SMS-618, Particle Dynamics, Fall 2009 Modeling of suspended sediment transport From: http://www.usask.ca/geology/classes/geol243/243notes/243week3b.html What is sediment transport? Why does sediment transport take place? When does sediment transport take place? Where does sediment transport take place? Where is the sediment coming from? Types of sediment transported: Total sediment load Wash load Bed material load Mov ing as su spended load Mov ing as bed load Today’s topic What information do we need to know to model sediment transport? Flow field (‘physics’) Particle field (‘sedimentology’) Properties of flow away from Bed and wash material bottom boundary (wave, mean characteristics (e.g. density, size current), and of the water (e.g. ν). distribution, shape) Within the BBL the two are coupled: •Stress on bottom due to flow imparts the force that resuspends particles. •Flow is affected by added water density due to suspension of particles. •Flow is affected by settling particles. •Flow is affected by bottom morphology (e.g. ripples). What information do we need to know to model sediment transport? Flow field (‘physics’) Particle field (‘sedimentology’) Properties of flow away from Bed and wash material bottom boundary (wave, mean characteristics (e.g. density, size current), and of the water (e.g. ν). distribution, shape) The concept of a boundary layer: Southard, 2000 As flow encounter a wall the velocity right next to the wall has to vanish (no slip). Away from the wall the velocity is the free flow velocity. In between we get a ‘boundary layer’. Movie Laminar boundary layer: U U νx ∂u ∂w Scaling: U ~ν → δ = + = 0 L δ 2 U ∂x ∂z ÅNon-linear Blasius equ. Can be solved numerically. ∂u ∂u ∂ 2u u + w =ν ∂x ∂z ∂z 2 A more ‘realistic’ view: Southard, 2000 A hierarchy of BBLs: Southard, 2000 Example: boundary layer in a pipe: Comparison of laminar (i) and turbulent (ii) velocity profiles in a pipe for: (a) The same mean velocity (b) th e sam e drivin g f or ce (pr essur e difference). Figure 22.16 from Tritton, D.J. 1977. Physical Fluid Dynamics. Van Nostrand Reinhold, NY. p. 277. Æ BBL is smaller and more dissipative in the turbulent case Turbulent BBL structure: Mann and Lazier, 1996 The stress in the BBL varies from being dominated by the Reynolds stress away from the wall to being dominated by viscous stress right next the bed. Example: horizontal flow field in a bottom boundary layer Geostrophy/ buoyancy/ tidally Driven (frequency ω) Transition layer, du/dz~u*/z(~20% of BBL). Laminar flow, Re<500 Constant viscous stress layer, u~z, ‘smooth bottom’ (~1% of BBL) Jumars, 1993 κ~0410.41, von Karman ’s cons tan t. Bottom effect parameterized through z0 and 1/2 u*=(τ0/ρ) . τ0=νdu/dz+ρ<u’w’> Æ Two parameter fit to velocity profile. Example: horizontal flow field in a bottom boundary layer Southard, 2000 u du u * u * ∝ = dz z κz τ 0 u z u = * ln + u’ Ln z κ z0 u* κ Ln(z0) Equivalent to assuming: Keddy=κu*z Slope ~ u*/κ Intercept ~ u* lnz0 /κ Zo-roughness length. Example: horizontal flow field in a bottom boundary layer Predicting z0 from D or u*. Knowing z0, BBL depth and u∞ Æ u*. Jumars & Nowell. D-bottom grain size 1984. Am. Zool. 24: 45-55 Example: horizontal flow field in a bottom boundary layer http://cvu.strath.ac.uk/courseware/calf/CALF/bl/equations/eq5a.html Turbulent boundary layer is more dissipative; Applies more resistance to the flow. Sets up faster. Example: horizontal flow field in a bottom boundary layer Data for sand tracked by an epifaunal bivalve: Burst-Sweep cycle: Nowell, A.R.M., P.A. Jumars and J.E. Eckman. 1981. Effects of biological activity on the entrainment of marine sediments. Mar. Geol. 42: 155-172. Gravity waves: Effect s: Changes the mean flow field. Change bottom shear stress. Wave boundary layer is very shallow, δ~(4πνT)1/2, for a 4sec wave, δ~0.7cm. Orientation relative to mean current is important: 2 4 2 2 4 u *cw = u*c + 2u*cu*w cosφcw + u*w Finally, we get to particles… Rouse (1937 ) approach: Conservation of particle mass (sources and sinks comes as BCs): ∂C ∂C ∂(u C) ∂(v C) ∂(w C) + ∇ ⋅()uv C = + s + s + s = Κ∇2C ∂t s ∂t ∂x ∂y ∂z Assume no gradient in x and y, and divide into time-mean and fluctuations: CCC=+'; ws = wss + w' Equation for mean becomes: ∂(wC) ∂(wCss+ w'' C ) s ==0 ∂∂zz Rouse’s (1937) approach: Convert Reynolds’ stress flux into (eddy-)diffusive flux: ∂C KwC= − '' s ∂z s Combining we get: ∂∂⎡⎤C ⎢⎥wCss−= K 0 ∂∂zz⎣⎦ Issues: ws is a function of sediment size, excess weight, and shape. Ks is not necessarily the same as that of the fluid. Boundary conditions. Rouse’s (1937) approach; 1D balance: assume no net flux from top and bottom boundary (reduces to 1st order ODE). Solution (up to a constant of integration, C(z1)): Near the bottom: K = κu*z, KS = αK, 3 ≥ α ≥ 0.3: Defining R = w s κ u * we get: −R /α C (z) w z dz w z C (z) ⎛ z ⎞ ln = − s = − s ln ⇒ = ⎜ ⎟ C ()z ακu ∫ z ακu z C ()z ⎜ z ⎟ 1 * z1 * 1 1 ⎝ 1 ⎠ This profile fits well lower 30% of BBL. Higher up in the water column (post Rouse): K s = const. = ακu* H BBL w z w (z − z ) C ()z s s 1 ⎧ R()z − z1 ⎫ ln = − dz = − → C (z) = C (z1 )exp⎨− ⎬ C (z ) ακu H ∫ ακu H αH 1 * BBL z1 * BBL ⎩ BBL ⎭ This profile fits well upper 80% of BBL. Comparison with laboratory observations (See Allen, 2001) −R /α C ()z w z dz w z C (z) ⎛ z ⎞ ln = − s = − s ln ⇒ = ⎜ ⎟ C ()z ακu ∫ z ακu z C ()z ⎜ z ⎟ 1 * z1 * 1 1 ⎝ 1 ⎠ w z w (z − z ) C ()z s s 1 ⎧ R()z − z1 ⎫ ln = − dz = − → C ()z = C ()z1 exp⎨− ⎬ C ()z ακu H ∫ ακu H αH 1 * BBL z1 * BBL ⎩ BBL ⎭ The two solutions are matched at z~HBBL/5 Problem with the Rouse equations near the bottom when the sediment concentration is large. Taylor and Dyer (1977) approach; Add effects of sediment concentration on density. Let’s the eddy coefficient vary relative to that of the unstratified fluid. K K = s,strat z γ + β L Where the Monin-Obukov-length, L, is defined as (based on shear stress and buoyancy flux being the two fundamental processes): u 3 ρ L = * κgρ'w' β and γ are constant (e.g. 4.7-5.2 and 0.74 respectively, Styles and Glenn, 2000) and z/L is termed the stability parameter. Taylor and Dyer’s (1977) approach; Add effects of sediment concentration on density. Let’s the eddy coefficient vary relative to that of the unstratified fluid. n-size bin. ()ρ − ρ sn 0 (ρsn − ρ0 ) ρ'= C'n ⇒ ρ'w' = w'C' ∑ ρ ∑ n n 0 n ρ0 ⎡ ()ρsn − ρ0 ⎤ ρ = ρ0 ⎢1+ ∑ Cn ⎥ ⎣ n ρ0 ⎦ Since: u 3 ρ L = * κgρ'w' Assuming all sediment classes have the same density: z zκg()ρs − ρ0 = 3 wsn 'Cn ' L u* ρ0 ρ Taylor and Dyer’s (1977) approach; Add effects of sediment concentration on density. Let’s the eddy coefficient vary relative to that of the unstratified fluid. Denoting by A = β z L (with β=5.2 and z/L from above) and Rouse number Rn = wsn κu* and for small rouggghness length scale z0,,, close to the bed, where K=κu*z, the analytical solution for the flow and particle concentration is: ⎡ ⎧ 1−Rn ⎫⎤ ⎛ C()z ⎞ z 1 ⎪ AR ⎡⎛ z ⎞ ⎤⎪ ⎜ ⎟ ⎢ n ⎢⎜ ⎟ ⎥ ⎥ ln⎜ ⎟ = −Rn ln + ln⎨1+ ⎜ ⎟ −1 ⎬ C ⎢ z R ()1− R ⎢ z ⎥ ⎥ ⎝ 0 ⎠ ⎣ 0 n ⎩⎪ n ⎣⎝ 0 ⎠ ⎦⎭⎪⎦ ⎛ ⎧ ⎡ 1−Rn ⎤⎫⎞ u* ⎜ z 1 ⎪ ARn ⎛ z ⎞ ⎪⎟ u()z = ln + ln⎨1+ ⎢⎜ ⎟ −1⎥⎬ κ ⎜ z R ()1− R ⎢ z ⎥ ⎟ ⎝ 0 n ⎩⎪ n ⎣⎝ 0 ⎠ ⎦⎭⎪⎠ Note that both flow and concentration field is affected and that for an unsorted sediment there is a need to find a way to characterize the effect of all size classes on velocity (through Rn), eg:e.g.: ~ ws = ∑Cn wsn ∑Cn n n Boundary conditions (needed when there is no continuous field data) To solve the particle concentration equations we need to BCs. ∂∂⎡⎤C ⎢⎥wCss−= K 0 ∂∂z ⎣⎦z The top B C i s l ess im port ant (in th e limit of infinit e ocean , CÆ00eeosao there. For shallow waters specify no flux. Can incorporate flux from a productive ML if needed. BC at (near) the bottom (τcn-critical shear stress): ⎧ γ S C δ − ,t 0 n S > 0 Concentration BC (Sn=(τd- τcn)/τcn ): + ⎪ n ( ) n Cn ()δ ,t = ⎨ 1+ γ S 0 n otherwise ⎩⎪ 0 A problem with this approach is that γ0 varies by 3 orders of magnitudes across studies and by 2 orders of magnitude within a single study over a short time. Boundary conditions Flux BC: ∂Cn − − − wsn Cn − K s = J ei (0 ,t)+ J di (0 ,t) ∂Z z=0+ − ⎧Ce S n S n > 0 − + J ei (0 ,t) = ⎨ , J di (0 ,t) = − pn wsnCn (0 ,t) ⎩ 0 otherwise with Ce an empirically determined erosion rate coefficient and pn the probability that a falling particle makes contact with the bed and remains there. Two models for pn are used: pn=1 for all shear stresses, in which case erosion balances diffusion in the BC. The second model assumes: ⎧1−τ 0 τ dn τ 0 > τ dn pn = ⎨ ⎩ 0 otherwise The depositional shear stress of class n, τdn, defines the stress below which sediment is able to remain on the bed after contacting it.