User:Tclamb/Journal/Kim and Moin

Projection methods split each timestep into fractional steps. The first step advances the momentum equation, providing a candidate velocity. Subsequent steps enforce the continuity equation. This is accomplished by making use of a Helmholtz decomposition for the velocity field, splitting it into a divergence-free component and a curl-free component.

=Weak Formulation= There are three steps involved in Kim and Moin's method:
 * Burgers' Equation (Kim & Moin, Eq. 3)
 * Poisson's Equation (Kim & Moin, Eq. 8)
 * Scalar potential (Kim & Moin, Eq. 4)
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$$ \displaystyle \frac{\mathbf{\hat u} - \mathbf{u^n}}{dt} = \frac12 \left( 3 \mathbf{H^n} - \mathbf{H^{n-1}} \right) + \frac12 \frac1{Re} \nabla^2\left(\mathbf{\hat u} + \mathbf{u^n}\right) \quad \text{where } \mathbf{H} = -\mathbf{u} \cdot \nabla \mathbf{u} $$ (KM.1)
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$$ \displaystyle \nabla^2 \phi^{n+1} = \frac1{dt}\nabla \cdot \mathbf{\hat u} $$ (KM.2)
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$$ \displaystyle \frac{\mathbf{u^{n+1}} - \mathbf{\hat u}}{dt} = - \nabla \phi^{n+1} $$ (KM.3)
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By inspection, we see that we will be solving Eqn. KM.1 for $$\mathbf{\hat u}$$. We will solve Eqn. KM.2 for $$\phi^{n+1}$$, and Eqn. KM.3 for $$\mathbf{u^{n+1}}$$. We also note that this method has reduced Navier-Stokes' nonlinear system of PDEs to three sequential linear PDEs. In the following sections, we will derive a weak formulation of these three equations.

Step One: Burgers' Equation
I have previously worked out the variational form of Burgers' Equation in an in-development FEniCS tutorial. From that page, we have the following:
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$$ \displaystyle \int_\Omega \frac{\mathbf{u}_{n} - \mathbf{u}_{n-1}}{dt} \cdot \mathbf{v} + \mathbf{u}_n \cdot \left(\nabla \mathbf{u}_n\right) \cdot \mathbf{v} + \nu \left(\nabla \mathbf{u}_n\right) : \left(\nabla \mathbf{v}\right) - \mathbf{f} \cdot \mathbf{v} \, dx - \oint_{\partial \Omega} \nu \mathbf{v} \cdot \left(\nabla \mathbf{u}_n\right) \cdot \mathbf{n} \, ds = 0 $$ (BE.9)
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From this, we can simply change our variables to match our problem, keeping in mind the origin of each term. We note that this equation may be split into bilinear and linear forms, but retyping the equation would be too tedious.
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$$ \displaystyle \int_\Omega \frac{\mathbf{\hat u} - \mathbf{u^n}}{dt} \cdot \mathbf{v} + \frac32 \mathbf{u^n} \cdot \left(\nabla \mathbf{u^n}\right) \cdot \mathbf{v} - \frac12 \mathbf{u^{n-1}} \cdot \left(\nabla \mathbf{u^{n-1}}\right) \cdot \mathbf{v} + \frac12 \frac1{Re} \left(\nabla \left(\mathbf{\hat u} + \mathbf{u^n} \right)\right) : \left(\nabla \mathbf{v}\right) \, dx - \oint_{\partial \Omega} \frac12 \frac1{Re} \mathbf{v} \cdot \left(\nabla \left(\mathbf{\hat u} + \mathbf{u^n}\right)\right) \cdot \mathbf{n} \, ds = 0 $$ (KM.4)
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Step Two: Poisson's Equation
Again, I have already worked out this variational form in the FEniCS Tutorial. From that page, we have the following:
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$$ \displaystyle \oint_{\partial \Omega} \left(\nabla u \cdot \mathbf{n}\right) v \, ds - \int_\Omega \left(\nabla u\right) \cdot \left(\nabla v\right) \, dx = \int_\Omega f v \, dx $$ (PE.3)
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From this, we can again simply change our variables to match our problem.
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$$ \displaystyle \int_\Omega \left(\nabla \phi^{n+1}\right) \cdot \left(\nabla \chi\right) \, dx - \oint_{\partial \Omega} \left(\nabla \phi^{n+1} \cdot \mathbf{n}\right) v \, ds= - \int_\Omega \frac1{dt} \left(\nabla \cdot \mathbf{\hat u}\right) \chi \, dx $$ (KM.5)
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Step Three: Scalar potential
As this equation already consists of only first-order derivatives, the only step that remains in the weak formulation is to take the dot product of a test-function and integrate over all space.
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$$ \displaystyle \int_\Omega \mathbf{u^{n+1}} \cdot \mathbf{v} \; dx = \int_\Omega \left(\mathbf{\hat u} - dt \nabla \phi^{n+1} \right) \cdot \mathbf{v} \; dx $$ (KM.6)
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=References=