1.5 DEVELOPMENT OF THE DEPTH AVERAGED GOVERNING EQUATIONS

Transcription

1 CE Topic.5 - Spring Revied Feruary 3, :20 pm.5 DEVELOPMENT OF THE DEPTH AVERAGED GOVERNING EQUATIONS General Conideration We are developing a hierarchy of equation averaged over variou cale. Molecular Scale: Navier-Stoke Equation + Continuity Equation u,v,w,p Valid for all flow down to the molecular cale We can not reolve down to the vicou diipation cale for normal computation Turulent Scale: Reynold Equation + Continuity Equation u, v, w, p and u'u', u'v', u'w', v'v', v'w', w'w' Need to elect a turulence cloure model (i.e. a et of contitutive relationhip) Valid for all flow down to the turulent averaging cale (in time T and related pace cale) p..5. CE Topic.5 - Spring Revied Feruary 3, :20 pm Let u examine patial averaging over the depth dimenion River and channel flow Etuarine and coatal ocean flow depth averaged flow A key aumption for depth averaging i that the flow in the vertical direction i mall. Thi implie that all term in the z-direction Reynold Equation are mall compared to the gravity and preure term. Thu the z-direction Reynold Equation reduce to p ρg z (.5.) Thi implie that the preure ditriution over the vertical i hydrotatic. p..5.2

2 CE Topic.5 - Spring Revied Feruary 3, :20 pm Depth Integrated Continuity Equation Conider the geoid to e defined at z0, the free urface (water-air interface) at z, and the ottom (water-ediment interface) at z-h. p..5.3 CE Topic.5 - Spring Revied Feruary 3, :20 pm Depth averaged velocitie are defined a: u ---- u dz H (.5.2) v ---- v dz H (.5.3) Where total water column height i defined a: H +h (.5.4) Flow rate over the vertical i defined a: qx u dz u H (.5.5) qy v dz v H p..5.4 (.5.6)

3 CE Topic.5 - Spring Revied Feruary 3, :20 pm Auming incompreile turulent time averaged flow, the appropriate form of the continuity equation i: u v w y z (.5.7) Now let vertically average: --- H u v dz dz H y H w dz 0 z (.5.8) Multiplying through y H and evaluating the lat integral: ( x, y, h( x, y) u dz + ( x, y, h( x, y) v dz + w( ) w( h) 0 y (.5.9) Uing Leinitz Rule, we know that: B( x, y, A( x, y) f dz B B A f z f d z B f z A A (.5.0) p..5.5 CE Topic.5 - Spring Revied Feruary 3, :20 pm Sutituting in for the remaining integral term uing Leinitz Rule and re-arranging: udz v z u v y d w + u ( h) + v ( h) w 0 y y z z h (.5.) Velocitie at the free urface and ottom may e expreed a: D w z Dt z u z v z y (.5.2) w z D h h ( ) Dt z h ( h) ( h) ( h) u z h v z h y (.5.3) h( x, y) Re-writing aove and noting that u v w y z (.5.4) u ( h) + v ( h) w y z h 0 (.5.5) p..5.6

4 CE Topic.5 - Spring Revied Feruary 3, :20 pm Sutituting thee expreion into the continuity equation we have: udz vdz 0 y (.5.6) However y definition we have: udz vdz q x q y (.5.7) (.5.8) Again utituting: q x + q y 0 y (.5.9) Noting that Hũ, q y Hṽ where ũ, ṽ equal the vertically averaged velocitie: q x ( ũh) ( ṽh) y (.5.20) p..5.7 CE Topic.5 - Spring Revied Feruary 3, :20 pm Depth Integrated Reynold Equation Conider the x-direction Reynold equation: u u u v u w u y z ---- p τ xx τ yx τ zx y z ρ 0 ρ 0 ρ 0 ρ 0 (.5.2) where τ xx µ u ρ (.5.22) 0 u'u' τ yx µ u ρ y 0 u'v' (.5.23) τ zx µ u ρ z 0 u'w' (.5.24) Now conider the hydrotatic preure equation: p ρg z (.5.25) p..5.8

5 CE Topic.5 - Spring Revied Feruary 3, :20 pm Integrating thi equation etween the free urface at z and ome level z p( x, y, z) p ( x, y) p z z ρgz (.5.26) where preure at the free urface (.5.27) p Auming that denity i contant: p p ( ρgz ρg) (.5.28) p p + ρg ρgz (.5.29) The preure gradient term in the x-direction Reynold equation ecome p ρ -- p g ρ (.5.30) Auming that urface preure doe not vary patially: p g ρ (.5.3) p..5.9 CE Topic.5 - Spring Revied Feruary 3, :20 pm Thu the x-direction Reynold equation with the aumption of contant denity flow, and hydrotatic preure ditriution and contant atmopheric preure ecome: u u u v u w u g y z -- τ xx τ yx τ zx ρ ρ y ρ z (.5.32) Now add u time the continuity equation to the aove equation: u u u v u w u u u u v u w y z y z g τ xx τ yx τ zx + ρ ρ y ρ z (.5.33) Re-arranging u u 2 ( uv) ( uw) y z g τ xx τ yx τ zx + ρ ρ y ρ z (.5.34) p..5.0

6 CE Topic.5 - Spring Revied Feruary 3, :20 pm Vertically averaging thee equation: g u u 2 ( uv) dz dz dz + y dz + τ xx dz + -- ρ dz + ρ y τ yx ( uw) dz z τ zx dz ρ z (.5.35) Apply Leinitz rule a follow: ( x, y, h( x, y) f dz ( h) ---- f z f d z f z h (.5.36) where f u, u 2, uv,, τ xx, τ yx (.5.37) Furthermore, we note that: g dz g g z z g z h (.5.38) where g uw, τ zx (.5.39) p..5. CE Topic.5 - Spring Revied Feruary 3, :20 pm Sutituting and re-arranging we have: ---- udz u 2 + dz uvdz y u u v w + u ( h) + u ( h) + v ( h) w y y z z h g----- z g g ( h) τ d + + ρ xx dz τ ρy yx z ρ -- τ + d xx τ yx τ y zx z ( h) ( h) -- τ ρ xx τ yx τ y zx z h (.5.40) p..5.2

7 CE Topic.5 - Spring Revied Feruary 3, :20 pm However a we noted efore: u v w y z 0 (.5.4) ( h ) + u ( h) + v ( h) w y z h 0 (.5.42) By performing a tre alance at the urface, it can e hown that tre in the x-direction and parallel to the urface. τ x applied urface τ x τ xx t m τ yx τ y zx z (.5.43) Similarly at the ottom: τ x τ xx t m ( h) ( h) τ yx τ y zx z h (.5.44) p..5.3 CE Topic.5 - Spring Revied Feruary 3, :20 pm Sutituting reduce the x-momentum equation to: ---- udz u 2 + dz uvdz y g----- z g g ( h) τ ρ xx z τ ρy yx z ---- τ x d + + d + d ρ ρ τ x (.5.45) Let u define the depth averaged variale a α --- αdz H (.5.46) The Reynold averaged quantity i then defined a the um of the depth averaged variale and the deviation from the depth averaged variale α α + αˆ (.5.47) p..5.4

8 CE Topic.5 - Spring Revied Feruary 3, :20 pm Thu we define velocitie in term of the depth averaged quantity and the deviation from the depth averaged quantity Thu patial averaging i applied a: udz Hũ vdz Hũ (.5.48) (.5.49) Furthermore we let: u( x, y, z, ũ( x, y, + û( x, y, z, v( x, y, z, ṽ( x, y, + vˆ ( x, y, z, (.5.50) (.5.5) p..5.5 CE Topic.5 - Spring Revied Feruary 3, :20 pm Thi implie that: ûdz 0 and vˆ dz 0 (.5.52) Hence u 2 d z ( ũ 2 + ũû + û 2 ) dz u 2 d z ũ 2 dz + 2ũ ûdz + u 2 d z Hũ 2 + û 2 dz û 2 dz (.5.53) (.5.54) (.5.55) Similarly uv d z ( ũṽ + ũvˆ + ûṽ + ûvˆ ) dz ũṽh + ûvˆ dz (.5.56) p..5.6

9 CE Topic.5 - Spring Revied Feruary 3, :20 pm Finally dz dz H (.5.57) Sutituting and re-arranging: ( Hũ) ( Hũ 2 ) ( Hũṽ) y g ( H) g g h τ xx û 2 d z ρ y t τ m yx ûvˆ τ d z x ---- ρ ρ ρ τ x (.5.58) Expanding the term involving gravity: g ( H) + g g h g H H ( + h) (.5.59) g ( H) + g g h gh (.5.60) p..5.7 CE Topic.5 - Spring Revied Feruary 3, :20 pm We alo note that the acceleration term can e expanded a: ( Hũ) ũ H H----- ũ ( Hũ) ũ ( + h) + H ũ ( Hũ) ũ H----- ũ (.5.6) (.5.62) (.5.63) ( Hũ ) ũ ( Hũ) + Hũ----- ũ (.5.64) ( Hũṽ) y ũ ( Hṽ) + Hṽ ũ y y (.5.65) p..5.8

10 CE Topic.5 - Spring Revied Feruary 3, :20 pm Sutituting in the gravity and acceleration term re-arrangement: H ũ Hũ ũ Hṽ ũ ũ y ( Hũ) ( Hṽ) y gh τ xx û 2 + dz ρ y τ yx ûvˆ τ τ d z x ---- x ρ ρ ρ (.5.66) It i clear that the depth averaged continuity equation i emedded in the previou equation and therefore drop out. Dividing through y H reult in the depth averaged conervation of momentum equation in non-conervative form (refer to u having altered the form of the acceleration term). ũ ũ----- ũ + ṽ ũ g y H τ xx û 2 dz ρ H y τ yx ûvˆ z ρ --- τx τ x d H ρ H ρ (.5.67) p..5.9 CE Topic.5 - Spring Revied Feruary 3, :20 pm The Shallow Water Equation are collectively written a: ( ũh) ( ṽh) y (.5.68) ũ ũ----- ũ + ṽ ũ g y H τ xx û 2 dz ρ H y τ yx ûvˆ z ρ --- τx d H ρ τx H ρ (.5.69) ṽ ũ----- ṽ + ṽ ṽ g y y H τ xy ûvˆ dz ρ H y τ yy vˆ2 z ρ --- τy d H ρ τy H ρ (.5.70) p..5.20

11 CE Topic.5 - Spring Revied Feruary 3, :20 pm The Shallow Water Equation were etalihed in 775 y Laplace. The Momentum Conervation Statement are quite imilar to the Reynold equation with the following exception: Variale are now depth averaged quantitie. The z-dimenion ha een eliminated. There are convective inertia force caued y the flow deviation from the depth averaged velocitie ũ, ṽ. Thee equation have uilt into them 3 level of averaging: Averaging over the molecular time/pace cale Averaging over the turulent time/pace cale Averaging over the depth pace cale The latter two produce momentum tranport term that are intimately related to the convective term. p..5.2 CE Topic.5 - Spring Revied Feruary 3, :20 pm There are now three mechanim of momentum tranfer uilt into thee equation: ν u dz type term are the vicou tree and repreent the averaged effect of molecular motion. Thee term are neceary ince we do not directly imulate the momentum tranfer via molecular level colliion. u'u' dz type term are the turulent Reynold tree and repreent the averaged effect of momentum tranfer due to turulent fluctuation. Thee term are neceary ince we are not directly imulating momentum tranfer via turulent fluctuation. ûûdz type term repreent the preading of momentum over the water column. Thi proce i known a momentum diperion. Thee term are neceary ince we are no longer directly imulating thi proce via the actual depth varying velocity profile. They pread momentum laterally. p..5.22

12 CE Topic.5 - Spring Revied Feruary 3, :20 pm Momentum diperion can e conceptualized y following a pollutant pread over the water column. Non-depth average imulation at t t 0 Non-depth average imulation at t t Depth averaging the reult of the non-depth averaged imulation at t t p CE Topic.5 - Spring Revied Feruary 3, :20 pm Depth averaged imulation at t t 0 Depth averaged imulation at t t without diperion term Thu, diperion term are neceary! p..5.24

13 CE Topic.5 - Spring Revied Feruary 3, :20 pm The Shallow Water equation greatly implify flow computation in free urface water odie. Reduce the numer of p.d.e. from 4 to 3. Reduce the complexity of the variale ũ( x, y,, ṽ( x, y,, ( x, y, intead of (.5.7) u( x, y, z,, v( x, y, z,, w( x, y, z,, p( x, y, z, (.5.72) Built-in poitioning of the free urface oundary which i typically unknown when applying the Reynold equation. The Shallow Water equation include 0 additional unknown a compared to the Navier-Stoke equation: ( u'u' ) ( u'v' ) ( v'v',, ) Lateral turulent momentum diffuion ûû, ûvˆ, vˆvˆ Lateral momentum diperion related to vertical velocity profile, τy Applied free urface tre τ x τ x, τy Applied ottom tre. It i related to the vertical velocity profile, momentum tranport through the water column, ottom roughne. p CE Topic.5 - Spring Revied Feruary 3, :20 pm Thee 0 additional unknown require that 0 contitutive relationhip are provided in order to cloe the ytem. A very imple cloure model for the comined lateral momentum diffuion (due to turulence) and diperion (due to averaging out vertical velocity profile) i: τ xy τ xx û 2 Hũ dz E ( ) ρ xx τ yy vˆ 2 Hṽ dz E ( ) ρ yy y ( Hũ) ( Hṽ) ûvˆ d z Exy ρ y (.5.73) (.5.74) (.5.75) E xx, E yy and E xy are called the eddy diperion coefficient. Thi model aume that the diperion proce dominate the turulent momentum diffuion proce which dominate the molecular momentum diffuion proce. In a typical gravity-driven open channel flow, the lateral momentum diperion term do not play a major role in the momentum alance equation - they can e neglected. p..5.26

14 CE Topic.5 - Spring Revied Feruary 3, :20 pm In cae where a trong vertical velocity profile exit (i.e. wind driven flow in deep water), the lateral diperion term may e important due to the u, v contriution. The importance of thee term i tied to the advective term. In cae where very trong lateral gradient in the horizontal velocity profile exit (e.g. trong flow emerging into a receiving water ody), the lateral diperion term may e important due to the u', v' contriution. The importance of thee term i related to the convective term. The eddy diperion model provide equation for 6 unknown (ince unknown were conolidated). We till need contitutive relationhip for 4 unknown. p CE Topic.5 - Spring Revied Feruary 3, :20 pm Surface tre i very mall unle wind i lowing over the water. Wind tre relationhip depend on ea urface roughne and 0 m wind peed. They are not related to the water peed. Bottom tre i cloed y empirical relationhip: τx c f ( u + v ) u ρ (.5.76) τy c f ( u + v ) v ρ (.5.77) c f friction factor (.5.78) c f --- f DW Darcy Weiach 8 (.5.79) g c f Chezy c (.5.80) where 2 n g - Manning c f h p (.5.8)