Transport: Non-diffusive, flux conservative initial value problems and ...

More documents

Recommendations

Info

66 a concentration b concentration 3.5 3 2.5 2 1.5 1 0.5 3.5 3 2.5 2 1.5 1 0.5 t=100 t=0 0 2 4 6 8 10 distance t=100 t=0 0 2 4 6 8 10 distance Figure 5.5: (a) Evolution of gaussian initial condition (amplitude 3, width 1., 257 grid points) using a staggered leapfrog scheme with α = 0.5. (b) Gaussian is narrower (width=.5) but α = .5 For well resolved problems with α close to one however, this is an easy and accurate scheme.
Transport equations 67 j-1 F(j-1/2) F(j+1/2) j Figure 5.6: A simple staggered grid used to define the control volume approach. Dots denote nodes where average values of the control volume are stored. X’s mark control volume boundaries at half grid points. tive of the average value of the control volume, then we can replace the first integral by cj∆x. The second integral is the surface integral of the flux and is exactly j+1 � cV · dS = cj+1/2Vj+1/2 − cj−1/2Vj−1/2 s (5.5.11) which is just the difference between the flux at the boundaries F j+1/2 and F j−1/2. Eq. (5.5.11) is exact up to the approximations made for the values of c and V at the boundaries. If we assume that we can interpolate linearly between nodes then c j+1/2 = (cj+1 + cj)/2. If we use a centered time step for the time derivative then the flux conservative centered approximation to is c n+1 j − cn−1 j ∂c ∂cV + ∂t ∂z = 0 (5.5.12) ∆t � � = − Vj+1/2(cj+1 + cj) − Vj−1/2(cj + cj−1) ∆x (5.5.13) or if V is constant Eq. (5.5.13) reduces identically to the staggered leapfrog scheme. By using higher order interpolations for the fluxes at the boundaries additional differencing schemes are readily derived. The principal utility of this sort of differencing scheme is that it is automatically flux conservative as by symmetry what leaves one box must enter the next. The following section will develop a slightly different approach to choosing the fluxes by the direction of transport. 5.5.3 Upwind Differencing (Donor Cell) The fundamental behaviour of transport equations such as (5.5.13) is that every particle will travel at its own velocity independent of neighboring particles (remember the characteristics), thus physically it might seem more correct to say that if the flux is moving from cell j − 1 to cell j the incoming flux should only depend on the concentration upstream. i.e. for the fluxes shown in Fig. 5.6 the upwind differencing for the flux at point j − 1/2 should be F j−1/2 = � cj−1Vj−1/2 Vj−1/2 > 0 cjVj−1/2 Vj−1/2 < 0 (5.5.14)
Page 1 and 2: Chapter 5 Transport: Non-diffusive,
Page 3 and 4: Transport equations 55 the particle
Page 5 and 6: Transport equations 57 easily taken
Page 7 and 8: Transport equations 59 x = i∆x (n
Page 9 and 10: Transport equations 61 concentratio
Page 11 and 12: Transport equations 63 or dividing
Page 13: Transport equations 65 a concentrat
Page 17 and 18: Transport equations 69 concentratio
Page 19 and 20: Transport equations 71 a c concentr
Page 21 and 22: Transport equations 73 An example o
Page 23 and 24: Transport equations 75 c getweights
Page 25 and 26: Transport equations 77 return end i
Page 27 and 28: Transport equations 79 would be ext
Page 29 and 30: Transport equations 81 (which works
Page 31: Transport equations 83 they are mor

Transport: Non-diffusive, flux conservative initial value problems and ...

Create successful ePaper yourself

Delete template?

Save as template?