Transport Equation. Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

Transkript

1 Transport Equation home/lehre/vl-mhs--e/folien/vorlesung/8_transport/cover_sheet.tex page of 22. p./22

2 Table of contents. Introduction 2. Transport Equation 3. Analytical Solution 4. Discretization of the transport equation home/lehre/vl-mhs--e/folien/vorlesung/8_transport/toc_8.tex page 2 of 22. p.2/22

3 Material / substantial derivation - Substantial acceleration - De Dt }{{} = e t }{{} 2 e ˆ= ρ (continuity) e ˆ= ρv (momentum eqn.) e ˆ= ρu (energy eqn.) + v e }{{} 3 : substantial acceleration 2 : local acceleration 3 : advective acceleration home/lehre/vl-mhs--e/folien/vorlesung/8_transport/mat_subst_der.tex page 3 of 22. p.3/22

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--e/folien/vorlesung/8_transport/navier_stokes.tex page 4 of 22. p.4/22

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 length, e.g length of the domain home/lehre/vl-mhs--e/folien/vorlesung/8_transport/transp_eqn.tex page 5 of 22. p.5/22

6 Transport properties Typical transport properties in selected hydrosystems V D M,D L Pe = vl D X s [m] t River [m/s] 25 [m 2 /s] [s] Estuary 0.05 [m/s] 0 [m 2 /s] [s] Groundwater [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--e/folien/vorlesung/8_transport/comparison.tex page 6 of 22. p.6/22

7 D - Transport-Equation Let us consider one of the representative model equations, for instance the advection-diffusion equation, written here as follows: u t }{{} + v u x }{{} 2 x D u }{{ x} 3 = 0 where: 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--e/folien/vorlesung/8_transport/d_transp_eqn.tex page 7 of 22. p.7/22

8 Derivation of the analytical solution Consider the steady state transport equation v du dx D d2 u dx 2 = 0 () Take exponential ansatz u = e λx. Insert it vλe λx Dλ 2 e λx = 0 (2) and solve the eigen-value problem (vλ Dλ 2 )e λx = 0 λ(v Dλ) = 0 λ 0 = 0 λ = v D. home/lehre/vl-mhs--e/folien/vorlesung/8_transport/derivation.tex page 8 of 22. p.8/22

9 Derivation of the analytical solution The general solution is: u(x) = M e λ0x + M 2 e λ x = M + M 2 e v D x With the boundary conditions, we get u(0) = u i M + M 2 = u i u(l) = u j M + M 2 e v D L = u j M = u i M 2 u i M 2 + M 2 e v D L = u j M 2 = u j u i e v D L M = u i u j u i e v D L home/lehre/vl-mhs--e/folien/vorlesung/8_transport/derivation2.tex page 9 of 22. p.9/22

10 Derivation of the analytical solution This yields the solution for our problem This can be rearranged u(x) = u i u j u i e v D L + u j u i e v D L e v D x. u u i u j u i = e v D L + e v D x e v D L = e v D x e v D L = ep e x L e P e. home/lehre/vl-mhs--e/folien/vorlesung/8_transport/derivation3.tex page 0 of 22. p.0/22

11 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) h u j u i exp(p e), { 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--e/folien/vorlesung/8_transport/analyt_sol.tex page of 22. p./22

12 Analytical Solution II eplacements u j Pe << Pe = Pe =0 Pe = u i i P e = v + P e = 0 j u j u i x i = 0 Pe >> x u i x j = h v v = 0 i j u j h home/lehre/vl-mhs--e/folien/vorlesung/8_transport/analyt_sol_2.tex page 2 of 22. p.2/22

13 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--e/folien/vorlesung/8_transport/discr_transp_.tex page 3 of 22. p.3/22

14 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+ i c n i t + v cn+ i+ cn+ i 2 x D cn+ i+ 2cn+ i + c n+ i q x 2 i = 0, Note: In this case the advective term may be discretized incorrectly. For steady state : look at the stability of the scheme home/lehre/vl-mhs--e/folien/vorlesung/8_transport/discr_transp_3.tex page 4 of 22. p.4/22

15 steady state, no sources/sinks Stability v c i+ c i D c i+ 2c i + c i = 0 2 x x 2 v x D (c i+ c i ) 2(c i+ 2c i + c i ) = 0 v x with D = P e (P e 2)c i+ + 4c i (P e + 2)c i = 0 boundary conditions: c 0 = ; c Nx = 0 (with N x the number of points) home/lehre/vl-mhs--e/folien/vorlesung/8_transport/stability.tex page 5 of 22. p.5/22

16 Stability Now replace c i with the ansatz e λih (P e 2)e λ(i+)h + 4e λih (P e + 2)e λ(i )h = 0 : e λ(i )h (P e 2)e 2λh + 4e λh (P e + 2) = 0 (P e 2) ( e λh) 2 + 4e λh (P e + 2) = 0 ; e λh = β (P e 2)β 2 + 4β (P e + 2) = 0 Solve the quadratic equation for β with the quadratic formula. home/lehre/vl-mhs--e/folien/vorlesung/8_transport/stability2.tex page 6 of 22. p.6/22

17 Stability β /2 = b± b 2 4ac 2a = 4± 6 + 4(P e 2)(P e + 2) 2P e 4 = 4± 6 + 4P e 2 6 2P e 4 = 4±2P e 2P e 4 = 2±P e P e 2 2 P e β = β P e 2 2 = 2+P e P e 2 β = (P e+2) P e 2 β 2 = home/lehre/vl-mhs--e/folien/vorlesung/8_transport/stability3.tex page 7 of 22. p.7/22

18 Stability Calculate M and M 2 in the general solution c i = M + M 2 β i 2 with the boundary conditions. c i (x = 0) = = M + M 2 M = M 2 c i (N x ) = 0 = M + M 2 β N x Solve for M 2 : 0 = M 2 + M 2 β N x = M 2 (β N x ) M 2 = β N x and M : M = + β N x = βn x β N x home/lehre/vl-mhs--e/folien/vorlesung/8_transport/stability4.tex page 8 of 22. p.8/22

19 Stability This yields the solution c i = β (N x β N x ) βi β N x = βn x β i β N x i N x = β β N x = β (N x i) β N x. Examine the solution. If the Peclet number is greater than 2, then β is smaller than zero. As the exponent N x i will take alternating odd and even values, the contribution will be positive and negative for odd and even nodes. This will lead to oscillations. The scheme is therefore only stable, if Pe < 2 is satisfied. home/lehre/vl-mhs--e/folien/vorlesung/8_transport/stability4.tex page 9 of 22. p.9/22

20 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--e/folien/vorlesung/8_transport/discr_transp_2.tex page 20 of 22. p.20/22

21 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+ i c n i t dg + Γ ( vc n+ i D cn+ 2 x i+ cn+ i ) dγ q i dg = 0. Note: In this case the advective term is approximated correctly, if we pick the correct concentration, depending on the flow direction. home/lehre/vl-mhs--e/folien/vorlesung/8_transport/discr_transp_4.tex page 2 of 22. p.2/22

22 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--e/folien/vorlesung/8_transport/diff_adv_flux.tex page 22 of 22. p.22/22