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Pressure Gradient and Velocity Divergence

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Old   April 28, 2010, 21:39
Default Pressure Gradient and Velocity Divergence
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the approximation of the pressure gradient and the divergence of the velocity must be compatible if the kinetic energy conservation is to hold.
Once either approximation is chosen, the freedom to choose the other is lost.

I want to know the meaning of this word " compatible"
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Old   May 18, 2010, 10:10
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I suppose you are referring to the Navier-Stokes equations for an uncompressible fluid.

In that case, at least for FEM, there is a well established theory that explains when two discretizations are compatible. You should look for LBB or inf-sup condition. It should be explained in any recent text books on uncompressible CFD.
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Old   May 25, 2010, 17:11
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You can read these papers...

There is also a lecture of Gilbert Strang about the LBB condition at the MIT section of iTunes U.
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Old   May 29, 2010, 00:07
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The frequently-heard statement which Alvin Xu refers to is incorrect as it stands. However, if the word "pressure" is replaced, a very similar statement is true. The resolution is, as one author states, "not everything denoted by the symbol p is a pressure". Further, the statement is restricted to a particular class of projection methods.

The governing equation (Navier-Stokes) for incompressible flow is a restatement of Newton's law for conservation of momentum, perhaps after some simplifying assumptions about the fluid density. With those incompressibility assumptions, the velocity field is divergence-free. Then the governing equation begins dv/dt=... . Since the velocity is divergence-free, so the right hand side must be.

Now, according to Helmholtz and his successors, a vector field can be orthogonally decomposed into divergence-free (solenoidal) and irrotational parts, subject to appropriate boundary conditions. One can see immediately that solutions of Laplace's equation are both divergence-free and irrotational. One can embrace this ambiguity or eliminate it through boundary conditions.

A term like grad p is patently irrotational, and cannot appear on the RHS. That is - the fluid flow is independent of the pressure. Such a statement will not surprise those who use stream function-vorticity formulation (which does not contain a pressure) to compute fluid flow. But to be correct, one must project out the irrotational part of the RHS. There are several ways of doing this.

Doing this projection by the usual mathematical analysis leads to an integro-differential equation. I don't know anyone who has used this approach for practical computation.

The most common projection method is to subtract off an irrotational function in the form of the gradient of a potential to cancel the errant terms on the RHS. When one does this, the resulting equation looks just like the original form except that the subtracted term is not the pressure. It looks like a pressure, but has different boundary conditions, and the functional form must satisfy the LBB/inf-sup conditions.

Once one has computed an approximate (weak) divergence-free velocity, one can compute the pressure with the "pressure Poison" equation (as the stream function-vorticity practitioners do) using whatever functional form you wish, with appropriate b.c. for the pressure (and no LBB).

A third approach (which I advocate) is to expand the velocity in terms of an appropriately conforming divergence-free basis, i.e. using divergence-free finite elements. These velocity elements are the curl of a Hermite-type stream function element in 2D. Such elements can be found by modifying one of the many Hermite plate-bending finite elements found in the literature. Projection is accomplished by the orthogonality of solenoidal and irrotational functions a la Helmholtz when computing the element matricies.

Much of this carries over to 3D, but there is no shortcut to finding satisfactory elements as in 2D.

Note that there is always a scalar function that appears: the projection gradient, the 2D vorticity, or the stream function.
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Old   May 29, 2010, 11:11
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Originally Posted by walli View Post
You can read these papers...

There is also a lecture of Gilbert Strang about the LBB condition at the MIT section of iTunes U.
dear verdana,
i am looking for velicity profile over naca0012 airfoil at incompressible mach number.
please send me any document that may help me.
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