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


~exp(iw0t) (forced)


Velocity ratio, l

l=(U2U1) / (U2 + U1)


L, length

(U2+U1) / 2w0


t, time

L / (U2U1) =1/ 2lw0

L /Umax =1 / 2aUmax

V, velocity



p, pressure



T, temperature






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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