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Some questions about incompressible flow,thx!

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Old   December 28, 2010, 21:52
Default Some questions about incompressible flow,thx!
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Aladdin
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Hi guys,
i got some words on a book, but i cannot understood them very well, it said "In the low speed fluid and gas,they are treated as imcompressible flow.There was no change in the density,so there is no relationship between momentum equations and mass conservation equations".
Well ,my question is ,why?
THX!
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Old   December 29, 2010, 03:59
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Quote:
Originally Posted by aladdincham View Post
Hi guys,
i got some words on a book, but i cannot understood them very well, it said "In the low speed fluid and gas,they are treated as imcompressible flow.There was no change in the density,so there is no relationship between momentum equations and mass conservation equations".
Well ,my question is ,why?
THX!
It is true. It's possible to explain it so.
It is known from compressible gasdynamic theory that
density depends on Mach number M as
\rho = \rho_0 (1+\frac{\gamma -1}{2}M^2)^{-\frac{1}{\gamma-1}}
where \gamma is the specific heat ratio,
\rho_0=const is the stagnation density,
M = \frac{u}{c}
where u is the velocity and c is the speed of sound.

Thus, when the maximum M (and maximum u) tends to zero, density becomes independent of u and M and may be treated as constant. In practice when maximum M < 0.1 flow may be considered as incompressible.
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Old   December 29, 2010, 22:21
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Quote:
Originally Posted by ignat View Post
It is true. It's possible to explain it so.
It is known from compressible gasdynamic theory that
density depends on Mach number M as
\rho = \rho_0 (1+\frac{\gamma -1}{2}M^2)^{-\frac{1}{\gamma-1}}
where \gamma is the specific heat ratio,
\rho_0=const is the stagnation density,
M = \frac{u}{c}
where u is the velocity and c is the speed of sound.

Thus, when the maximum M (and maximum u) tends to zero, density becomes independent of u and M and may be treated as constant. In practice when maximum M < 0.1 flow may be considered as incompressible.
Thanks for your reply, i think i make some mistake,the words are“In the low speed fluid and gas,they are treated as imcompressible flow.There was no change in the density,so momentum equations & mass conservation equations are no relationship with energy equations
WHY?
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Old   December 30, 2010, 07:01
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With density constant, momentum and conservation depend only on velocities and pressure. In 3-d flow, that's (for example) u, v, w, and p. With 3 momentum eqs. and continuity, that's 4 equations and 4 unknowns. They can (and are) solved without reference to Temperature, internal energy, etc.

Knowing u, v, w, p, etc, the energy equation is one equation in one unknown (temperature, internal energy, enthalpy, etc depending on the form you select for the equation).

It might be better to say "the momentum equations are independent of the temperature" but temperature depends on the momentum and pressure.

Some other folks might say the coupling between momentum-continuity and energy is one-way only.

That thing about 'no relationship' is sort of confusing.

Hope this helps,

OTD
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Old   December 30, 2010, 07:46
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Quote:
Originally Posted by aladdincham View Post
There was no change in the density,so momentum equations & mass conservation equations are no relationship with energy equations
WHY?
It follows from non-dimensional form of energy equation.
Let non-dimensional pressure be:
p=\frac{p-p_0}{\rho_r u_r^2}
where p_0 is the constant background pressure, \rho_r is the reference density and u_r is the reference velocity.

then the energy equation in non-dimensional form can be written as

M_r^2 \{  \frac{\partial }{\partial t} [p+(\gamma-1)\rho u^i u^i/2 ]    + [u^k (\gamma p+(\gamma-1)\rho u^i u^i/2) ]_{,k}\}+u_{,k}^k=0

Here
M_r=\frac{u_r}{\sqrt{\gamma p_0 / \rho_r}} is the reference Mach number.
If M_r tends to zero then the energy equation coincides with continuity equation div(u)=0!

Thus energy equation in itself "disappears" for incompressible flows (M->0)!

Therefore "momentum equations & mass conservation equations are no relationship with energy equations"

Details see for example in Wesseling's papers
1. van der Heul DR, Vuik C, Wesseling P. A conservative pressure correction method for compressible flow at all speeds.// Int. J. Numer. Meth. Fluids, 2002, v. 40, pp. 521-529.

2. I. Wenneker, A. Segal and P. Wesseling. A Mach-uniform unstructured grid method. // Int. J. Numer. Meth. Fluids, 2002, v. 40, pp. 1209-1235.
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Old   December 30, 2010, 23:34
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thanks for your reply, i got the quetion`s answer, your answer is very clear.
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