Instabilities in stratified mixing layers

Gelfgat & Kit, JFM, 2006, vol. 552, pp. 189-227

 

Temporal formulation



 

 

 

 

 

 

 

Perturbation: u’,v’,p’,T’ ~ exp(iax)

 

Governing parameters:   Re,  Ri,  Pe, R=dv/dT

 

Calculation of 2D case:      animation

 

Calculation of 3D case:     animation1    animation2

 

Spatial  formulation

 

 

 

 

 

 

 


Spatial wavenumber of excitation:    

Spatial wavenumber of perturbation:   

Length scale:                

Governing parameters:   Re,  Ri,  Pe,  l, , R=dv/dT

 

Calculation of 2D case:    animation

 

 

Scales and governing parameters:

 

Spatial problem

Temporal problem

Perturbation:

~exp(iw0t) (forced)

~exp(iax)

Velocity ratio, l

l=(U2U1) / (U2 + U1)

 

L, length

(U2+U1) / 2w0

a-1

t, time

L / (U2U1) =1/ 2lw0

L /Umax =1 / 2aUmax

V, velocity

U2U1

2Umax

p, pressure

r(U2U1)2

4rU2max

T, temperature

T2T1

T2T1

Governing

parameters:

 

Re = VL/n

(U2U1) (U2+U1) / 2w0n

2Umax / an

Ri = gbDTL/V2

gb(T2T1) / 2w0l(U2U1)

gb(T2T1) / 4aU2max

Pe = VL/c

(U2U1) (U2+U1) / 2w0c

2Umax / ac

 

General form of perturbation:       A(y)exp[i(ax+azz+wt)] 

 

Criterion for the temporal instability:   a=a0, Im(w)<0

 

Criterion for the spatial instability:       w=w0, Im(a)<0   

                                                                or   Im(b)<0  

 

See mathematical formulation

 

See iterative procedure

to solve for both temporal and spatial instabilities

 

 

Profiles of temporal and spatial perturbations for isothermal flow

 

Dependencies of w and b on the mixing layer thickness x

 

Dependencies of w and b on the Richardson number Ri

 

Dependencies of w and b on the Reynolds number Re

 

 Stability diagrams in xRe planes

 

 

Possibility for 3D initial growth

temporal                                 spatial

 

 

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