[blackpingu]

Hi!

I am struggling with calculating the vorticity thickness in temporal mixing layers. Which is defined as





Where U_1 and U:2 are the lower and upper free stream velocities in opposite directions.
A typical nondimensionalized velocity profile is U/U_1=tanh(y).

To demonstrate the problem:
A simple example is if U_1=1 and U_2=-1. The maximum gradient of tanh(y) is 1. Which yileds a vorticity thickness of 2. But I am supposed to get vorticity thickness of order 1e-4, based on a couple of papers, what am I doing wrong?

[blackpingu]

Vorticity thickness =(U_1-U_2) / (du/dy)_max

[FMDenaro]

Originally Posted by blackpingu

Hi!

I am struggling with calculating the vorticity thickness in temporal mixing layers. Which is defined as





Where U_1 and U:2 are the lower and upper free stream velocities in opposite directions.
A typical nondimensionalized velocity profile is U/U_1=tanh(y).

To demonstrate the problem:
A simple example is if U_1=1 and U_2=-1. The maximum gradient of tanh(y) is 1. Which yileds a vorticity thickness of 2. But I am supposed to get vorticity thickness of order 1e-4, based on a couple of papers, what am I doing wrong?

Are you working with dimensional or non-dimensional variables? The Reynolds number has the vorticity thickness as characteristic lenght, therefore in non-dimensional variables this thickness is O(1).

[blackpingu]

Originally Posted by FMDenaro

Are you working with dimensional or non-dimensional variables? The Reynolds number has the vorticity thickness as characteristic lenght, therefore in non-dimensional variables this thickness is O(1).

Dimensional. For example, one paper claims to have vorticity Reynolds number of 100 with 0.2 mach, which for air is about 70 m/s. which gives a vorticity thickness of 140/70 which is 2. In order to fulfill reynolds number of 100 the viscosity has to be 0.35 which doesn't make any sense.

100=((U_1-U_2)*d)/vis


vis=100/(140*2)=0.35

where d is viscosity thickness.

[FMDenaro]

Originally Posted by blackpingu

Dimensional. For example, one paper claims to have vorticity Reynolds number of 100 with 0.2 mach, which for air is about 70 m/s. which gives a vorticity thickness of 140/70 which is 2. In order to fulfill reynolds number of 100 the viscosity has to be 0.35 which doesn't make any sense.

100=((U_1-U_2)*d)/vis


vis=100/(140*2)=0.35

where d is viscosity thickness.

vorticity thickness of 2 meters????

[blackpingu]

Originally Posted by FMDenaro

vorticity thickness of 2 meters????

Yes, I know it doesn't make any sense. There should be something I have misunderstood.

[FMDenaro]

Originally Posted by blackpingu

Yes, I know it doesn't make any sense. There should be something I have misunderstood.

If you consider that U1=1m/s and U2=-1m/s and the initial velocity is


u(y)= U1 * tan h (2*y/delta1) the derivative is


du/dy = 2*U1 / (delta1 *cos h (2*y/delta1))


Therefore the delta1 value is the initial condition you have to prescribe.


We did that in Sec.5.2.1 https://www.researchgate.net/publica...-uniform_grids

[blackpingu]

Originally Posted by FMDenaro

If you consider that U1=1m/s and U2=-1m/s and the initial velocity is


u(y)= U1 * tan h (2*y/delta1) the derivative is


du/dy = 2*U1 / (delta1 *cos h (2*y/delta1))


Therefore the delta1 value is the initial condition you have to prescribe.


We did that in Sec.5.2.1 https://www.researchgate.net/publica...-uniform_grids

Thank you, but what if the delta is the vorticity thickness? Which looks like the paper I read did?

[FMDenaro]

Originally Posted by blackpingu

Thank you, but what if the delta is the vorticity thickness? Which looks like the paper I read did?

But the vorticity thickness evolves starting from some initial condition... If you are interested in the temporal mixing you have to consider the evolution delta(t) but the thickness at t=0 is what you prescribe from your initial velocity profile

[blackpingu]

Originally Posted by FMDenaro

But the vorticity thickness evolves starting from some initial condition... If you are interested in the temporal mixing you have to consider the evolution delta(t) but the thickness at t=0 is what you prescribe from your initial velocity profile

Ok thanks, so that means I can just reverse the calculations in order to find out what initial delta1 they used ?

[FMDenaro]

Originally Posted by blackpingu

Ok thanks, so that means I can just reverse the calculations in order to find out what initial delta1 they used ?

from the Re, prescribing velocity and viscosity

[blackpingu]

Originally Posted by FMDenaro

from the Re, prescribing velocity and viscosity

Problem is, they dont list their viscosity

[FMDenaro]

Originally Posted by blackpingu

Problem is, they dont list their viscosity

don't worry about the viscosity value, the key is to use the same Reynolds number...

otherwise, you can simply do the simulation using the non-dimensional equations

[blackpingu]

Thank you