environmental fluid mechanics iii model applications · 2 20.04.2010 institute for hydromechanics...

KIT – University of the State of Baden-Wuerttemberg and

National Research Center of the Helmholtz Association

INSTITUTE FOR HYDROMECHANICS

www.kit.edu

Environmental Fluid Mechanics III Model Applications

Dong-Guan Seol

Integral Plume Modeling

Institute for Hydromechanics 2 20.04.2010

Agenda

Definition of jets and plumes

Challenges in modeling

Density stratification

Integral plume model

Numerical solution of the integral plume model

Problem definition

D.-G. Seol


Definition of jets and plumes

Jets: source of energy - momentum

D.-G. Seol


Plumes: source of energy - buoyancy

D.-G. Seol

CO2 injection:

Norway Sleipner Oil drilling rig

Oil well blowout:

IXTOC I Oil well


Challenges in modeling

Cross-flow

Density-stratification

Buoyant Jet

in Crossflow

D.-G. Seol


Challenges in modeling (continued)

Density-stratification

D.-G. Seol


Density stratification

Temperature, salinity, particular material

D.-G. Seol

• Buoyancy force

• Buoyancy frequency

• Reduced gravity



D.-G. Seol

Basic assumptions

• Buossinesq approximation

• Self-similarity of the mean vertical velocity (w) and buoyancy force (g’)

• Entrainment hypothesis



D.-G. Seol

Buossinesq approximation:

: Plume density

: Ambient density

: Reference density



D.-G. Seol

Self-similarity: Gaussian profile



D.-G. Seol

Entrainment hypothesis

Entrainment velocity:



D.-G. Seol

Governing equations

• Mass conservation

• Momentum conservation

• Buoyancy conservation



D.-G. Seol

Mass conservation

• [Mass flux exiting control volume]

=[Mass entering control volume]+[Mass entraining from side]

Introducing Buossinesq approximation



D.-G. Seol

Momentum conservation conservation

• [Momentum flux exiting control volume]

=[Momentum entering control volume] + [Momentum entraining from side]

+ [Upward buoyancy]

[Upward buoyancy] =[weight of displaced water]-[actual weight of plume segment]



D.-G. Seol

Momentum conservation conservation (continued)

• [Momentum flux exiting control volume]

=[Momentum entering control volume] + [Momentum entraining from side]

+ [Upward buoyancy]

Introducing Buossinesq approximation



D.-G. Seol

Buoyancy conservation

[Buoyancy flux exiting control volume]

=[Buoyancy flux entering control volume]

+[Influx of buoyancy entraining from side]

(from the mass conservation)



D.-G. Seol

Buoyancy conservation (continued)

Multiply both sides with g/ r and introduce buoyancy frequency N2



D.-G. Seol

Governing equations

• Mass conservation

• Momentum conservation

• Buoyancy conservation

For simplicity, let us put


Numerical solution

D.-G. Seol

Model setup

where

Initial conditions for pure plume (only buoyancy) at the plume bottom (z=0)

Assume linear stratification,

Institute for Hydromechanics 20 20.04.2010 U. Mohrlok, D.-G. Seol

Numerical solution

Matlab imprementation

clear all, close all, clc,

% define constants global alpha N;

alpha = 0.125; %entrainment coefficietn N =1.; % Buoyancy frequency

dz= 0.1; % integration interval Z = 5.2; % Total height of integration

z=[0:dz:Z]; % height increment

% specify initial value W0=0;

V0=0.00001; % non-zero for non-trivial solution F0=1;

Nz = length(z); % number of integration steps

% initialize the numerical integration nz=1; W=W0;

V=V0; F=F0;

while (nz < Nz)

zi=z(nz);

% non-dimensionalize the variables using F0 and N zp=zi/(F0^0.25/N^(3/4));

Wp=W/(F0^0.5/N^(5/2)); Vp=V^0.25/(F0^(3/4)/N^(5/4));

Fp=F/F0;

plot(Wp,zp,'.g‘‚Vp,zp,'.r‘,Fp,zp,'.b');

axis([-1 3 0 6]) legend('volume flux','momentum flux','buoyancy flux',4);

ylabel('Height z'); xlabel('Plume parameter');

[Wn,Vn,Fn]=func_rk4(W,V,F,dz);

nz = nz + 1;

W = Wn; V = Vn; F = Fn;

end


Matlab imprementation (functions)

function [Wn,Vn,Fn]=func_rk4(W,V,F,dz)

k1=dz*myfunc1(W,V,F); l1=dz*myfunc2(W,V,F);

m1=dz*myfunc3(W,V,F);

k2=dz*myfunc1(W+k1/2,V+l1/2,F+m1/2);

l2=dz*myfunc2(W+k1/2,V+l1/2,F+m1/2); m2=dz*myfunc3(W+k1/2,V+l1/2,F+m1/2);

k3=dz*myfunc1(W+k2/2,V+l2/2,F+m2/2); l3=dz*myfunc2(W+k2/2,V+l2/2,F+m2/2);

m3=dz*myfunc3(W+k1/2,V+l1/2,F+m2/2);

k4=dz*myfunc1(W+k3,V+l3,F+m3); l4=dz*myfunc2(W+k3,V+l3,F+m3); m4=dz*myfunc3(W+k3,V+l3,F+m3);

Wn = W + (k1 + 2*k2 + 2*k3 + k4)/6;

Vn = V + (l1 + 2*l2 + 2*l3 + l4)/6; Fn = F + (m1 + 2*m2 + 2*m3 + m4)/6;

return

Runge-Kutta 4th order method: A numerical technique used to solve ordinary differential

equation of the form

where


Matlab imprementation (functions)

function [dWdz] = myfunc1(W,V,F)

global alpha N;

dWdz=2*alpha*V^0.25;

return

function [dVdz] = myfunc2(W,V,F)

global alpha N;

dVdz=2*F*W;

return

function [dFdz] = myfunc3(W,V,F)

global alpha N;

dFdz=-N^2*W;

return

Mass conservation

Momentum conservation

Buoyancy conservation


Results

Momentum

overshooting


Problem definition By using a blower and some preheating, one can adjust both the upward

velocity and buoyancy of fumes exiting from the top of a

smokestack. Specifically, two scenarios are being considered, on with

more velocity and one one with more buoyancy, as follows:

Scenario 1: Average exit vertical velocity = 12 m/s

Average exit buoyancy = 0.01 m/s2

Scenario 2: Average exit vertical velocity = 1 m/s

Average exit buoyancy = 0.12 m/s2

In each case, the exit diameter is 1.5 m and the entrainment coefficient

is taken as 0.115. Assume linear stratification (N=1 s-1).

• Which of the two scenarios gives the highest vertical velocity at the

center of the plume 20 m above the smokestack?

• Briggs (1969) proposed following formula for the rise of the plume:

Compare the model results with above formula for different stratification

conditions (N=[0:0.1:1] 1/s). Assume pure plume of scenario 2.


House-keeping: Access to IfH computers

D.-G. Seol


References

Briggs, G. A. (1965). “A plume rise model compared with observations.”

J. Air Poll. Control Assoc., 15(9), 433-438.

Lee, J. H. W., and Chu, V. H. (2003). Turbulent jets and plumes.

Springer.

Morton, B. R., Taylor, G., and Turner, J. S. (1956). “Turbulent

Gravitational Convection from Maintained and Instantaneous Sources.”

Proceedings of the Royal Society of London. Series A, Mathematical

and Physical Sciences, 234(1196), 1-23.

D.-G. Seol

environmental fluid mechanics iii model applications · 2 20.04.2010 institute for hydromechanics...

Documents