Transport Equation. Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung. . p.1/21. home/lehre/vl-mhs-1-e/cover_sheet.

Transkript

1 Transport Equation home/lehre/vl-mhs-1-e/cover_sheet.tex. p.1/21

2 Table of contents 1. Introduction 2. Transport Equation 3. Analytical Solution 4. Discretization of the transport equation 5. Explicit schemes 6. Implicit schemes home/lehre/vl-mhs-1-e/toc_8.tex. p.2/21

3 Material / substantial derivation - Substantial acceleration - De Dt }{{} 1 = e t }{{} 2 e ˆ= ρ (continuity) e ˆ= ρv (momentum eqn.) e ˆ= ρu (energy eqn.) + v e }{{} 3 1 : substantial acceleration 2 : local acceleration 3 : advective acceleration home/lehre/vl-mhs-1-e/mat_subst_der.tex. p.3/21

4 Navier-Stokes Equation The Navier-Stokes equation can be seen as a transport equation for velocities and is a vector-equation. L(v, p) := ρ v t + v } {{ v } Advection σ = τ pi }{{} σ + f = 0 Viscosity The surface forces σ are the sum of hydrostatic pressure p plus the viscous stresses τ. f are the body forces, which is in our case gravity. For a newtonian fluid the viscous stresses are proportional to the rate of strain and the viscosity. For example: τ xx = 2µ v x x. home/lehre/vl-mhs-1-e/navier_stokes.tex. p.4/21

5 Transport Equation The classical transport equation is a scalar-equation for e.g. a concentration. L(c) := c t + v } {{ c } (D c) +r = 0 }{{} Advection Diffusion The dimensionless number that describes the ratio between the advection and the diffusion is the Peclet number. P e = Advection Diffusion P e = v L D [ ] (Peclet Number) L ˆ= characteristic lenght, e.g length of the domain home/lehre/vl-mhs-1-e/transp_eqn.tex. p.5/21

6 Transport properties Typical transport properties in selected hydrosystems V D M,D L Pe = vl D X s [m] t River 1 [m/s] 25 [m 2 /s] [s] Estuary 0.05 [m/s] 10 [m 2 /s] [s] Groundwater 1 [m/d] 50 [m 2 /d] [d] V ˆ= Typical transport velocity D M ˆ= Typical longitudinal diffusion coefficient Pe ˆ= Convection diffusion ratio for a mixing length of 50m X s ˆ= Width of distribution for a mixing length of 50m t ˆ= Time after solute is mixed home/lehre/vl-mhs-1-e/comparison.tex. p.6/21

7 1D - Transport-Equation Let us consider one of the representative model equations, for instance the advection-diffusion equation, written here as follows: where: u t }{{} 1 + v u x }{{} 2 x D u }{{ x} 3 = 0 1 ˆ= accumulation term 2 ˆ= advective / convective term 3 ˆ= diffusive term and u is the unknown function of (x,t), v is the convective speed and D the hydrodynamic dispersion tensor. home/lehre/vl-mhs-1-e/1d_transp_eqn.tex. p.7/21

8 Analytical Solution The Peclet-number is equal to: P e = vh D The analytic solution for the steady state transport equation, i.e. u = 0 is t u u i = exp(p e x) 1 h u j u i exp(p e) 1, { u = u i for x i = 0 u = u j for x j = h The solution of this equation is displayed on the next slide for different values of the Peclet number. home/lehre/vl-mhs-1-e/analyt_sol.tex. p.8/21

9 Analytical Solution II eplacements u j Pe << 1 Pe = 1 Pe =0 Pe = 1 u i i P e = v + P e = 0 j u j u i x i = 0 Pe >> 1 x u i x j = h v v = 0 i j u j h home/lehre/vl-mhs-1-e/analyt_sol_2.tex. p.9/21

10 Discretization of the Transport Equation c t + {vc D c} q = 0 One possibililty to discretize this equation is to separate the advection and the diffusion term c t + v c + c v (D c) q = 0. The third term is zero, because of the continuity equation c t + v c (D c) q = 0. Note: This discretization is not very well suited for IFD/FVmethods. home/lehre/vl-mhs-1-e/discr_transp_1.tex. p.10/21

11 Discretization of the Transport Equation c t + {vc D c} q = 0 The second possibility is to formulate the equation in a general integral form G c t dg + G {vc D c} dg and then apply the theorem of Gauss c t dg + (vc D c) n dγ G Γ G G q dg = 0, q dg = 0. home/lehre/vl-mhs-1-e/discr_transp_2.tex. p.11/21

12 Discretization of the Transport Equation The first discretization scheme is very well suited for the FDM, as the differential equation can be directly transferred to a difference equation. For example: implicit time discretization and a central difference scheme in space. c n+1 i c n i t + v cn+1 i+1 cn+1 i 1 2 D cn+1 i+1 2cn+1 i + c n+1 i 1 q 2 i = 0, Note: In this case the advective term is discretized incorrectly. home/lehre/vl-mhs-1-e/discr_transp_3.tex. p.12/21

13 Discretization of the Transport Equation The second discretization scheme is very well suited for the IFDM and the FVM, as the flux over the boundary of the control volumes can be approximated. For example: implicit time discretization G c n+1 i c n i t dg+ Γ (v cn+1 i+1 cn+1 i D cn+1 i+1 ) cn+1 i 1 dγ q i dg = 0, Note: In this case the advective term is approximated incorrectly. home/lehre/vl-mhs-1-e/discr_transp_4.tex. p.13/21

14 Diffusive and advective fluxes Diffusive fluxes Advective fluxes yield diagonal dominating matrices with positive coefficients the resulting system of equations can be solved without stability problems. (e.g. Laplace-equation) Transport properties of the advective flux have to be described correctly by choosing the right discretization method. the differential operator describing convection is not symmetric and e.g. nonlinear for the Navier-Stokes equation. home/lehre/vl-mhs-1-e/diff_adv_flux.tex. p.14/21

15 Explicit schemes for advective transport Unstable scheme t unknown k+1 known t k j 1 j j +1 s The partial derivations of the differential equation are replaced by differences quotients: u s uk j+1 u k j 1, 2 u t uk+1 j u k j t. The function values are taken from the known time plane: u u k j. home/lehre/vl-mhs-1-e/unstab_method.tex. p.15/21

16 Lax Scheme I t unknown k+1 known t k j 1 j j +1 s The diffusive (lax) method distinguishes itself from the unstable model insofar as that for the time derivation not u k j is used, but a mean from u k j 1 and u k j+1. home/lehre/vl-mhs-1-e/lax_scheme_1.tex. p.16/21

17 Lax Scheme II u x uk j+1 u k j 1 2 (as in the unstable method) u t 2uk+1 j u k j 1 u k j+1 2 t (new compared to the unstable method) or, more general: u t uk+1 j [ αu k j + (1 α) ( ) ] u k j 1 + u k j+1 /2 t (0 α 1; for α = 1 results the unstable model). home/lehre/vl-mhs-1-e/lax_scheme_2.tex u u k j or u uk j 1 + u k j+1 2. p.17/21

18 Upstream Difference Method u x u x uk j u k j 1 uk j+1 u k j This method takes into account that during the course of the observed time step, quantities of state and their derivatives are transported from a particular direction to the point j. u t uk+1 j u k j, u u k j t k+1 k flow from left to right flow from right to left t t j 1 j j +1 v > 0 unknown known s home/lehre/vl-mhs-1-e/upstream_scheme.tex. p.18/21

19 Implicit schemes for advective transport t k+1 unknown known t k j 1 j j + 1 s In the implicit methods, the influence of the unknown quantities of state u k+1 j 1, uk+1 j+1 of the neighboring grid points to the new time level is considered. An equation system for the vector of the unknown quantity u k+1 i results. The time step t is not limited from the start. home/lehre/vl-mhs-1-e/implicit_transp.tex. p.19/21

20 Preissmann Scheme t k+1 Θ t ( 1 Θ ) t unknown known approximated k i x ij 2 x ij 2 j s ij The Preismann schems differs from the before mentioned schemes in that it not only uses point information but also includes the derivatives in the approximation. The first derivatives are depictable in the point + in the space and time directions. home/lehre/vl-mhs-1-e/preismann_1.tex. p.20/21

21 Preissmann Scheme II u + x = u+ Θuk+1 x j u k+1 i u + t = u+ t = 1 2 t 1 t ( u k+1 j ij ( u k+1 j + u k+1 i 2 u k j + uk+1 i u k i ). + (1 Θ) uk j u k i, ij ) uk j + u k i 2 For the quantity of state in the calculation point, the mean value is established: u + Θ uk+1 j + u k+1 i + (1 Θ) uk j + u k i. 2 2 The method is stable for Θ > 0, 5. yields best results between: Θ 0, 55 0, 6. and creates numeric damping for larger values. home/lehre/vl-mhs-1-e/preismann_2.tex. p.21/21