Low mach number approximation

[s_desai]

I have some doubt in governing equation regarding Low mach number
approximation.
looking forward for your support.

we decompose the total pressure ( p ) in back ground pressure (p_m) &
pertubation pressure (p_p) and scaling analysis is carried out

1) It suggest the del.p_m =0 and hence del . p = del. p_p which should
be retained in momentum equation but in equation 3.18 of tech guide
del.p is defined as del.p=del.p_p + (rho_n*g) from where does this
additional term come?

2) In equation B.28 spatial variation of back ground pressure is
considered (which is against low mach number approximation). Material
derivation dp_m/dt=del.p_m/del.t - (w*rho_i*g).while low macnumber
approximation suggest the del.p_m =0.

Thans in advance.


Saumil

[rmcdermo]

Saumil,

Your question comes across as if you are questioning whether fire is
indeed a low-Mach flow. Do you really question this? I am not going
to go through the details here, but for Mach number (M) < 0.3 or so,
the flow can be considered "low-Mach". See any elementary fluids
text.

As to your specific questions:

1. You are incorrect in your decomposition of the total pressure. We
decompose the pressure into

p = p0(t) + rho_n*g*z + \tilde{p}

where rho_n is the background density for zone n and \tilde{p} is the
fluctuating hydrodynamic pressure (I think you called it p_p). So grad
(p) = rho_n*g + grad(\tilde{p}).

2. Again, you are incorrect. Spatial variation of the background
pressure does not preclude the low-Mach assumption. Consider a
quiescent atmosphere. The pressure changes with height above ground
level. Would you not consider this a low-Mach flow? It is in fact
static! Again, this error is because of the way you decomposed the
pressure.

Please take the time to go through the derivations and ask whether any
of the math is incorrect. Most of these physics questions can be
answered by studying a good fluids text like Panton "Incompressible
Flow".

Cheers,
Randy

[saumil desai]

Thanks for the clarification.
 
As said while decomposing total pressure , back ground pressure po(t) is function of time only  and hence grad of total pressure will result in grad (p) = rho_n*g + grad(\tilde{p}).
but in divergence equation total derivative of background pressure  po is defined as dpo/dt= delpo/delt+u_i del.u_i (equation B.28) i.e po is function of time and space.this is very confusing point for me . pls help.

regards..
Saumil.

--
Regards...
Saumil Desai

[drjfloyd]

read section 3.1.2 of the Theory manual:

Note that the background pressure is a function of z, the vertical
spatial coordinate, and time. For most
compartment fire applications, pm changes very little with height or
time. However, for situations where the
pressure increases due to a fire in a tightly sealed enclosure, or
when the height of the domain is significant,
pm takes these effects into account [26].

What do you not understand about this statement?

[s_desai]

Probably I have understood the concept now.I would like to know,for
open pool fires,Where del po /del t is zero. how po is updated. po
is required for determining temperature field through the EOS,as well
as for determination of density of each species.

Regards...
Saumil

On Oct 6, 9:38 pm, drjfloyd <drjfl...@gmail.com> wrote:
> read section 3.1.2 of the Theory manual:
>
> Note that the background pressure is a function of z, the vertical
> spatial coordinate, and time. For most
> compartment fire applications, pm changes very little with height or
> time. However, for situations where the
> pressure increases due to a fire in a tightly sealed enclosure, or
> when the height of the domain is significant,
> pm takes these effects into account [26].
>
> What do you not understand about this statement?
>
> On Oct 6, 11:51 am, saumil desai <saumil2...@gmail.com> wrote:
>
> > Thanks for the clarification.
>
> > As said while decomposing total pressure , back ground pressure po(t) is
> > function of time only  and hence grad of total pressure will result in grad
> > (p) = rho_n*g + grad(\tilde{p}).
> > but in divergence equation total derivative of background pressure  po is
> > defined as dpo/dt= delpo/delt+u_i del.u_i (equation B.28) i.e po is function
> > of time and space.this is very confusing point for me . pls help.
>
> > regards..
> > Saumil.
>
> > On Mon, Oct 5, 2009 at 6:10 PM, rmcdermo <randy.mcderm...@gmail.com> wrote:
>
> > > Saumil,
>
> > > Your question comes across as if you are questioning whether fire is

> > > indeed alow-Machflow.  Do you really question this?  I am not going
> > > to go through the details here, but forMachnumber(M) < 0.3 or so,

> > > the flow can be considered "low-Mach".  See any elementary fluids
> > > text.
>
> > > As to your specific questions:
>
> > > 1. You are incorrect in your decomposition of the total pressure.  We
> > > decompose the pressure into
>
> > > p = p0(t) + rho_n*g*z + \tilde{p}
>
> > > where rho_n is the background density for zone n and \tilde{p} is the
> > > fluctuating hydrodynamic pressure (I think you called it p_p).  So grad
> > > (p) = rho_n*g + grad(\tilde{p}).
>
> > > 2. Again, you are incorrect. Spatial variation of the background

> > > pressure does not preclude thelow-Machassumption.  Consider a

> > > quiescent atmosphere.  The pressure changes with height above ground

> > > level.  Would you not consider this alow-Machflow?  It is in fact

[dr_jfloyd]

For any variable phi(x,t) if dphi(x,t)/dt = 0, then the variable phi
(x,t) does not change as a function of time. If the variable does not
change as a function of time, then that means phi(x,t) = phi(x,0).
Randy has already explained in an earlier reply the spatial variation
of pressure due to the hydrostatic gradient.