Mass Consider mass balance over a control volume 0 Then ipdV 7 pvndA7 51139 0 an W Time rate of change Net mass flugtlt across of total mass m 0 boundary of Q where o p is the fluid density kgm3 u o v v is the fluid velocity ms and W o n is the outward unit normal vector on 80 Conservation of Mass Differential Form Applying Gauss39s Theorem gives d E pdViQVpvdVi Assuming that p is smooth we can apply the Leibniz integral rule 93 i clV 0 Q l at v Ml Since 0 is arbitrary ln mermaan H Conservation of Linear Momentum Next consider linear momentum balance over a control volume 0 Then d 7 pvdV 7 pvvndA pde TnclA7 51139 0 an 0 an V V v V Tlme rate of Net momentum flugtlt Body forces External contact change of total across boundary of Q actmg on Q forces aCtlng mentum In 0 where o b is the body force density per unit mass acting on the fluid Nkg and o T is the Cauchy stress tensor Nmz See 5 6 for more details and an existence proof Conservation of LInear Momentum Differential Form Applying Gauss39s Theorem again and rearranging gives 1pvdVVpvvdV7pdeiVTclVOl 51139 Q Q Q 0 Assuming pv is smooth we apply the Leibniz integral rule again 8 7pvVpvvipb7VT dVOl 9 6139 Since 0 is arbitrary pvVpvvipb7VTO r I m n39 Conservation Laws Differential Form Combining the differential forms of the equations for conservation of mass and linear momentum we have 93 i 0 at V W 8 3PVV39WV bV39T To obtain the Navier Stokes equations from these we need to make some assumptions about our fluid sea water about the density p and about the body forces b and stress tensor T ln nemam Sea water Properties and Assumptions o It is incompressible This means that p does not depend on p It does not necessarily mean that p is constant In ocean modeling p depends on the salinity and temperature of the sea water a Salinity and temperature are assumed to be constant throughout our domain so we can just take p as a constant So we can simplify the equations V v O7 8 EperVpvv pbVT 0 Sea water is a Newtonian fluid This affects the form of T r I m n39 Body Forces and Stresses in the Momentum Equati We know that gravity is one body force so Pb pg pbothers where o g is the acceleration due to gravity msz and o bothers are other body forces eg the Coriolis force in rotating reference frames Nkg We will neglect for now For a Newtonian fluid T ipl T where p is the pressure Pa and T is a matrix of stress terms r I w m The Navier Stokes Equations So our final form of the NavierStokes equations in 3D are V v 07 a EperVpw7VerJngVT7 The Navier Stokes Equations Written out 3W 907112 907W BUWW 8TXX 7 P g 977x By 82 8139 8X By 82 8X 2 aw am am 6pvwaw 6mm an ataxayaz ax 6y 7 3 907W 907W BUNW 903W 7 8a 87y 60227 at 6x By 62 7pg aiy 62 4 A ypical Water Column o tXy is the elevation m of the free surface relative to the geoid o b bXy is the bathymetry measured positive downward from the geoid 0 H HtXy is the total depth m of the water column Note that H C b I l MW in quot an 11 5 41 239 i Derivation of the SWE mm mm A T 7 iiwl Caribbean plate 1 2391 at 39i i1 v K I 1 yquot lfgtrr I 39 if I rquot 39 1 1 i quot 4 rf q 1 F 4 i I Bathymetry of the Atlantic Trench Image courtesy USGS C M irabito The ghg iilikgw Wetter Egy m ln mermaan H Boundary Conditions We have the following BCS 0 At the bottom 2 7b o Noslip u v0 a No normal flow 6b 6b ua Va w 7 O 5 a Bottom shear stress 6b 6b Tbx 7 TXXE Txyaiy TX 6 where Tbx is specified bottom friction similarly for y direction 9 At the free surface 2 4 o No relative normal flow 6C 6 6C 7 6ruaxvay W70 7 o p O done in a Surface shear stress BC 5 7 67 M 8 Tax Txx ln Deriwanmn z momentum Equation Before we integrate over depth we can examine the momentum equation for vertical velocity By a scaling argument all of the terms except the pressure derivative and the gravity term are small Then the z momentum equation collapses to Q 62 Pg implying that p pglt 7 Z This is the hydrostatic pressure distribution Then 8p 64 i i 9 6X pgax with similar form for 3 Il The 2D SWE Continuity Equation We now integrate the continuity equation V v 0 from z 7b to z 4 Since both b and 4 depend on t X and y we apply the Leibniz integral rule 4 O V v dz 7b 74 Jr dzwi wi 7 b 8X 8y 24 24 6 4 a 4 64 ab iaibuderaiyibvdzi ltuiz4ampuizbampgt 8 7 Viz i Viz7b Wiz 7 Wiz7b 46y 8y 4 D E The Continuity Equation Cont Defining depthaveraged velocities as 1 4 1 4 uipibuclz7 vipibvclz7 we can use our BCs to get rid of the boundary terms So the depthaveraged continuity equation is 6H 8 9 EampHuaiyHv 70 10 LHS of the X and y Momentum Equations If we integrate the lefthand side of the Xmomentum equation over depth we get 4 a a a a i i 2 i i Liar ax ay quot 62UW dz 8 8 2 8 Diff adv EHu ampHu Huv terms 11 The differential advection terms account for the fact that the average of the product of two functions is not the product of the averages We get a similar result for the lefthand side of the y momentum equation r I m n39 RHS of X and y Momentum Equations Integrating over depth gives us PEHTSX TbxfEbTgtoltffb7 Xy 12 a a C a C PgHaiy Tsy 739er E 137er E LbTyy At long last Combining the depthintegrated continuity equation with the LHS and RHS of the depthintegrated X and y momentum equations the 2D nonlinear SWE in conservative form are 6H 6 a Etawwiywv a 62 aui 6lt1 EHu amp Hu Huv 7 igHaX rsx 7 Tbx i FX 6 an 647 6lt1 EHvampHuvaiy Hv 7 igHaier EiTsy Tby i Fy The surface stress bottom friction and FX and Fy must still determined on a casebycase basis O Gonzalez and J Li UTiAustin 29 October 2008 Problem statement Consider an experiment in which a 2 component solution solute i solvent is separated in a centrifuge rotor h 5pm 0 cell i solute solvent 5 olvem i i J elapsed 3 time solute t 0 gtgt 0 cloudy mixture separatedmxxture n Derive model for concentration of solute particles 1 Use model to estimate shape from concentration data 1 Begin with simple case of spherical particles 1 Generalize to particles of arbitrary shape Simplifying assumptions In a coord system attached to and rotating with the rotor assume I time averaged velocity V of a solute particle due to centrifugal force is small and in radial direction I solvent is macroscopically stationary solute particle Assume solute concentration p and velocity V depend only on radial coordinate r and time t p pr t V Vr t hltlt1 qgtltlt1 Assume particles are spherical and independent with velocity proportional to centrifugal force Stokes law solute particle v radius m bouyant mass u viscosity T abs temperature f 7 f m wzr gt V 5qu7 5 L 67m Assume mass flux J of solute particles across radial surface at r can be decomposed into two parts diffusion l convection Jr t 70 p l 8r Mass Area Time kT For spherical particles in a fluid D Stokes Einstein 67m Governing equations Consider the region R between two radial surfaces 90 and 91 where r0 lt r1 are arbitrary For the region R we have 2 mass of solute 7 mass flow 7 mass flow at inside of R 7 in to R out of R n va i39 16 For the region R we have 2 mass of solute 7 mass flow 7 mass flow at inside of R 7 in to R out of R fr or7 tAreaQ dr Jr07 tAreaQ0 7 Jr17 tAreaQ1 For the region R we have 2 mass of solute 7 mass flow 7 mass flow at inside of R 7 in to R out of R er P fAreaQr dr Jro fAreaQ0 7 Jr1 t AreaQ1 fr1prtAreaQ dr if JU7 t AreaQ dr For the region R we have 2 mass of solute 7 mass flow 7 mass flow at inside of R 7 in to R out of R fr or7 t AreaQ dr Jr07 t Areamm 7 Jr17 t AreaQ1 fr or7 tAreaQ dr if J07 tAreaQ dr Substituting AreaQ h gtr and rearranging gives 1 8 8 r0 hwyg i a lth gtrJgt dr 07 Vro lt r1 Assuming the integrand is continuous we deduce by the arbitrariness of re and r1 8p 3 i gthr87 i Elt gthrJgt 7 07 r3 lt r lt rb t Assuming the integrand is continuous we deduce by the arbitrariness of re and r1 8p 3 i gthr i Elt gthrJgt 7 07 r3 lt r lt rb Substituting J 70 i pV and V Swzr gives after cancelling factors 8t gltw2r2pi r030 07 r3 lt r lt rb 1 7 7 r r Solute particles cannot cross surfaces at ra and rb solute solvent rb Jrt7 pVO7 rrarb