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fe m Fri Feb1508:15:072002 1. 693e-57 6 viscosity of air at 3000m mp =917*4*pi/3*a 3: t mass of ice particle 8 Initial conditions u0 -0.001: 8 Use a small initial velocity to avoid problems with Re=o and CD z0- 0: 8 Initial particle locat et time length of integration, and number of steps t Start iterative loop for n 2: N+1 8 Calculate drag at n-l 6/(1+sqrt(Re)) te right-hand sides at n-1 8 Update using Forward Euler :,n-1)+dt·f t Plot results linspace [0, Tmax, N+ subplot (211) abel('time): ylabel[ 'z' subplot (212) xlabel ('time'): ylabel('u')
Fri Feb1508:15:122002 rhog =0.9:8 Density of air at 3000m mu =1. 693e-5: t Viscosity of air at 3000m b Check on conver 9.8: t Gravi 7*4pi/3a; t mass of ice particl fprintf( Maximum number of Newton sub-iterations occurred\n')i vI:, n)=w z0= 0: 6 Initial particle locatio t Set time length of integration, and number of steps B Plo t- linspace 10, Tax, N+l) dt = Tmax/N ewton-Raphson convergence parameters ubplot (211) t Initialize vector for oDE integration title(Ba Euler integration' 212); xlabel('time'l: ylabel('u 8 Start iterative loop for n =2: N+1 t Begin sub-iteration loop ng this forces at least one sub-iteration to occur while((m Mmax)&(curres restol)) u"w(} rhog,u·2*a}/mu; pRe_pu rhog* (2*a)/mu =24/Re+6/1+sqrt(Re))+0.4 pCD pRe=-24"Re^(-2)-6*(1+sqrt{Re)^(-2}*0.5/sqrt(Re) 0.5·hogu^2pi"a^2·cD; pDpu=0,5*rhog*u^2p1·a^2· pCD_ pRepRe_pu+rhog"pia^2“uCD; late right-hand sides us linearization of f with respect to w 1)=-pD_pu/mp: i this is pfu dual with respect to w 5 Calculate dw from pres_dwdw = -res
drop befd Fri Feb1508:32:462002 号 Initia1 Ize some pa rhog=0.9: t Dens air at 3000m 钅Incr sub-iteration counter 9.8:tGravity 917*4*pi/3*a^3:8四s8 g check on convergence t st or ater us fprintf('Maxiaum number on sub-iterations occurred\n') t Initial conditions l1 initial velocity to avoid problems with Re=o and CD z0- 0: 8 Initial particle location 6 Set time length of integration, and number of steps g plot result linspace(0, at Tmax/N l0-4i t Finite difference step size for pf_pw calculation 2·V(,;); t Set Newton-Raphson convergence parameters title(Backward Euler integration with finite-c Iced pf/pw'I 6 Initialize vector for ODE integration 22121 8 start iterative loop for n =2: N+1 ures restol +1: t Doing this forces at least one sub-iteratio while((m Mmax)& (curres >restol ); curres norm(res) t Calculate linearization of f with respect to w using finite differences w(iiss w0+ eps; Perturb ii state by eps fp= drop rhs ( w,p》 w(ii)= w0-eps: Perturb ii state by -eps m=drop_rhs(w,p》; (ii)= wO: t Restore ii state to original value Ifp- fm)/(2*eps): g Finite differenc B Calculate linearization of residual with respect to w es_pw a eye(size(pf pw))-dt'pf_Pw
drop rhs. I Fx⊥Feb1508:32:032002 function [f]= drop_rhs(w and the parameters of the problem p t Get velocity =24/Re+6/(1+sgrt(Re))+0.4 m0.5*rhog*u,2*pi*a 2*CD i Calculate right-hand sides