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aleahsommersaleahsommers
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New code used in revised manuscript
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‎clean_circular_turb_jokhul.m‎

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% Heat equation in radial coordinates (circular pipe) to understand Nusselt number in turbulent flow
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% Can set up for heated walls or with internal dissipation profile
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clear all
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close all
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Re_list=[1000000];
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Nu_list=zeros(size(Re_list));
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for Rei=1:length(Re_list)
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r0=0.2; % Pipe radius (m)
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% Typical 0 deg conditions used in ice sheet subglacial modeling
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k=0.558; % Thermal conductivity of water (W/m/K)
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cw=4.22e3; %specific heat capacity of water (J/kg/K)
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mu=1.787e-3; % Dynamic viscosity of water at 0 degC (Pa s)
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rhow=1000; % Density of water (kg/m3)
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kappa=k/(rhow*cw); % Thermal diffusivity
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nu=mu/rhow; % Kinematic viscosity (m2/s)
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Q=1; % Set to 0 to turn off internal heat generation, 1 for internal dissipation
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dr=0.00001*r0;
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J=ceil(r0/dr+1);
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z=0:dr:r0;
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% Get velocity profile
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y=0:dr:r0;
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Re=Re_list(Rei); % Prescribe bulk Reynolds number
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vonK=0.4; % Von Karman constant
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B=5; % Constant to use in log-law
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Cf=fzero(@(Cf) -sqrt(2/Cf)+1/vonK*log(Re*sqrt(Cf)/2/sqrt(2))-1/vonK+B,0.009); % Friction factor (from log-law, A/A eqn 1.5)
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u_b=Re*nu/(2*r0); % Bulk mean velocity -- A/A uses 2h, where h is half-width, but Re should depend on hydraulic radius?
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tau_w=Cf*rhow*u_b^2/2; % Wall shear stress
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u_tau=(tau_w/rhow)^0.5; % Friction velocity (inner velocity scale)
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yplus=y.*u_tau./nu; % Non-dimensional y coordinate near wall
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uplus(yplus<=20)=yplus(yplus<=20)-1.2533e-4*yplus(yplus<=20).^4+3.9196e-6.*yplus(yplus<=20).^5; % Velocity smooth profile (transition at yplus=20)
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uplus(yplus>20)=1/vonK*log(yplus(yplus>20))+B; % Log-layer
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u=flip(uplus)'.*(u_tau);
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ub=2/r0^2*trapz(u.*z')*dr; % Mean bulk velocity for circular pipe
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% Dissipation profile (based on Abe and Antonia, missing interior portion)
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% depth-integrated E=<epsilon_bar>+<nu*(du/dz)^2>=(2.54.*log(yplus)+2.41).*u_tau^3;
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% eps here epsilon_bar, NOT depth-integrated!
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outer=find((r0-abs(z))/r0>0.2);
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eps(outer)=(2.45./((r0-abs(z(outer)))./r0)-1.7).*u_tau^3/r0;
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inner=find((r0-abs(z))/r0<=0.2);
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eps(inner)=(2.54./((r0-abs(z(inner)))./r0)-2.6).*u_tau^3/r0;
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% Inner portion of dissipation function (very near wall)
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hplus=u_tau*r0/nu;
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[ii,jj]=find(abs(y/r0-30/hplus)==(min(abs(y/r0-30/hplus))));
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wall=find((r0-abs(z))/r0<30/hplus);
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eps(wall)=eps(min(wall)-1);
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% % Get eddy diffusivity
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tau=tau_w.*z./r0; % Laminar shear stress distribution - linear
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nu_T=-(tau(1:end-1)+tau(2:end))./2./(rhow.*diff(u')./dr)-nu; % Eddy viscosity (at half-nodes)
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nu_T(nu_T<0)=0;
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kappa_T=nu_T; % Eddy diffusivity from Reynolds' analogy (at half-nodes)
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dx=r0;
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nx=5000/dx; % Number of x steps (iterations)
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z_half=(z(1:end-1)+z(2:end))./2; % Half-node radial coordinates
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alpha=(kappa+kappa_T').*dx./dr'.^2.*z_half';
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% Define vectors (coefficients for interior nodes)
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aconst=-alpha./2./z(2:end)';
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bconst=(u(2:end-1)+alpha(2:end)./2./z(2:end-1)'+alpha(1:end-1)./2./z(2:end-1)');
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cconst=-alpha./2./z(1:end-1)';
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% Build a,b,c,r vectors
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a=[0;aconst];
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b=[0;bconst;0];
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c=[cconst;0];
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r=zeros(J,1);
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% Boundary condition at pipe center (symmetry condition)
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% Type II at z=0 (no-flux)
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a(1)=0;
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b(1)=-1;
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c(1)=1;
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r(1)=0;
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if Q==0 % Type I at wall (heated)
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a(J)=0;
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b(J)=1;
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c(J)=0;
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r(J)=1;
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end
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if Q==1 % Type I at wall (T=0)
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a(J)=0;
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b(J)=1;
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c(J)=0;
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r(J)=0;
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end
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% % Type II at top (no-flux)
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% a(J)=-1;
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% b(J)=1;
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% c(J)=0;
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% r(J)=0;
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% Initial condition u(x,0)=0
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T=2*ones(J,1);
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T_CN=[]; % Pre-allocate matrix to store plot values for each t
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% Loop through distance along x
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for it=1:nx
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% Interior nodes
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r(2:J-1)=alpha(1:end-1)./2./z(2:end-1)'.*T(1:J-2)+(u(2:J-1)-alpha(1:end-1)./2./z(2:end-1)'-alpha(2:end)./2./z(2:end-1)').*T(2:J-1)+alpha(2:end)./2./z(2:end-1)'.*T(3:J)+...
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Q/rhow/cw*dx.*(eps(2:J-1)'.*rhow+mu.*((u(3:J)-u(1:J-2))./dr).^2);
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Tnext=thomas(a,b,c,r);
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T=Tnext';
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% Tbar(it)=trapz(T)*dr/r0;
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Tbar(it)=2/(ub*r0^2)*trapz(u.*T.*z')*dr;
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% Save to plot for selected downstream distances
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if it*dx==1 | it*dx==5 | it*dx==10 | it*dx==20 | it*dx==50 | it*dx==100
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T_CN=[T_CN T];
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end
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% % Brinkman number
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Br_lam(it)=mu.*ub.^2./(k.*(T(end)-Tbar(it)));
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Phi=2*pi.*((mu*trapz(((u(2:J)-u(1:J-1))'./dr).^2.*(z(2:J)+z(1:J-1))/2))+trapz(eps.*z)*rhow)*dr;
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h_coeff=Phi/(2*pi*r0*(Tbar(end)-T(J)));
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Br(it)=Phi./(2*h_coeff*(2-T(J)));
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end
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% Plotting - turn off for timing in problem 3
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figure(1)
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hold on
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plot(T_CN,z./r0,'--','linewidth',1.2)
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% axis([0 1 0 1])
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legend('x=0.01','x=0.1','x=0.2','x=0.5','x=1','x=2','x=5','x=10','x=20','x=50','x=100','x=200','x=500','x=1000')
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xlabel('T')
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ylabel('r/r_{0}')
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figure(2);hold on;
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x=(0:(nx-1)).*dx;
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plot(x./r0,log(Tbar-T(end)),'linewidth',2);
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xlabel('x/r_0','fontsize',14);ylabel('ln(T_b-T_w)','fontsize',14)
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if Q==0 % For wall-heating case
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p=polyfit(x,log(T(end)-Tbar),1);
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Nu=-p(1)*rhow*cw*u_b*r0^2/k;
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elseif Q==1 % For internal dissipation case -- with constant T=0 walls
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Phi=2*pi.*((mu*trapz(((u(2:J)-u(1:J-1))'./dr).^2.*(z(2:J)+z(1:J-1))/2))+trapz(eps.*z)*rhow)*dr;
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h_coeff=Phi/(2*pi*r0*(Tbar(end)-T(J)));
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Nu=h_coeff*2*r0/k;
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end
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% Comparison with Siegel and Sparrow for internal heat generation (fig. 2)
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% -- make sure to set wall b.c. as no-flux for comparison
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SS=(T(J)-Tbar(end))./(Q*r0^2/k); % Siegel and Sparrow, fig. 2
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% Plot Brinkman number
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figure(3);hold on;plot(x./r0,Br)
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xlabel('x/r_0','fontsize',14);ylabel('Brinkman number')
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% Plot non-dimensional Brinkman
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figure(4);hold on;plot(x./r0,log((Tbar-T(J))./(2-T(J))),'linewidth',2)
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xlabel('x/r_0','fontsize',14);ylabel('ln((T_b-T_w)/(T_0-T_w))','fontsize',14)
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end
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‎plot_ReNu_circular_2020revisions_DBonly.m‎

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clear all;close all
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% Circular pipe flow - wall heating - matches well with D-B correlation
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Re=[10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 200000 300000 400000 500000 750000 1000000];
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Nu=[89.2587 165.3526 236.3955 304.3768 370.2091 434.4064 497.2896 559.0866 619.9480 680.0172 1.2511e+03 1.7895e+03 2.3086e+03 2.8093e+03 4.0257e+03 5.1997e+03]; % With finer mesh, from circular_heatedwalls_data
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Pr=13.5146;
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% figure(1);hold on;plot(Re,Nu,'-o','linewidth',2)
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figure(1);semilogx(Re,Nu,'-o','linewidth',2)
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% Circular pipe flow with internal dissipation with Abe/Antonia dissipation profile - epsilon*rhow in Phi
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% calculation (compare Phi/rhow with E) - WITH NEAR-WALL EPSILON (CONSTANT from y/h<30/hplus)
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Re=[10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 200000 300000 400000 500000 750000 1000000];
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Nu=[43.3028 85.510 126.5970 166.9143 206.6587 245.9381 284.8349 323.4069 361.6920 399.7326 763.6548 1.1172e+03 1.4616e+03 1.8287e+03 2.6788e+03 3.5119e+03]; % With finer mesh, from circular_dissipation_data
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% figure(1);hold on;plot(Re,Nu,'-o','linewidth',2)
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figure(1);hold on;semilogx(Re,Nu,'-o','linewidth',2)
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Pr=13.5146;
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Re2=10000:10000:1000000;
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DB=0.024.*Re2.^0.8.*Pr^0.4;
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Re2=10000:10000:1000000;
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figure(1);hold on;semilogx(Re2,DB,'--','linewidth',2)
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Gni=0.012.*(Re2.^0.87-280).*Pr^0.4;
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figure(1);semilogx(Re2,Gni,'--','linewidth',2)
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figure(1);set(gca,'FontSize',18)
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axis([10000 1000000 0 6000])
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xlabel('Re');ylabel('Nu');
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legend('Heated Walls','Dissipation','Dittus-Boelter','Gnielinski')

‎plot_ReNu_combined_2020revisions.m‎

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clear all;close all
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% Circular pipe flow - wall heating - matches well with D-B correlation
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Re=[10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 200000 300000 400000 500000 750000 1000000];
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Nu=[89.2587 165.3526 236.3955 304.3768 370.2091 434.4064 497.2896 559.0866 619.9480 680.0172 1.2511e+03 1.7895e+03 2.3086e+03 2.8093e+03 4.0257e+03 5.1997e+03]; % With finer mesh, from circular_heatedwalls_data
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Pr=13.5146;
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figure(1);semilogx(Re,Nu,'-o','linewidth',2)
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% Circular pipe flow with internal dissipation with Abe/Antonia dissipation profile - epsilon*rhow in Phi
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% calculation (compare Phi/rhow with E) - WITH NEAR-WALL EPSILON (CONSTANT from y/h<30/hplus)
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Re=[10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 200000 300000 400000 500000 750000 1000000];
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Nu=[43.3028 85.510 126.5970 166.9143 206.6587 245.9381 284.8349 323.4069 361.6920 399.7326 763.6548 1.1172e+03 1.4616e+03 1.8287e+03 2.6788e+03 3.5119e+03]; % With finer mesh, from circular_dissipation_data
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figure(1);hold on;semilogx(Re,Nu,'-o','linewidth',2)
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% Restart color order
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ax = gca;
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ax.ColorOrderIndex = 1;
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% Channel flow - wall heating
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Re=[10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 200000 300000 400000 500000 750000 1000000];
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Nu=[162.4 304.0 436.5 563.6 686.8 807.1 925.0 1041 1155.3 1268.1 2.3430e+03 3.3585e+03 4.3391e+03 5.2859e+03 7.5885e+03 9.8136e+03]; % With finer mesh, from channel_heatedwalls_data
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figure(1);hold on;semilogx(Re,Nu,'-.x','linewidth',2)
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% Channel flow - with Abe/Antonia dissipation profile - epsilon*rhow in Phi
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% calculation (compare Phi/rhow with E) - WITH NEAR-WALL EPSILON (CONSTANT from y/h<30/hplus)
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Re=[10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 200000 300000 400000 500000 750000 1000000];
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Nu=[91.1845 178.9022 264.1219 347.7000 430.0565 511.4583 592.0762 672.0347 751.4468 830.0000 1.5861e+03 2.3193e+03 3.0395e+03 3.8071e+03 5.5886e+03 7.3389e+03]; % With finer mesh, from AA_dissipationintegral_data
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E=[0.0026 0.0173 0.0530 0.1176 0.2186 0.3632 0.5580 0.8099 1.1253 1.5105];
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Phi=[4.98 33.29 101.75 224.15 415.74 684.34 1048 1512.7 2097.6 2794.3];
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figure(1);semilogx(Re,Nu,'-.x','linewidth',2)
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figure(1);set(gca,'FontSize',18)
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axis([10000 1000000 0 10000])
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xlabel('Re');ylabel('Nu');
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legend('Circular conduit - heated walls','Circular conduit - dissipation','Sheet - heated walls','Sheet - dissipation')

‎plot_fig11a.m‎

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% Plot Figure 11
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clear all;close all;
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load Re1e5_jokhul_r2.mat
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figure(2);hold on;
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x=(0:(nx-1)).*dx;
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plot(x(1:nx/2),log(Tbar(1:nx/2)-T(end)),'linewidth',2);
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xlabel('x (m)','fontsize',14);ylabel('ln(T_b-T_w)','fontsize',14)
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clear all;load Re1e5_heatedwalls_r2.mat
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figure(2);hold on;
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x=(0:(nx-1)).*dx;
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plot(x(1:nx/2),log(Tbar(1:nx/2)-T(end)),'--','linewidth',2);
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xlabel('x (m)','fontsize',14);ylabel('ln(T_b-T_w)','fontsize',14)
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clear all;load Re1e5_dissipation_r2.mat
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figure(2);hold on;
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x=(0:(nx-1)).*dx;
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plot(x(1:nx/2),log(Tbar(1:nx/2)-T(end)),'--','linewidth',2);
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xlabel('x (m)','fontsize',14);ylabel('ln(T_b-T_w)','fontsize',14)
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figure(2);legend('T_0=2 case','Heated wall case','Dissipation case','fontsize',14)
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% title('Re=10^6','fontsize',16)

‎plot_fig11d_smallr0.m‎

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% Plot Figure 11
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clear all;close all;
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load Re1e4_jokhul_r2.mat
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figure(2);hold on;
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x=(0:(nx-1)).*dx;
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plot(x(1:15000),log(Tbar(1:15000)-T(end)),'linewidth',2);
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xlabel('x (m)','fontsize',14);ylabel('ln(T_b-T_w)','fontsize',14)
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clear all
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load Re5e4_jokhul_r2.mat
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figure(2);hold on;
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x=(0:(nx-1)).*dx;
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plot(x(1:15000),log(Tbar(1:15000)-T(end)),'linewidth',2);
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xlabel('x (m)','fontsize',14);ylabel('ln(T_b-T_w)','fontsize',14)
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clear all
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load Re1e5_jokhul_r2.mat
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figure(2);hold on;
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x=(0:(nx-1)).*dx;
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plot(x(1:15000),log(Tbar(1:15000)-T(end)),'linewidth',2);
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xlabel('x (m)','fontsize',14);ylabel('ln(T_b-T_w)','fontsize',14)
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clear all
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load Re5e5_jokhul_r2.mat
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figure(2);hold on;
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x=(0:(nx-1)).*dx;
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plot(x(1:15000),log(Tbar(1:15000)-T(end)),'linewidth',2);
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xlabel('x (m)','fontsize',14);ylabel('ln(T_b-T_w)','fontsize',14)
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clear all
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load Re1e6_jokhul_r2.mat
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figure(2);hold on;
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x=(0:(nx-1)).*dx;
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plot(x(1:15000),log(Tbar(1:15000)-T(end)),'linewidth',2);
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xlabel('x (m)','fontsize',14);ylabel('ln(T_b-T_w)','fontsize',14)
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figure(2);legend('Re=10^4','Re=5x10^4','Re=10^5','Re=5x10^5','Re=10^6','fontsize',14)
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% title('Re=10^6','fontsize',16)

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